MATHEMATICS AND ART Marc Frantz

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Department of Mathematical Sciences

IUPUI

402 North Blackford Street

Indianapolis, Indiana 46202-3216

mfrantz@math.iupui.edu

c by Marc Frantz 1997. All rights reserved.

NATIO N AL

S CI ENCE

F O U NDA T I O N

n f

This project was supported, in part,

by the

National Science Foundation

Opinions expressed are those of the authors

and not necessarily those of the Foundation

Linear Perspective

1. Basic Results

Figure 1(a). Figure 1(b).

Take a good look at the images in Figure 1. Figure 1(a) is a detail from an illumi-

nated manuscript called the Tr_es Riches Heures, painted by the Limbourg brothers, Paul,

Hermann and Jean, in the _fteenth century. Figure 1(b) is a detail from an eighteenth

century oil painting by the Italian artist Canaletto (Giovanni Antonio Canal), entitled

View of the Grand Canal towards the Pallazzo Contarini dagli Scrigni. Of all the things

one might notice about these works, we would like to focus on one speci_c aspect, namely

the illusion of depth in three-dimensional space, which is far more e_ective and believable

in the painting on the right. Given that the artists in either case were considered very

competent in their time, it would appear that something happened in Europe between

the _fteenth century and the eighteenth century that added a powerful new tool to the

artist's toolbox. The \something" was the Renaissance, a period of vigorous and creative

activity in the arts and sciences. The powerful new tool was a mathematical technique

called \linear perspective," or just \perspective."

An illusion of depth is not necessarily the most important aspect|or even a necessary

aspect|of a successful painting. But whenever the e_ect has been needed, it has been

found that a knowledge of mathematics is an indispensable tool in achieving it. In fact, this

is why we have chosen to compare two paintings from Europe, for nowhere was the e_ect

of perspective upon art more dramatic. Moreover, the use of mathematical perspective is

perhaps more important today than ever before, if one considers modern creative media

such as _lmmaking, computer graphics, and virtual reality that have been made possible

by advances in technology.

1

Figure 2. Computer-aided perspective helps bring to life a

landscape with dinosaurs in Jurassic Park.

The idea behind the concept of \linear perspective" is illustrated in Figure 3. We

imagine a viewer using only one eye, with the eye located at E(0; 0;􀀀d) in a three di-

mensional coordinate system, looking in the direction of the positive z-axis. The point

P(x; y; z) on the vase in the _gure is visible to the viewer, and light reected from that

point travels in a straight line to the viewer's eye, piercing the \picture plane" z = 0 at

the point P0(x0; y0; 0). We think of the picture plane as the surface of a canvas on which

we wish to paint a realistic picture of the vase; thus the point P0 on the canvas should

be painted as a small dot of the same color as the light reected from the point P on the

vase, so that the viewer will see the same thing from that direction that he or she would

see if the vase were actually there.

x

y

z

Px,y,z

P'x',y',0

picture plane z=0

eye of viewer

at E(0,0,–d)

EP'=x',y',d

EP=x,y,z+d

Figure 3. P0 is the perspective image of P.

2

The point P0 is called the perspective image (or sometimes the central projection)

of P. If we identify the vase with the set of all visible points on its surface, then the

perspective image of the vase (represented by the small vase image in the _gure) is the set

of perspective images of those visible points.

We only need the two coordinates (x0; y0) to locate a point in the picture plane. From

now on a pair (x0; y0) of primed coordinates will indicate a point in the picture plane (the

coordinates of this point in space would be (x0; y0; 0)). To determine x0 and y0 in Figure 3,

notice that the vectors 􀀀E􀀀P!0 = hx0; y0; di and E􀀀!P = hx; y; z + di are parallel, and hence

there is a positive number t such that

thx0; y0; di = hx; y; z + di:

Equating components and solving for u, v, and t, we get

x0 =x=t; y0 = y=t; t = (z +d)=d:

Substituting for t in the expressions for x0 and y0, we get

x0 =

dx

z + d

and y0 =

dy

z + d

: (1)

Thus, if we know the coordinates (x; y; z) of a point on the vase|or any other object|

and if we know the distance d from the viewer's eye to the picture plane, we can determine

the coordinates (x0; y0) of the corresponding perspective image in the picture plane using

the formulas in (1).

Example 1. Assume we have the setup in Figure 3, with the viewer 3 units from the

picture plane. If P(2; 4; 5) is a point on an object we wish to paint, _nd the picture plane

coordinates coordinates (x0; y0) of the perspective image of P.

Solution. We have d = 3, x = 2, y = 4, and z = 5. Thus, by (1),

x0 =

(3)(2)

(5) + (3)

=

6

8

=

3

4

and y0 =

(3)(4)

(5) + (3)

=

12

8

=

3

2

:

Notice in Figure 3 that we have put the vase on the opposite side of the picture

plane from the viewer; that is, the z-coordinates of all the points on the vase are positive.

Although it is possible to do the same kind of perspective treatment when the object

is between the viewer and the picture plane (in which case the image gets larger than

the object), we shall assume from now on that any object we wish to make a picture of

consists of points whose z-coordinates are positive. Thus, in a sense, the image will always

be smaller than the object. (We say \in a sense" because we have not been clear about the

meaning of \smaller." To be more precise, if P1 and P2 are two points on the object, then

3

the distance between their corresponding perspective images P0

1 and P0

2 will be smaller

than the distance between P1 and P2; Exercise # gives you some hints and asks you to

prove this.)

Of course, we would like to do problems that are more interesting than Example 1, but

on the other hand, we don't want to knock ourselves out doing lots of tedious computations.

To obtain some nice shortcuts that will help us do interesting drawings, we will derive

important drawing rules from (1). To start with, we would like to say that the perspective

image of a straight line is also a straight line. Unfortunately, that's not quite true. For

example, a line that goes through the viewer's eye, such as the line through E and P in

Figure 3, would be represented by a single point in the picture plane (which makes sense,

because a thin pencil lead, seen end-on, looks like a dot, not a line segment). However, the

following theorem gives us some help in this regard. For convenience, we stick to our rule

that any object we might want to depict consists of points with positive z-coordinates.

Theorem 1. If S is a subset of a straight line L in space (S might be a line segment, a

point, a ray, etc.), and if all the points of S have positive z-coordinates, then the perspective

image S0 of S is a subset of a straight line Ax0 + By0 = C in the picture plane.

Proof. Let us assume that L is given parametrically by

x = x0 + at; y = y0 + bt; z = z0 + ct;

where 􀀀1 < t < 1 and x0; a; y0; b; z0; c are constants. Now let P be any point of S.

Then P = (x0 +at; y0 + bt; z0 + ct) for some value of t, and by (1), the perspective image

P0 = (x0; y0; 0) of P satis_es

x0 =

d(x0 + at)

(z0 + ct) + d

and y0 =

d(y0 + bt)

(z0 + ct) + d

:

(Since z = z0 + ct is positive, we are not dividing by zero.) If we de_ne the constants A,

B, and C by

A = bz0 + bd 􀀀 cy0; B=cx0 􀀀 ad 􀀀 az0; C=d(bx0 􀀀 ay0);

it is straightforward (but tedious) to check that Ax0 + By0 = C. Since the point P of S

was arbitrary, the theorem is proved.

Although it's unfortunate that we had to say something technical instead of \The

perspective image of a straight line is also a straight line," there are some useful drawing

tricks related to Theorem 1. One useful related fact is that the perspective image of a line

segment P1P2 is not only a subset of a straight line, it is (not surprisingly) either a point

or a line segment P0

1P0

2, where P0

1 and P0

2 are the corresponding perspective images of P1

and P2 (Figure 4).

4

E

x

y

z

P2

picture plane

P1

P'2

P'1

Figure 4. The perspective image of P1P2 is P0

1P0

2.

Instead of proving this fact, let's use it to do

Example 2. The visible corners of a cube in space are

M(4;􀀀2; 3); N(6;􀀀2; 3); P(4;􀀀4; 3); Q(6;􀀀4; 3); R(4;􀀀2; 5); S(6;􀀀2; 5); T(4;􀀀4; 5):

If the viewer is located at E(0; 0;􀀀3), make a line drawing to show the cube the way the

viewer sees it.

Solution. Our _rst step is to use (1) to determine the x0y0-coordinates of the corresponding

perspective images M0, N0, P0, Q0, R0, S0, T0. Since the z-coordinate of E is 􀀀3, we have

d = 3, and we can, for example, compute the x0y0-coordinates of T0 as follows:

x0 =

(3)(4)

(5) + (3)

=

12

8

=

3

2

and y0 =

(3)(􀀀4)

(5) + (3)

= 􀀀12

8

= 􀀀

3

2

:

Proceeding in this fashion, we can specify each perspective image with an ordered pair

(x0; y0) and get the points

M0(2;􀀀1); N0(3;􀀀1); P0(2;􀀀2); Q0(3;􀀀2); R0_3

2

;􀀀

3

4_; S0_9

4

;􀀀

3

4_; T0_3

2

;􀀀

3

2_:

Now the edges of the cube are line segments in space, and by our comment above, their

perspective images are line segments in the picture plane. Thus, to make a line drawing of

the cube, we plot the primed points that represent the corners, and connect appropriate

corners with line segments (Figure 5).

5

1 2 3

-1

-2

M' N'

P' Q'

R' S'

T'

x'

y'

-3 -2 -1

1

Figure 5. A cube in perspective.

Although Figure 5 is not exactly a masterpiece, we can make some interesting obser-

vations about it. First however, test your grasp of the geometry involved by answering

these questions:

(i) There is one corner point U(x; y; z) of the cube which is not visible. What are its

coordinates? Deduce them by thinking about the coordinates of M, N, P, Q, R, S,

and T.

(ii) In Figure 5, use a pencil to indicate where you think the perspective image U0 of U

would be if the cube were transparent. What are its picture plane coordinates (x0; y0)?

(iii) Check your answers to questions (i) and (ii) for consistency by applying (1) to your

coordinates for U to determine x0 and y0.

Now let's make some observations about Figure 5. We know that the object depicted

there is a cube, and hence angles such as 6 QPM and 6 TPM are in reality right angles.

However, if we look at the corresponding angles 6 Q0P0M0 and 6 T0P0M0 in Figure 5, we see

that 6 Q0P0M0 is a right angle, while 6 T0P0M0 is not. Similarly, each face of the cube is in

reality a 2_2 square, but the only visible face of the cube whose image is a square is the face

MNPQ, whose image is the 1_1 square M0N0P0Q0 in Figure 5. If you answered Questions

(i) and (ii) correctly, you found that the hidden corner U had coordinates (6;􀀀4; 5) and

its image U0 had picture plane coordinates (9=4;􀀀3=2). Thus the image R0S0T0U0 of the

hidden \back face" RSTU is also a square, and its dimensions are 3=4 _ 3=4. In other

words, the two faces RSTU and MNPQ of the cube have perspective images R0S0T0U0

and M0N0P0Q0 that are miniature, but undistorted, versions of the original square faces.

It also happens that these two faces of the cube are parallel to the picture plane z = 0.

This can be checked by noticing that the points R, S, T, U all have a z-coordinate of 5,

and the points M, N, P, Q all have a z-coordinate of 3. Thus R, S, T, U belong to the

plane z = 5, while M, N, P, Q belong to the plane z = 3.

6

This state of a_airs is not an accident; in fact, it reflects a more general principle

which artists have taken advantage of for hundreds of years, namely, that any shape which

lies in a plane parallel to the picture plane will have a perspective image that is an exact,

undistorted miniature of the original (Figure 6). Such shapes are relatively easy to draw,

because they don't have to be \distorted" mathematically to give an illusion of depth.

CAT

FOOD

CAT

FOOD

y'

x'

surface parallel to picture plane

picture plane

perspective

image

viewer

P1 P2

P'1 P'2

Figure 6. Since the cat food label is parallel to the picture

plane, its perspective image is undistorted.

You probably have a pretty good idea of what we mean by an \undistorted miniature"

by looking at Figure 6, but we should be precise about the idea, since it turns out to be very

convenient for artists. As a _rst step in making things more precise, we present Theorem 2:

while reading it, think of S as the set of points on the cat food label in Figure 6, and think

of P1 and P2 as any two points on the label.

Theorem 2. Let S be a set of points in a plane z = z0 (z0 > 0) parallel to the picture

plane, and let the viewer be located at E(0; 0;􀀀d) with d > 0. Then there exists a constant

_, with 0 < _ < 1, such that for any two points P1(x1; y1; z0) and P2(x2; y2; z0) in S, their

corresponding perspective images P0

1 (x0

1; y0

1; 0) and P0

2(x0

2; y0

2; 0) are exactly _ times as far

apart as P1 and P2, and the vector 􀀀􀀀! P0

1 P0

2 is parallel to the vector P􀀀􀀀1P!2. In other words,

��􀀀! P0

1 P0

2 = _P􀀀􀀀1P!2.

7

Proof. From (1) we can compute

x0

1 =

dx1

z0 + d

; y0

1=

dy1

z0 + d

; x0

2=

dx2

z0 + d

; y0

2=

dy2

z0 + d

:

If we de_ne the constant _ by _ = d=(z0 + d), then 0 < _ < 1 since d and z0 are positive

constants, and we can rewrite the above equations as

x0

1 = _x1; y0

1=_y1; x0

2=_x2; y0

2=_y2:

Thus

􀀀􀀀! P0

1 P0

2 = hx0

2􀀀x0

1; y0

2􀀀y0

1; 0i = h_x2􀀀_x1; _y2􀀀_y1; 0i = _hx2􀀀x1; y2􀀀y1; z0􀀀z0i = _P􀀀􀀀1P!2:

To grasp the meaning of Theorem 2, suppose that the viewer is 3 units from the

picture plane (that is, let d = 3), and consider an object, say, an equilateral triangle with

sides of length 2, that lies in the plane z = 4. Then the number _ in Theorem 2 is given

by _ = d=(z0 + d) = 3=(4 + 3) = 3=7. Theorem 2 says that the images of any two points

in the plane z = 4 will be 3=7 as far apart as the actual points; thus the image of each

side of the equilateral triangle will be a line segment of length (3=7) _ 2 = 6=7. It follows

that the image of the equilateral triangle is also an equilateral triangle, and the angles are

preserved (their images are still 60_ angles). It's not hard to prove in general that any

angle in a plane z = z0 will have an image with the same degree measure (Exercise #).

What we have said so far means that \shapes are preserved." But what about orientation?

That is, is it possible that the image of an equilateral triangle is an upside down version

of the original triangle, or tilted at some strange angle? The answer is no, because the

vector P􀀀􀀀1P!2 connecting any two points P1, P2 (such as two vertices of the triangle) in the

plane z = 4 will have a perspective image 􀀀􀀀! P0

1 P0

2 = (3=7)P􀀀􀀀1P!2 that is parallel to it, so the

image is not rotated in any way.

This non-distortion e_ect can be seen in Figure 7. In this picture, we are look-

ing at a picture plane that is vertical with respect to the (locally flat) earth. Several

buildings on the right side of the painting have walls that are parallel to the picture

plane, and hence their images, as well as the images of their doors and windows, are

undistorted. As we mentioned earlier, this makes it easier for the artist to render these

parts of the painting, and indeed many paintings and drawings of buildings feature walls

that are parallel to the picture plane. However, convenience is not the only reason for

drawing buildings in this way. Remember that most paintings are done on a rectangu-

lar canvas, and hence objects in the painting that have a rectangular shape will help

organize the composition and keep it in harmony with its border. (Several lines that

parallel the edges of the canvas have been indicated in the _gure.) Even if the face of

a building is not parallel to the picture plane, the edges of the buildings|which are

8

vertical lines in space|will each lie in some plane that is parallel to the picture plane,

and hence by Theorem 2 the images of the lines will also be vertical; this adds still more

lines that are parallel to edges of the canvas, such as the two vertical lines indicated at the

left side of the picture. One compositional use of such lines is illustrated by the vertical

edge of the tall building that can just barely be seen at the far right edge of the canvas.

As the viewer's eye scans the rooftops from left to right, the eye tends to be led o_ the

edge of the painting. This vertical line serves to \stop the eye" and bring the viewer's

attention back to the center of the painting. This little trick is a direct application of the

rule 􀀀􀀀! P0

1 P0

2 = _P􀀀􀀀1P!2 derived in Theorem 2: The vector P􀀀􀀀1P!2 is the actual vertical edge of

the building, and the line segment in the painting is the vector 􀀀􀀀! P0

1 P0

2 , which is also vertical,

and _ times smaller.

Figure 7. A larger view of Canaletto's painting in Figure 1a. Horizontal and

vertical lines help organize the composition and relate the objects

in the painting to the border of the canvas.

Besides the non-distortion phenomenon, there is another important idea we can grasp

by looking at the cube from Figure 5, which has been reproduced in Figure 8 without all the

labels. Take a look at the cube we drew. Does it really look like a cube? If you're looking

at it from a typical reading distance, it probably seems too elongated in the direction of

9

your line of sight|more like the shape of dumpster or a cedar chest, than a die or a sugar

cube.

1 2 3

-1

-2

x'

y'

-3 -2 -1

1

Figure 8. The drawing from Figure 5. At a natural viewing distance (as op-

posed to the correct 3 units) the cube looks too long in the direction

of the line of sight.

1 2 3

-1

-2

x'

y'

-3 -2 -1

1

-3

-5 -4 4 5

Figure 9. The same cube and picture plane, with the correct viewing distance

increased to a more natural 21 units.

There's a good reason{or perhaps we should say a bad reason|for this. Remember

that the correct viewing distance from the picture plane is d = 3 units. If you hold a

scrap of paper up to the x0-axis, mark o_ 3 units, and use the paper to place your eye 3

units from the origin, your nose will almost touch the page! At this distance you won't

be able to focus properly, but if you glance o_ to the side towards the image, you should

be able to notice that it now looks like a (blurry) cube. Thus, even though our drawing

is mathematically correct, it's not a good drawing, because during the planning stage we

failed to take into account the actual _nished size of the drawing and the natural distance

10

at which a viewer would want to look at the picture. If we use the same cube and picture

plane, but change the viewing distance to a more reasonable value of 21 units (that is,

leave everything the same as in Example 2, except move the viewer back to E(0; 0;􀀀21))

we get the drawing in Figure 9, which looks much more cube-like from a normal viewing

distance.

Now you may argue that the only thing wrong with Figure 8 is that the drawing is too

small. After all, if it were so big that the units were, say, in feet, then the viewing distance

would be 3 feet, which is far enough away that one could at least focus on the image. To

see if this approach works, we have made an enlarged detail of Figure 8 in Figure 10.

1 2 3

-1

-2

Figure 10. Enlarged detail of Figure 8.

Of course, we couldn't make it big enough so that the viewing distance is 3 feet, but you

should be able to focus on the image with your eye 3 units from the origin. At this point,

however, a new problem becomes evident: the image of the box is so far from the center of

view (the origin) that it is awkward to look that far o_ to the side. In fact, no matter how

much we enlarge the picture, anyone viewing from 3 units away from the origin (measured

perpendicularly from the page) would have to look o_ to the side at more than a 45_ angle

to see the lower right corner of the box. (Prove it!) It is generally accepted that a drawing

or painting done in perspective should not require the viewer to have a \cone of view" of

11

more than about 60_; that is, the viewer should not have to look away from the center of

view more than about 30_ in any direction to see an object in the painting, so our drawing

is not that easily _xed.

Before going further, let's highlight some important points.

_ A drawing done in perspective should be planned so that the correct viewing distance

is practical and comfortable for the viewer (unless, of course, unusual e_ects are

deliberately intended).

_ With the same quali_cation, the viewer should not be required to look away from the

center of view more than about 30_.

_ The enlightened viewer (You!) should be aware that there is exactly one correct point

in space from which to view a work done in perspective.

Is it possible for the \enlightened viewer" to determine the correct viewing location?

In many cases, the answer is yes, and we'll be doing that later. Right now, have some fun

with Figure 10.

While looking at Figure 10, close one eye and look at the picture from arm's length.

Now move the page closer until your eye is about 3 units directly in front of the origin.

Move the page back and forth like this and notice that, as we said earlier, the box seems

elongated in the direction of your line of sight when viewed from far away, and more

cube-like when viewed closer in. From any viewing distance, the image is a perfectly good

drawing of a rectangular box with its sides parallel to the coordinate planes (we'll show why

this is true later). But, due to the elongation and shortening e_ects we have just described,

it is a correct representation of a cube only when viewed from the correct viewing distance.

Hence what we perceive from too far away (or too close) is not just a reduction (or an

enlargement) of the image the artist wants us to see, it is a distortion. This applies to any

work done in perspective. We'll prove it for images of rectangular boxes with their faces

parallel to the coordinate planes, and then easily generalize it to any kind of image.

x'

y'1

y'2

y'3

y'

picture plane

(front view)

image of box

(x,y,z) (x,y,z+a)

(x,y-b,z)

a

b

z

d

edge of picture

plane z=0

y'1

y'2

y'3

y

actual box

(side view)

Figure 11.

12

Theorem 3. Let B be a rectangular box in space, with positive z-coordinates and sides

parallel to the coordinate planes. Suppose also that at least one face of B not parallel to

the picture plane is visible to a viewer located at (0; 0;􀀀d). If this face has dimensions

a _ b, where a is the length in the z-direction, then the viewing distance d is directly

proportional to the \shape ratio" a=b .

Proof. For de_niteness, let us suppose that a face of the box parallel to the yz-plane is

visible to the viewer (Figure 11). Choose corner points (x; y; z +a), (x; y; z),(x; y 􀀀b; z) of

this face as in Figure 11, and let y0

1 , y0

2 , y0

3 be the y0-coordinates of their respective images

in the picture plane. From (1) we have

y0

1 =

dy

(z + a) + d

; y0

2=

dy

z + d

; y0

3=

d(y􀀀b)

z+d

:

Solving these equations for a, (z+d), and b, respectively, we get

a =

dy

y0

1 􀀀 (z + d); (z + d) =

dy

y0

2

; b=y􀀀

y0

3

d

(z+d):

Substituting dy=y0

2 for (z + d) in the solutions for a and b and simplifying, we get

a = dy

y0

2 􀀀 y0

1

y0

1 y0

2

and b = y

y0

2 􀀀 y0

3

y0

2

;

and hence

a

b

= d

y0

2 􀀀 y0

1

y0

1(y0

2 􀀀 y0

3)

;

or equivalently,

d =

y0

1(y0

2 􀀀 y0

3 )

y0

2 􀀀 y0

1 _a

b _:

Thus d is directly proportional to a=b. The case in which a face of the box is parallel to

the xz-plane can be handled similarly.

Here is what Theorem 3 tells us. Suppose we are given a drawing of a rectangular

box like that described in Figure 11. The drawing has already been done, so all the x0y0-

coordinates are _xed, and hence the quantity y01(y02 􀀀y03)

y02 􀀀y01

is a constant, which we will call K.

We can then write

d = K _a

b _: (2)

(In the case in where we consider a visible face of the box parallel to the xz-plane, we

still get an equation of this form, but the constant K depends on x0-coordinates instead

of y0-coordinates.) As we mentioned earlier, from any viewing distance, the drawing can

be considered an accurate image of some box with sides parallel to the coordinate planes.

But Equation (2) says that if d is doubled, then the shape ratio a=b of a particular side of

13

the box must be doubled also. In other words, if you view the picture at twice the correct

distance, you'll perceive a box which is twice as long in the z-direction as the box the artist

intended you to see. Similarly, if you change from d to d=3 (i.e., if you view too closely),

then the box you think you see will have the shape ratio of that same side changed to

(a=b)=3; that is, the box will appear too short in the z-direction by a factor of 1=3. Thus

Theorem 3 is just a precise quantitative description of what you experience when you view

Figure 10 from di_erent distances. You've already noticed that the box looks too elongated

when you view its image from a distance of 6 units instead of the correct 3 units; now you

know exactly how much it's elongated. The mathematics has quanti_ed your experience!

It is easy to generalize the implications of Theorem 3 to arbitrary shapes. Suppose you

have, say, a sculpture of a human head you want to draw in perspective. Suppose further

that the sculpture is made from tiny cubes|such as salt crystals, for instance|with their

faces parallel to the coordinate planes you are using for your perspective setup. Theorem 3

says that if your drawing is viewed from too far away (or too close), then each cube will

appear too elongated (respectively, too compressed) in the direction of the z-axis. But if

this applies to each cube, it also applies to the entire sculpture, so the entire image will

be appear distorted if viewed from the wrong distance. The funny thing is, we see this

kind of distortion so often that we don't even notice it. For example, hold your _st out

at arm's length with the thumb side towards you and look at it. If you were to take a

photograph of what you're seeing and then view the photo from very close up, you might

say to yourself, \Well, I guess this is what my _st looks like from close up." But it isn't!

To verify this, move your _st so close to your eye that the bottom knuckle of your thumb

almost touches your eye. (Remove your glasses if necessary.) Notice that your _st almost

entirely disappears: all you can see is your thumb! This is very di_erent from taking at

a picture of the whole _st and then looking at the picture close up; the fact that you can

still see the whole _st means you're seeing a distortion of reality.

On the other hand, Theorem 3 is not all bad news. If the viewing distance of a picture

is say, 40 inches, and you view the picture from a couple of inches too far away, then your

viewing distance is 42=40 = 1:05 times greater than it should be. Theorem 3 says that an

object in the picture will be perceived as being 1:05 times too elongated in the direction

of the z-axis, an error of only 5%. Thus Theorem 3 gives us a valuable drawing tip:

_ Keep the viewing distance fairly large, so that small errors by the viewer in choosing

a viewing distance will not cause noticeable distortions.

Of course, the viewing distance should also be \practical and comfortable for the viewer"

as we said earlier.

******** SUMMARY OF RESULTS & DRAWING TIPS ********

******** EXERCISES ********

2. Vanishing Points and Vanishing Lines

14

Our results so far have given us some useful tips for setting up , executing, and

viewing perspective drawings and paintings, but the execution part probably seems rather

technical to you. There are too many formulas, and too much reliance on coordinates:

what we would like to do is grab paper, pencil, and maybe a straightedge, and begin

making nice pictures, while keeping the math hidden in the background and concentrating

on the art. That's exactly what the concepts of vanishing points and vanishing lines will

help us to do. We'll start with vanishing points.

Suppose we have a straight line L in space with parametric equations

x = x0 + at; y = y0 + bt; z = z0 + ct;

where 􀀀1 < t < 1 and x0; a; y0; b; z0; c are constants and c > 0. The condition c > 0

means that limt!1 z(t) = 1, so the line gets in_nitely far away from the viewer in the

positive z-direction as the parameter t approaches in_nity. As usual, we consider the viewer

to be located at (0; 0;􀀀d), with d > 0. Theorem 1 tells us that the perspective image of

L (technically, the image of the part of L with positive z-coordinates) is a subset of a

straight line. However, the image is not an entire straight line: in particular, it vanishes

abruptly at a point called the vanishing point! To see why this should happen, look at

Figure 12. A viewer looks at a line L extending in_nitely in the positive z-direction. As

the eye gazes at more and more distant points on L, the line of sight eventually becomes

parallel to L, and the line L seems to disappear, because the two parallel lines (the line of

sight and L) cannot intersect. At the instant this happens, the viewer is looking directly

at the vanishing point. The vanishing point of a line can be anywhere in the picture plane,

depending on the direction of the line.

line visible

line still visible

line still visible

line vanishes

infinite straight line L

edge of picture plane

vanishing point

Figure 12.

With the line L given parametrically, it is easy to locate its vanishing point. By (1),

the parametric equations of the image of L in the picture plane are

x0 =

d(x0 + at)

(z0 + ct) + d

=

(da)t + (dx0)

(c)t + (z0 +d)

and y0 =

d(y0 + bt)

(z0 + ct) + d

=

(db)t + (dy0)

(c)t + (z0 +d)

;

where we have arranged terms in the numerators and denominators in descending powers

of t. This makes it easy to take limits as t goes to in_nity:

lim

t!1

x0 = da=c and lim

t!1

y0 = db=c:

15

That is, the x0y0-coordinates of the vanishing point are (da=c; db=c).

Now we present three results which will help us draw good perspective drawings

without explicitly using a lot of technical mathematics. The _rst result just con_rms what

we assumed in the above argument, and the other two follow from the _rst.

Theorem 4. Let L be a line in space not parallel to the picture plane, with parametric

equations

x = x0 + at; y = y0 + bt; z = z0 + ct;

where c > 0. Then the line of sight from the viewpoint E(0; 0;􀀀d) to the vanishing point

of L is parallel to L.

Proof. If V denotes the vanishing point of L, then the space coordinates of V are

(da=c; db=c; 0), and the line of sight is parallel to E􀀀!V . But

E􀀀!V = hda=c; db=c; di = (d=c)ha; b; ci;

and ha; b; ci is the direction vector of L.

Theorem 5. If two lines L1 and L2 are parallel to each other, but not parallel to the

picture plane z = 0, then they have the same vanishing point.

Proof. If V1 and V2 denote the respective vanishing points (in space coordinates) of L1

and L2, then by Theorem 4, E􀀀􀀀V!1 and E􀀀􀀀V!2 must also be parallel. This can only happen if

V1 = V2.

Theorem 6. If a line L in space is parallel to the z-axis, then the vanishing point V of L

is the origin (0; 0) of the picture plane z = 0.

Proof. The proof is left to the reader.

It's quite common to combine these two results in a painting or drawing. One example

is Figure 13, which features Piero Della Francesca'a painting, Ideal City, executed in the

15th century. Like many Renaissance artists, Piero was knowledgeable in mathematics,

and wrote treatises on solid geometry and perspective. The perspective setup here is our

standard one, with the viewer's line of sight (the z-axis) normal to the picture plane and

parallel to the plane of the ground. The principal straight lines of the buildings fall mainly

into two categories. There are vertical and horizontal lines in planes parallel to the picture

plane: by Theorem 2 their images are also vertical or horizontal, respectively. The other

category consists of lines which are parallel to the z-axis, and by Theorems 5 and 6, their

images all converge to the origin of the picture plane, as indicated in Figure 14. This

situation is so common in art, illustration, and architecture that it has a name. When

one point serves as the vanishing point for all, or almost all, of those lines in a painting or

drawing which have a vanishing point, it's called \one-point perspective."

16

Figure 13. Piero Della Francesca, Ideal City

Figure 14. Ideal City, with lines converging to a vanishing point.

Of course, a little more was required in this painting. For instance, the two octagonal

structures in the foreground have some lines which do not fall into the two categories just

mentioned, and the problem of drawing the cylindrical building in the center had to be

solved. We'll deal with this type of problem later on.

A famous example of the use of one-point perspective is The Last Supper by Leonardo

da Vinci (Figures 15 and 16). In this work, the head of Christ is placed near the single

vanishing point, so that lines of the architecture parallel to the z-axis (in our terminology)

will lead the viewer's eye to the central _gure. Thus artists not only use perspective as

a means to achieve an illusion of depth, but also as a compositional device to direct the

viewer's eye.

Figure 15. Leonardo da Vinci, The Last Supper

17

Figure 16. The Last Supper, with lines converging to a vanishing point.

Of course, one vanishing point is often not enough to represent scenes realistically. The

building in the photograph in Figure 17, when viewed as shown, requires two vanishing

points for an adequate representation. When just two vanishing points are used, the

siuation is often referred to as \two-point perspective."

Figure 17. A scene with two vanishing points.

Notice that lines of the building which are vertical in space have images that are also

vertical|and more importantly, parallel|in the picture. This indicates that the picture

plane is parallel to these lines; otherwise, by Theorem 4, they would converge to a vanishing

point.

An example of just such a situation is the photograph in Figure 18. The photo was

taken looking upward at a skyscraper, and hence the picture plane is no longer parallel

to the vertical lines of the building. As a result, the images of these lines converge to a

vanishing point. There are three principal vanishing points in this image|a fairly common

occurence in architectural renderings. Naturally, this situation is referred to as \three-point

perspective."

18

Figure 18. A scene with three vanishing points.

In practice, a perspective drawing may have any number of vanishing points, depend-

ing on what kind of scene one is required to draw. A concept which is often helpful in

locating vanishing points is that of the vanishing line of a plane. To visualize this, look at

the aerial photograph in Figure 19. As shown in Exercise #, the part of the earth we can

see locally can be idealized as a plane, provided that the terrain is not too hilly and we

are not at too high an altitude.

Figure 19. Three vanishing points on a vanishing line.

19

Now notice that there are three pairs of of lines in the picture, each pair converging to

a di_erent vanishing point. The corresponding lines in the real world are architectural lines

of buildings or streets, and the two members of any pair are in actuality parallel to one

another, but not parallel to the picture plane; hence the convergence of the images of any

given pair to a single vanishing point is guaranteed by Theorem 5. The interesting thing

here is that all three vanishing points lie on the same line, which coincides with the horizon

in the picture. The reason this happens is that all the lines are parallel to the plane of the

ground. The horizon line in this instance is an example of what we call a vanishing line.

One way of looking at a vanishing line is to consider it a set of vanishing points for other

lines, such as the lines of streets and buildings we just mentioned. Another way of looking

at the vanishing line is to think of it as the place where an entire plane seems to vanish:

It turns out that if the earth were actually an in_nite flat plane, it would not appear that

di_erent to us at relatively low altitudes, and in fact, even though it would no longer be

curved, it would still appear to vanish at almost the exact location of the horizon line in

the picture.

To make some of these ideas more precise|and more useful to our drawing needs|we

present the next two results on vanishing lines.

Theorem 7. Let 􀀀 be a plane in space given by the equation

Mx+Ny +Pz +Q = 0;

where M and N are not both 0; i.e., 􀀀 is not parallel to the picture plane z = 0. If the

viewer is located as usual at (0; 0;􀀀d), then any line L in 􀀀 which is not parallel to the

picture plane has a vanishing point which lies on the line

Mx0 +Ny0 = 􀀀dP;

called the vanishing line of the plane 􀀀.

Proof. Let L be a line in 􀀀 not parallel to the picture plane, and let us consider a point

(x; y; z) on L. Assume that L is parameterized as hx(t); y(t); z(t)i, where z !1as t!1,

and let (x0v; y0v) be the vanishing point of L, so that x0v = limt!1 x0 and y0v = limt!1 y0.

Then

Mx0

x +Ny0

v = M lim

t!1

x0 + N lim

t!1

y0 = lim

t!1

(Mx0 +Ny0)

= lim

t!1_M

dx

d + z

+ N

dy

d + z_ = lim

t!1

d(Mx+Ny)

d+z

:

But L lies in 􀀀, so for any value of t, the point (x; y; z) does also. Hence (Mx + Ny) =

􀀀Pz 􀀀Q, and we can write

Mx0

x +Ny0

v = lim

t!1

􀀀dPz 􀀀 dQ

z + d

= lim

z!1

􀀀dPz 􀀀 dQ

z + d

= 􀀀dP:

It follows that (x0v; y0v) lies on the vanishing line.

20

Corollary 8. If 􀀀1 and 􀀀2 are parallel planes, not parallel to the picture plane z = 0,

then they have the same vanishing line.

Proof. If the equation of 􀀀1 is put in the form Mx + Ny + Pz + Q1 = 0, then 􀀀1 has

a normal vector hM;N;Pi. Since 􀀀2 is parallel to 􀀀1, it has the same normal vector, so

its equation can be put in the form Mx + Ny + Pz + Q2 = 0 for some number Q2. But

Theorem 7 says that the equation of the vanishing line depends only on M, N, P, and the

viewing distance d; that is, the two planes have the same vanishing line Mx0+Ny0 = 􀀀dP.

Figure 20 illustrates a common application of Theorem 7. To get the main idea, let's

not worry about viewing distance or object size. Our goal is to draw a structure with a

slanted roof, like a house, a children's playhouse, or a tool shed. The steps are as follows.

(a) With our line of sight level with respect to the ground, the plane of the ground is

parallel to the plane y = 0. Thus the equation of the ground plane can be put in

the form Mx + Ny + Pz + Q = 0, with M = P = 0 and N = 1. By Theorem 7,

the vanishing line of the ground plane (commonly called the horizon line) is the line

y0 = 0 (the x0-axis). Thus we choose a horizontal line as the horizon, and shade the

ground gray to make it stand out.

(b) Let's make the front wall of the building parallel to the picture plane z = 0, so that

it is an undistorted rectangle (a consequence of Theorem 2). Let's also make the

left wall visible. Since the left wall is perpendicular to the front wall, its top and

bottom edges, which are parallel to the ground plane, are also parallel to the z-axis.

Thus by Theorem 6, the vanishing point v0 for these lines is the origin (0; 0) in the

picture plane. This vanishing point must therefore be on the horizon line y0 = 0 and

somewhere to the left of the corner of the building, so that we can see the left wall.

Since we're not concerned about object size or viewing distance, we place v0 on the

horizon somewhat arbitrarily, and use it to draw the left wall, whose back edge we also

locate somewhat arbitrarily. To repeat, v0 is now the origin of the picture plane, and

thus our center of view is well to the left of the center of the picture in this exercise.

(c) In order to locate the peak of the roof, we _rst need to locate the center of the left

wall. Since the diagonals of the left wall cross at the center of the wall, and since their

perspective images are also line segments, we can locate the perspective image of the

center of the wall by simply drawing the images of the diagonals (dashed lines) and

noting their intersection point. From this point we draw a vertical line segment to

locate the peak of the roof at an arbitrary height. The question now is: How do we

correctly draw the rest of the roof?

21

a b

c

?

v0

v0

d

v0

v1

?

e

v0

v1

f

Figure 20. An application of Theorem 7.

(d) The answer is given by Theorem 7. The plane of the left wall has an equation of the

form Mx + Ny + Pz + Q = 0, where N = P = 0. (Why?) Thus by the Theorem 7,

the vanishing line of this plane is the line x0 = 0 (the y0-axis). In particular, the left

front edge of the roof has its vanishing point v1 on this line, so we locate it by simply

extending the line of the roof edge until it meets the vertical line x0 = 0.

(e) The right front edge of the roof is parallel to this line, so by Theorem 5 it has the

same vanishing point. Hence we can draw the right front edge, and then draw the top

edge, which must be horizontal since it is level with the ground and parallel to the

picture plane.

(f) Finally, we erase all the construction lines, add a little color to the roof, and we are

done.

Our building seems a little lonely, so let's add a road with a fence on either side of it

22

(Figure 21). To do this, we can again employ a step-by-step method which makes use of

Theorem 7.

a

v0

b

v0

v2

c

v0

v2

d

Figure 21. Using Theorem 7 to locate fenceposts.

(a) If we choose the road to be parallel to the left wall of the building, we can use our

previous vanishing point v0 (the origin of the picture plane). After shading the road,

we can locate the _rst fenceposts even with the front corner of the building by sketching

a dashed horizontal line from the corner of the building and making the posts have

equal height. The fences are parallel to the road, so their top and bottom edges have

v0 as their vanishing point, as indicated by the dashed lines. We locate the second

post on the right somewhat arbitrarily, and make the second post on the left even

with it by use of another horizontal dashed line. We can't be arbitrary about the rest

of the fenceposts, however, because fenceposts have equal spacing.

(b) In reality, the rectangular sections of fence between the fenceposts all have the same

shape, so their diagonals are parallel; thus the perspective images of these diagonals

will all have the same vanishing point v2. To locate v2, note that the plane of each

fence is parallel to the plane of the left wall of the building, so by Corollary 8 they

have the same vanishing line, which was previously determined to be the y0-axis. We

re-draw the y0-axis through v0 and extend the diagonal from the bottom of the _rst

fencepost through the top of the second fencepost until it intersects the y0-axis; this

locates v2. Next, we draw the dashed line from the bottom of the second fencepost to

v2, and the place where this line intersects the top edge of the fence determines the

top of the third fencepost. We the draw the third fencepost on either side.

23

(c) Continuing this process, we keep going until we're tired of drawing fenceposts. Actu-

ally, we shouldn't go too far unless we're going to draw our lines thinner and thinner

as we go into the distance.

(d) Finally, we erase our construction lines and add another rail to each fence, using the

vanishing point v0 at the \end" of the road.

The method we just used to locate fenceposts isn't the only one, or even the best

one. For instance, the vanishing point v2 was o_ the \canvas," and this is sometimes

inconvenient. Another method is presented in Exercise #.

******** EXERCISES ********

3. Analyzing Works Done in Perspective

The theorems we have presented so far are not just good for creating works in per-

spective. We can also use them to help us better understand and appreciate paintings

and drawings that other people have done in perspective. To begin with, let's practice on

the building we drew in Figure 20. We said we wouldn't worry about viewing distance or

object size, but now that the drawing is done, is there anything that can be said about

these things? The answer is yes. To prove it, we're going to play dumb and forget the

information we used to set up the drawing, and see what can be deduced just by looking.

As each deduction is stated, try to flesh out the argument or name the theorem which

justi_es it.

Let's start with the viewing distance. We saw in Theorem 3 that the viewing distance

is related to the shape of the object being drawn. If we consider our building to be, say,

a backyard toolshed, then we can make some reasonable assumptions about its shape and

size. To analyze the picture, we _rst notice that the front wall is apparently a perfect

rectangle (Figure 22(a)), so it is parallel to the picture plane (otherwise, at least one pair

of its edges would converge to a vanishing point instead of being parallel). Next, we extend

the lines of the top and bottom edges of the left wall to locate the vanishing point v0. Since

the left wall is in reality orthogonal to the front wall, its top and bottom edges are parallel

to our line of sight (the z-axis), so the vanishing point v0 must be the origin of the picture

plane. Thus, to view the picture correctly, we should place ourselves (rather, one eye)

directly in front of v0, but how far from the picture? To _nd out, we draw the y0-axis,

which is the vanishing plane of the left wall. We could make use of the left edge of the

roof, but to show that our technique doesn't depend on slanted roofs, we instead extend a

diagonal of the rectangular part of the left wall until it meets the y0-axis at a new vanishing

point v3, and we label the distance from v0 to v3 with the letter b.

24

b

d

b

l

h

picture plane

v0

v3

v0

v3

parallel

Figure 22(a). Figure 22(b).

Figure 22(b) shows a side-view schematic of the situation. In space, the diagonal and the

line from the viewer's eye through v3 must be parallel by Theorem 4, and hence by similar

triangles, we have d=b = l=h, where d is the viewing distance and l and h are the actual

length and height, respectively, of the wall. Thus we have

d = b(l=h):

Notice that we don't need to know l and h to compute d, but only their \shape ratio"

l=h. If we consider Figure 22(b) as a reasonable sketch of the wall of a toolshed, then

l=h _ 1:3. This means that the correct viewing distance is only about 1:3 times as long

as b, so our somewhat haphazard drawing has resulted in a viewing distance which is

uncomfortably small.

Now try this technique out for yourself by estimating the viewing distance for

Figure 13|you'll need a little cleverness!

Determining the viewing distance is an exercise which must be done on a case-by-

case basis, and sometimes it's not possible to do with much accuracy. For instance, a

photograph of a cloud would be di_cult to analyze without more sophisticated knowledge

of meterology, the speci_c camera used, etc. As we have seen, the presence of architecture

helps a lot. As an example of a somewhat more di_cult problem involving buildings,

consider the photograph in Figure 23.

This _gure is an aerial photograph of Franklin & Marshall College in Lancaster, Penn-

sylvania. The image was taken from the F&M home page at http://www.FandM.edu/.

The _rst problem we must face is the fact that it is not as easy to be sure of the center

of view|that is, the standard origin of the x0y0-coordinate system|in the picture plane.

A good guess would be the center of the photograph, which would be true if the photo

has not been cropped. However, we will not make this assumption, and we will instead

proceed as follows.

25

Figure 23. Franklin & Marshall College.

Figure 24. Locating vanishing points in the picture plane.

26

First, as illustrated in Figure 24, we put the photo into the picture plane (which can

be a large piece of paper) and extend pairs of parallel roof lines and wall lines until they

meet at the vanishing points P, Q, and R. Next, we mark o_ a pair of perpendicular axes

parallel to the edges of the photo, and choose them for convenience to contain two of the

vanishing points. Since these axes are almost certainly not the actual x0y0-axes, we label

them x00 and y00. We then choose a unit of length, indicated by the 1's on the axes, and

estimate the x00y00-coordinates of P, Q, and R. In order to proceed further, we need to

visualize the viewer and the picture plane in 3-dimensional space.

x''

y''

z

P0,18,0

Q7,0,0

picture plane z=0

eye of viewer at

E(a,b,–d)

in x'',y'',z-coordinates

R23,14,0

Figure 25. The points E, P, Q, and R in x00y00z-coordinates.

In Figure 25 we make use of the standard z-axis, with the picture plane being z = 0 as

usual. However, since we only have x00y00-coordinates available at this stage, we use x00, y00,

and z as our space coordinates. Thus the viewer's eye is located at the point E(a; b;􀀀d) in

x00y00z-coordinates, and it is our job to determine a, b, and d. By Theorem 4, the vectors

E􀀀!P, E􀀀!Q, and E􀀀!R must be parallel to the wall and roof lines whose images converge

respectively to P, Q, and R in the picture plane. But in space, these architectural lines

are either parallel or perpendicular to each other, and thus E􀀀!P, E􀀀!Q, and E􀀀!R are mutually

orthogonal. Hence we can write

E􀀀!P _ E􀀀!Q = 0; E􀀀!P _ E􀀀!R = 0; E􀀀!Q _ E􀀀!R = 0;

or, respectively,

h􀀀a; 18 􀀀 b; di _ h7􀀀a;􀀀b; di = 0

h􀀀a; 18 􀀀 b; di _ h23 􀀀 a; 14 􀀀 b; di = 0

h7􀀀a;􀀀b; di _ h23 􀀀 a; 14 􀀀 b; di = 0:

27

Carrying out the dot products, we get, respectively,

a2 􀀀 7a + b2 􀀀 18b + d2 = 0

a2 􀀀23a + b2 􀀀 32b + 252 + d2 = 0

a2 􀀀30a + 161 + b2 􀀀 14b + d2 = 0:

If we now subtract the second equation from the _rst, and then subtract the third equation

from the _rst, we get the simple system

16a + 14b = 252

23a 􀀀 4b = 161

whose solution is

a _ 8:45; b_8:34:

Finally, we can substitute these values into any of the equations involving d to get

d _ 8:3:

Actually, our work is not that accurate, so we would be more honest to write

a _ 8; b_8; d_8;

and conclude that the viewing distance is about 8 units for the small version of the photo.

The center of view (the true x0y0 origin) has been indicated in Figure 24 as a white dot at

the point (8:45; 8:34) in x00y00-coordinates. The fact that it is o_-center in the photo is most

likely an artifact our own experimental errors in drawing lines and estimating coordinates,

rather than a result of the photo being cropped.

If all these computations seem tedious, watch how easy it is in the computer algebra

program Maple. First, we load the linear algebra package, which handles vectors; at the

next prompt, we de_ne the vectors E􀀀!P, E􀀀!Q, and E􀀀!R; _nally, we solve the dot product

equations for a, b, and d, using the fsolve function (the \f" is for \floating-point" as in

decimal point).

> with(linalg):

> EP:=[0,18,0]-[a,b,-d]: EQ:=[7,0,0]-[a,b,-d]: ER:=[23,14,0]-[a,b,-d]:

> fsolve(fdotprod(EP,EQ)=0, dotprod(EP,ER)=0, dotprod(EQ,ER)=0g, fa,b,dg);

fa = 8.450777202, b = 8.341968912, d = 8.264792809g

WARNING: The results obtained here are very rough approximations. When you

start doing this type of analysis yourself, you'll see that no two people exactly agree on

how to extend roof or wall lines to vanishing points. If you do it twice, you may not

even agree with yourself! When the images of roof or wall lines are very nearly parallel|

and they frequently are|slight changes in the lines you draw to extend them can make

28

big di_erences in the estimated location of vanishing points, and correspondingly large

di_erences in the computed viewing distance. We have shown that the viewing distance is

computable in principle, but in practice, it can be tricky.

Find the center of view (the x0y0 origin) and compute the viewing distance for

Figure 18.

So far we have seen some examples of how to compute the correct distance from the

viewer to the picture plane, but how about the distance from the viewer to the objects

in the picture? Of course, the actual objects could be miles away or long vanished; they

may never have even existed, like the little shed we drew in Figure 20. Nevertheless, the

question can be meaningful in certain cases. For instance, whenever a work in perspective

is based on a real scene, it is reasonable to expect that the artist viewed the canvas from

what we would call the \correct" point of view, and if the scene appeared to the artist

as it does in the painting from that point, then we can attempt to \reverse engineer" the

picture and estimate how far from the objects in the picture the artist was located. In the

case of photographs, we can try to determine where the camera was located in the scene.

In some cases, the estimate can be made rather easily, but _rst we'll attempt the analysis

on the photograph in Figure 23 by placing it in the picture plane again (Figure 26).

Figure 26.

29

The photograph was apparently taken from a considerable height, so we can ask, How

far above the ground was the camera? At _rst it may seem that we couldn't say anything

about that; perhaps the photo was taken from a spy plane or satellite, using high-resolution

photography. But in fact|although we can't make a highly accurate determination, as we

indicated above|we can use our knowledge of perspective and some vector mathematics

to rule out any covert spying on Franklin & Marshall College! The _rst step is to pick

an object in the picture whose actual size we can reasonably estimate; we choose the

cylindrical turret to the right of the main entrance, and we label the images of points on

the top and bottom as M and N, respectively, in Figure 26. The coordinate system is

the same as that in Figure 24. The coordinates of M and N were actually estimated on

a higher resolution image, so we dare to name them to one decimal place. The point Q

is the same as in Figure 24; i.e., it's the vanishing point of the images of all the vertical

architectural lines, so it's collinear with M and N. Now, as we did earlier, we pass from

x00y00-coordinates to x00y00z-coordinates and view the whole setup in 3-dimensional space

(Figure 27.)









E

M

N

Q

h

a k

U

W V

turret

ground

picture

plane

viewer's

eye

180° – ()

Figure 27.

We have drawn the picture plane (seen edge-on) as being tilted, in order to have the turret

appear vertical. Since we still consider the picture plane to be the plane z = 0, the whole

coordinate system is therefore tilted in the picture. The known quantities are the height h

of the turret (which we will estimate later) and the points E, M, N, and Q. If we round

our previously estimated x00y00z-coordinates of E to one decimal place, we have

E = (8:5;8:3;􀀀8:3); M= (6:8;8:9; 0); N= (6:9;7:2; 0); and Q = (7;0; 0):

We denote by k the distance from E to V . The line from E through the vanishing point

Q of all vertical lines must itself be vertical, and hence perpendicular to the ground at W.

Thus the altitude a of the viewpoint E|the quantity we are after|is the distance from

30

E to W. Finally, you should convince yourself that our labeling of all the angles makes

sense.

Applying the law of sines to triangle EUV , and using the fact that sin(180_􀀀(_+_)) =

sin(_ + _), we have

k

sin(180_ 􀀀 (_ + _))

=

h

sin _

=)

k

sin(_ + _)

=

h

sin _

=) k = h _

1

sin _ _ sin(_ + _):

Using the right triangle EVW, we can now _nd a:

a = k cos _ = h _

1

sin _ _ sin(_ + _) _ cos _;

and the last three factors can all be written in vector notation:

a = h _ kE􀀀􀀀M!kkE􀀀􀀀!Nk

kE􀀀􀀀M!_ E􀀀􀀀!Nk

_ k􀀀􀀀! EM _ E􀀀!Qk

kE􀀀􀀀M!kkE􀀀!Qk

_

E􀀀􀀀!N _ E􀀀!Q

kE􀀀􀀀!NkkE􀀀!Qk

= hk􀀀􀀀! EM _ E􀀀!Qk(E􀀀􀀀!N _ E􀀀!Q)

kE􀀀􀀀M!_ E􀀀􀀀!NkkE􀀀!Qk2

:

Since kE􀀀!Qk2 = E􀀀!Q _ E􀀀!Q, it might be prettier to write a as

a = hk􀀀􀀀! EM _ E􀀀!Qk(E􀀀!Q _ E􀀀􀀀!N)

kE􀀀􀀀M!_ E􀀀􀀀!Nk(E􀀀!Q _ E􀀀!Q)

:

If we estimate the height h of the turret at about 60 feet, we can compute a in Maple:

> with(linalg):

> E:=[8.5,8.3,-8.3]: M:=[6.8,8.9,0]: N:=[6.9,7.2,0]: Q:=[7,0,0]:

> EM:=M-E: EN:=N-E: EQ:=Q-E:

> a:=60*norm(crossprod(EM,EQ),2)*dotprod(EQ,EN)/

(norm(crossprod(EM,EN),2)*dotprod(EQ,EQ));

a := 180.2357150

Thus the altitude a is about three times the value of h, or about 180 feet. However,

since our original measurments of the picture are very rough approximations, we should

be cautious and simply say that the camera was no higher than a few hundred feet o_ the

ground.

As we said earlier, estimating the distance to actual objects whose images appear

in a picture is not always so hard. For instance, the doorway to the cylindrical building

in Figure 13 is in a plane parallel to the picture plane (How do we know?) and its

image contains the central vanishing point, i.e., the center of view. Assuming you already

31

computed the viewing distance d, it is now easy to _nd the distance c from the viewer to

the doorway. A side view of the situation appears in Figure 28, where we have denoted the

points on the bottom and top of the image of the doorway directly below and above the

vanishing point by A0 and B0, respectively, and the actual bottom and top of the doorway

are denoted respectively by A and B. The height of the image of the doorway is h0 and

the actual height of the doorway is denoted by h.

d

B'

picture plane

A'

h'

A (bottom center of door)

B (top center of door)

E

c

h (door height)

Figure 28.

If we let E denote the location of the viewer, then triangles EA0B0 and EAB are

similar, and hence the ratios of corresponding altitudes to corresponding sides are equal;

that is, c=h = d=h0. It follows that

c =

dh

h0

;

or, in words,

(distance from viewer to door) =

(viewer-to-image distance)(actual door height)

(height of door image)

:

If you express d and h0 in inches, convert to feet, and then estimate h in feet, you should

get a reasonable estimate for c in feet. Try it!

Another case in which the viewing distance is not hard to compute is the case of two-

point perspective, such as in Figure 17 or Figure 29 below. By \two-point perspective" we

mean that a picture has been set up in such a way that the picture plane is perpendicular

to the plane of the ground or floor, and only two vanishing points on the horizon line

are needed to render buildings or other objects whose adjacent vertical walls or sides are

perpenducular to each other. This is a typical situation when dealing with architectural

subjects, both indoors and outdoors.

32

v1 v2

Figure 29.

Suppose the rectangular box in Figure 29 represents some kind of building, but we

don't know anything about its size or proportions. Can we say anything about the correct

location of the viewer? It turns out that we can. First, notice that we could guess by looking

that the picture plane is perpendicular to the plane of the ground|that is, vertical|

because the images of the vertical lines of the building are parallel to one another in the

picture, and hence do not converge to a vanishing point. This means that the actual vertical

lines of the building are parallel to the picture plane, so the picture plane is perpendicular

to the plane of the ground. From the discussion in Step (a) following Corollary 8, this

means that the horizon line is also the x-axis, so the center of view lies somewhere on

this line. Moreover, since our line of sight is level, the viewer location is somewhere in

a horizontal plane containing the horizon line of the picture. This horizontal plane is of

course the plane y = 0. This situation is typical of two-point perspective drawings. The

question is, where in the plane y = 0 should the viewer be located? (See Figure 30).

v1 v2

x

the picture plane

z=0

the plane

y=0

?

?

?

?

Figure 30.

Since the viewer location is unknown|as indicated by the question marks|we don't

yet know where to put the y-axis or the z-axis. In order to use coordinates to analyze the

problem, let us choose the midpoint of the two vanishing points v1 and v2 in Figure 30 as

an origin, and introduce a z00-axis through this origin and orthogonal to the picture plane,

and an x00-axis which coincides with the x-axis.. This is indicated in Figure 31, where, in

each of the two cases shown there, we are looking down onto the plane y = 0.

33

x''

z''

E

v1 v2

top of

building

parallel parallel

x''

z''

E

v1 v2

top of

building

parallel parallel

the plane y=0

Figure 31. Two possible cases for the viewer location E.

Figure 31 illustrates the fact that, however the viewer and the building are situated,

we can be sure of two things:

_ The lines of sight from E to the vanishing points v1 and v2 are parallel to the

horizontal edges of the building.

_ These lines of sight must be perpendicular at E, because adjacent horizontal

edges of the building are perpendicular.

This really narrows down the possibilities for E. To explain why, we will refer to Figure 32.

In the _gure we are again looking down onto the plane y = 0. We have labeled the vector

from the origin to v2 as r, and since we located the origin at the midpoint of v1 and v2, the

vector from the origin to v1 can be labeled 􀀀r. The vector from the origin to the viewer

location E is labeled a, and you should convince yourself that the vectors r􀀀a and 􀀀r􀀀a

are labeled correctly.

v1 v2 x''

z''

E

a

–r r

–r–a r–a

Figure 32.

34

Now we just observed that 􀀀r 􀀀 a and r 􀀀 a must be orthogonal, so we can write

(􀀀r 􀀀 a) _ (r 􀀀 a) = 0:

expanding the left side of this equation, we get

􀀀r _ r + r _ a 􀀀 a _ r + a _ a = 0:

Since a _ r = r _ a, and since r _ r = krk2 and a _ a = kak2, this simpli_es to

􀀀krk2 + kak2 = 0;

and hence

kak = krk:

Since kak is the distance from the viewer to the midpoint of v1 and v2, this means

that the correct viewer location lies somewhere on a semicircle of radius krk centered at

the midpoint of the vanishing points. This fact is interesting enough (and as we'll see later,

useful enough) to summarize with a picture and a theorem.

v1 v2

Figure 32. The viewpoint for two-point perspective lies on the semicircle.

Theorem 9. When viewing a work done in two-point perspective, the correct viewer

location lies on a horizontal semicircle whose diameter has the two vanishing points as

endpoints.

We still haven't completely solved the problem of the viewer location, and that's be-

cause we need further information. In fact, for any viewer location on the above semicircle,

except for the endpoints v1 and v2, there is a rectangular box (a building) whose perspec-

tive image matches that in the picture (an argument for this is indicated in the exercises).

However, if we know something about the actual proportions of the building, then we can

35

pin down the correct viewer location exactly (see Exercise# ). Of course, a reasonable

guess is that the center of view lies at the center of the picture; this would be the case for

an uncropped photograph. In this case, the correct viewpoint is the unique point E on the

semicircle directly opposite the center of the picture; that is, the line through E and the

center of the picture is orthogonal to the picture plane (Figure 33).

v1 v2

E

Figure 33. E is easy to locate if the center of view

coincides with the center of the picture.

This gives us a method for viewing such a picture correctly. First, estimate where the two

vanishing points are, and imagine the semicircle connecting them. Then walk around the

semicircle until your are directly opposite the center of the picture; you should be at just

about the right spot for appreciating the perspective e_ect intended by the artist!

Notice also that as the two vanishing points v1 and v2 get farther apart, the circle in

Figure 33 gets larger, forcing the viewer farther away from the picture. This is important,

because, as we have seen again and again, a poorly planned perspective drawing can result

in a viewpoint which is uncomfortably close to the picture. The moral for two-point

perspective drawings is: Keep those vanishing points far apart!

4. Measuring Points

In Section 2 we saw how the concepts of vanishing points and vanishing lines could

be used in making perspective drawings. Recall, however, that we said we wouldn't worry

about things like viewing distance or the size and shape of objects while constructing the

drawings; that's because it made it easier to illustrate how vanishing points and vanishing

planes can be applied to drawing. As a consequence, we just guessed the height of the

roof peak in Figure 20 and the distance between the _rst two fenceposts in Figure 21. In

fact, most art students do the same thing when they make their _rst perspective drawings;

some of the decisions are done by guesswork and intuition, and others are determined by

mathematically-based techniques like the use of vanishing points and vanishing planes.

36

In this section we'll introduce a technique which allows more accurate drawings

when the size, shape, and location of objects which we wish to draw are known.

***********************

5. Anamorphic Art

There are times when the correct viewer location for a painting may be surprisingly

di_erent than what we expect, and there are paintings which cannot be perceived correctly

by viewing in a conventional way. One class of such paintings, called anamorphic art,

became popular in the sixteenth century. An example of such a painting would be one

whose subject could only be correctly perceived by viewing the painting in a cylindrical

mirror. Another example is the painting in Figure 34, The French Ambassadors by Hans

Holbein the younger (1497{1547). When viewed from directly in front, the painting seems

perfectly normal, except for a strange, elongated object which seems to float above the

floor. If we change our viewpoint to one at an oblique angle at the extreme right of the

painting, the object is seen to be a human skull! The skull supposedly represents the

transience of life. The point here for us is that for this painting, there is no single correct

point of view.

Figure 34. Hans Holbein the younger, The French Ambassadors.

Another example of a painting which requires an unusual viewpoint is the anamorphic

portrait of Edward VI in Figure 35. In order to see the portrait correctly, we must view

the painting from so far to the right of the canvas that the picture frame had to be cut

away to keep from blocking our view!

37

Figure 35. Unknown artist, Anamorphic Portrait of Edward VI. 1546.

It seems certain that such a picture, when viewed from in front, would look very strange,

and indeed this the case, as can be seen in Figure 36.

Figure 36. Frontal view of the painting in Figure 36.

How did the artist _gure out how to do it? In a theoretical sense, perhaps, the trick

is no big deal. The canvas is still part of the picture plane|the only new feature is that

it lies completely to the left of the x0y0 origin. However, the human head is not usually

given in xyz-coordinates, and it does not consist of planes and straight lines, so simple

computation isn't enough. What is needed a combination of the artist's skill in painting

portraits in the usual way, along with some sound mathematical reasoning to achieve the

correct \distortion."

Let's try the trick ourselves, using a smiley face as our subject. Since our drawing

will apparently be rather elongated horizontally, let's start with a canvas which is twice

as wide as it is high. For later use, we'll cover it with 8 rows and 16 columns of squares

(Figure 37).

Figure 37.

Rather than try to draw the face with the correct distortion and hope it turns out right,

let's imagine what the _nished product should look like when seen from the right side at

38

an oblique angle. To help with the visualization, we'll draw our canvas in perspective: the

steps are shown in Figure 38.

(a)

v

(b)

v

(c)

v

(d)

v

Figure 38.

In step (a) we mark a vanishing point v: we'll take this as the center of view, although

this is not necessary. To draw the canvas in perspective, we make the sides vertical for

convenience, and make the lines determined by the top and bottom edges converge to v.

Since the sides of the canvas were located rather arbitrarily, how can we be sure that the

outline of the canvas corresponds to a rectangle seen in perspective which is twice as wide

as it is high? The answer is given by Theorem 3, which says that the \shape ratio" of

the canvas is proportional to the viewing distance. We have chosen the shape ratio _rst,

so the correct viewing distance is therefore completely determined, and from this distance

the outline will appear just as it should. (You should be able to estimate this viewing

distance!) To draw the horizontal lines between the squares, we make use of Theorem 2,

which says that the right edge of the canvas (which is parallel to the picture plane), and all

of the \features" on it, have undistorted images. The \features" we refer to are the right

endpoints of the lines between the rows of squares, and thus the images of these endpoints

will be equally spaced. Having located them, we then connect them with straight lines to

v, which must be their vanishing point (Theorem 5).

In step (b) we locate the center vertical line of the canvas, using the same technique and

reasoning that we used to locate the peak of the roof of the little house in in Figure 20 (c).

In step (c) we locate the vertical lines at the 1/2 and 3/4 points of the canvas using the

same technique, and continuing in this fashion we complete the process in step (d). We

deliberately chose the number of columns of squares to be a power of two, since it makes

this technique easy. Figure 38 (d) is now a perspective image of the gridded canvas in

Figure 37.

To _nish visualizing the picture, we should view the grid in Figure 38 (d) from the

correct point of view and draw the smiley face on it the way we wish it to appear, which

39

we have done in Figure 39.*

v

Figure 39.

Notice that we have marked many of the points where the curves of the drawing cross the

grid. In order to make the actual anamorphic drawing, we now locate the corresponding

version of each such point on the frontal view of the grid in Figure 40, and connect the

points with smooth curves. We have mainly used intersection points on the vertical lines

of the grid, since the proportions of features on these lines are preserved; i.e., if a point on

the side of a square is, say, two-thirds of the way up the side of the square in Figure 39,

then the same will be true of its counterpart in Figure 40.

Figure 40.

If you view the enlarged anamorphic drawing in Figure 41 from the extreme right side

of the page with one eye, you can see it as being undistorted. A similar technique could

be used by a portrait painter to achieve e_ects like that in Figure 34 or Figure 35.

* There is a _ne point here which we won't worry too much about. We decided that

the point v would be the center of view, and hence the smiley face should not be drawn

as a perfect circle in Figure 39, because then it would not appear to be a circle from this

viewpoint. However, we have drawn it as a circle anyway, because the resulting distortion

is not too great.

40

Figure 41.

Now try an anamorphic drawing yourself. Make a transparent photocopy of the per-

spective grid in Figure 42, and a paper photocopy of the grid in Figure 43. Place the

transparent grid from Figure 42 over a simple image (such as a _ve-pointed star or a

heart, or perhaps the face of a comic strip character), and use an erasable overhead pro-

jector pen to trace the image onto the grid. Then note where the tracing crosses the grid,

mark the corresponding points on the rectangular grid from Figure 43, and connect them

appropriately to make your anamorphic drawing, using color if you like. (The slight dis-

tortion mentioned in our previous footnote will again be introduced, but the result should

still be good.)

Figure 42.

41

Figure 43.

42

Fractal Geometry

1. The Geometry of Nature

Figure 1. Vine and Tablecloth

The graphic entitled Vine and Tablecloth in Figure 1 is a visual metaphor for the

contrast that many people feel exists between mathematics and the natural world. The

pattern of the tablecloth reminds us of the coordinate grid of the xy-plane, with its neat

rows and columns of little boxes. The vine has a completely di_erent character. It exhibits

a wild, scraggly, complex form, beautiful in a way that is both rugged and delicate. In no

way does it _t into the rigid order of the coordinate grid, and it seems to defy us to describe

it with mathematics. In fact, many forms in nature have this character, and this presents

a problem for us. After all, we are trying to link art with mathematics, and the complex

beauty of nature is one of the most important subjects of the visual artist. For example, the

Chinese painting in Figure 2 pays homage to precisely this kind of wild, complex beauty,

and as we will see, oriental artists have long admired those forms in nature which look

least like what we traditionally think of as geometry. The subject of the ancient Chinese

philosophy of Taoism is the tao, which can be interpreted as the \way" or the \course" of

nature, and the following quote by the popular philosopher Alan Watts on the tao neatly

summarizes our apparent di_culty in _nding a mathematical description of the geometry

of nature:

43

Figure 2.

The tao is a certain kind of order, and this kind of

order is not quite what we call order when we arrange

everything geometrically in boxes, or in rows. That

is a very crude kind of order, but when you look at a

plant it is perfectly obvious that the plant has order.

We recognize at once that is not a mess, but it is not

symmetrical and it is not geometrical looking. The

plant looks like a Chinese drawing, because they ap-

preciated this kind of non-symmetrical order so much

that it became an integral aspect of their painting. In

the Chinese language this is called li, and the charac-

ter for li means the markings in jade. It also means

the grain in wood and the _ber in muscle. We could

say, too, that clouds have li, marble has li, the human

body has li. We all recognize it, and the artist copies it

whether he is a landscape painter, a portrait painter,

an abstract painter, or a non-objective painter. They

all are trying to express the essence of li. The inter-

esting thing is, that although we all know what it is,

there is no way of de_ning it1.

At the time he wrote these words, Watts was

indisputably correct. However, Alan Watts died in

1973, and that is signi_cant, for only two years later

there appeared a book entitled Les objects fractals:

forme, hazard et dimension2, which later evolved into an English version entitled The

Fractal Geometry of Nature3. The author of these books, a mathematician named Benoit

Mandelbrot, had discovered a new kind of geometry, called fractal geometry, which would

radically change the way mathematicians and scientists|as well as many artists|viewed

the natural world. Mandelbrot coined the word \fractal" from the Latin adjective fractus,

meaning \fragmented" or \irregular," because of the forms fractal geometry describes. In

his introduction to The Fractal Geometry of Nature, Mandelbrot echoes the sentiments of

Watts concerning the discrepancies between traditional geometry and nature, but goes on

to announce that the scope of \geometry" has now been widened dramatically:

Why is geometry often described as \cold" and \dry?" One reason lies in

its inability to describe the shape of a cloud, a mountain, a coastline, or a tree.

Clouds are not spheres, mountains are not cones, coastlines are not circles, and

bark is not smooth, nor does lightning travel in a straight line.

More generally, I claim that many patterns of Nature are so irregular and

fragmented, that, compared with Euclid|a term used in this work to denote all of

1 Footnote 1.

2 Footnote 2.

3 Footnote 3.

44

standard geometry|Nature exhibits not simply a higher degree but an altogether

di_erent level of complexity. The number of distinct scales of length of natural

patterns is for all practical purposes in_nite.

The existence of these patterns challenges us to study those forms that Eu-

clid leaves aside as being \formless," to investigate the morphology of the \amor-

phous." Mathematicians have disdained this challenge, however, and have in-

creasingly chosen to flee from nature by devising theories unrelated to anything

we can see or feel.

Responding to this challenge, I conceived and developed a new geometry of

nature and implemented its use in a number of diverse _elds. It describes many of

the irregular and fragmented patterns around us, and leads to full-fledged theories,

by identifying a family of shapes I call fractals.

Figure 1 is our _rst piece of evidence that Mandelbrot's bold claim is valid, because

the vine in the _gure is a fractal: it was drawn with mathematics! Further evidence that

fractal geometry is appropriate for many natural forms can be seen by looking ahead at the

_gures in the text, including _gures that suggest that artists have intuitively used fractal

principles to depict nature for centuries. The best evidence, however, will be the beautiful,

never-seen-before fractals that you are going to make yourself, once you understand some

of the basic principles of fractal geometry. One of the best places to begin is with a

drawing technique called an iterated function system. The artistic possibilities (as well as

many other aspects) of iterated function systems were _rst investigated by mathematician

Michael Barnsley.

2. Iterated function systems

Let's play a game that makes an interesting picture, just by plotting points in the

plane. Let f; g : R2 ! R2 be a_ne transformations de_ned by

f(x) = A1x+b1; and g(x) = A2x+b2

where (1)

A1 = _1=2 􀀀1=2

1=2 1=2 _; b1=_0

0_; A2=_2=3 0

0 2=3_; b2=_1=3

0 _:

We'll also need a coin for the game, and a starting point x0 in the plane. It doesn't matter

what starting point we use; we'll pick x0 = [􀀀1=2 0]T. Now flip the coin. If it comes up

heads, we pick f, and if it comes up tails, we pick g. After each coin flip, we will plot a

point in the plane. To see how the points are determined, look at this example of how a

typical game might go (you should check a few of the steps to verify that the computations

45

have been done correctly):

COIN FLIP

RESULT

CHOICE OF

FUNCTION

POINT TO PLOT

Step 0 􀀀 􀀀 x0 =_􀀀1=2

0_

Step 1 Heads f x1 = f(x0) = _􀀀1=4

􀀀1=4_

Step 2 Tails g x2 = g(x1) = _ 1=6

􀀀1=6_

Step 3 Tails g x3 = g(x2) = _ 4=9

􀀀1=9_

Step 4 Heads f x4 = f(x3) = _5=18

1=6 _

...

...

...

...

Thus we wind up plotting a sequence fx0; x1; x2; . . .g of points in the plane. If we already

have a point xn, we get the next point xn+1 by using a coin flip to randomly choose f or

g, and then apply that function to xn. For instance, if we choose g, then xn+1 = g(xn).

The points x0; x1; x2; x3; x4 computed above have been graphed in Figure 3. The letters

f and g have been used to show which function was used to \move" the point xn to the

next point xn+1.

Figure 3.

46

So far it doesn't look like much|just a random-looking collection of dots. However,

look at Figure 4 to see what we get if we use a computer to continue the process!

Figure 4.

In Figure 4 we have reproduced the _rst _ve points x0; x1; x2; x3; x4 in red, and the

next 80,000 or so points in green. The green points make a fantastic image that seems to

be made of in_nitely repeated pine trees! The set S indicated by the green points is, of

course, a fractal, and the game we played to draw it is called an iterated function system.

To iterate means to do something repeatedly|in this case, to repeatedly get new points

by applying f or g to the previous point. When a fractal is made with an iterated function

system, or IFS, the fractal is called the attractor of the IFS. You can see why by noticing

that, although the points x0; x1; x2; x3 do not lie on S, they seem to be getting closer to it

(refer to the labels in Figure 3), and in fact, the point x4 seems to lie on|or very nearly

on|the attracting set S. After that, the green points x5; x6; . . . _ll out the attractor so

we can see it. The fractal in Figure 1 can be created in a similar way, and we'll show how

to draw it later.

In Figure 4 we have created an in_nitely complex \organic" shape with just two very

simple a_ne transformations; such a seemingly miraculous feat needs some explaining!

However, before we begin trying to understand this process, the are some points we should

make absolutely clear:

_ The range of shapes attainable with fractal geometry is endless: you can make trees,

plants, clouds, mountains, galaxies, flowers, snowflakes, lightning, rivers, coastlines,

and many other natural forms, as well as endless beautiful and strange \abstract"

shapes and textures. The artistic possibilities are awsome!

47

_ Despite the in_nite complexity of fractal forms, the basic principles of fractal geometry

are easy to understand and implement creatively.

_ Understanding fractal geometry does not make the magic go away. Whether you

study fractal geometry for weeks or years, the fascination of seeing fractal shapes

appear never diminishes, and there are always surprises.

_ Understanding fractal geometry will help you understand and draw Nature more ef-

fectively. However, this understanding of natural forms does not make Nature seem

technical, boring, or less magical. You cannot \kill" the beauty and mystery of Na-

ture by achieving greater understanding of it! In fact, it is well-known by scientists

that the opposite e_ect occurs: the realization of order amid complexity in Nature

makes the contemplation and experience of it so touching and awe-inspiring that the

e_ect is addictive! This is a Great Well-Kept Secret of Science and Mathematics: that

the process of understanding deepens and multiplies the mysteries. Fractal geometry

allows the ordinary person to take part in this process of wonder, in an authentic and

satisfying way.

So now that we have that straight, let's start investigating the process by which the

image in Figure 4 was drawn. First, let's look more closely at the functions f and g de_ned

in Equations (1). The matrix A1 is actually a scalar multiple of a rotation matrix. To see

this, observe that

A1 = _1=2 􀀀1=2

1=2 1=2 _=p2=2_p2=2 􀀀p2=2 p2=2 p2=2 _=p2=2_cos 45_ 􀀀sin 45_

sin 45_ cos 45_ _;

so A1 rotates vectors 45_ counterclockwise, and shrinks them by a factor of p2=2, or

approximately 0:7. Since b1 = 0, we have f(x) = A1x for each x 2 R2, so f also rotates

vectors 45_ counterclockwise, and shrinks them by a factor of _ 0:7. It should be clear

from the de_nition of g in (1) that g shrinks vectors by a factor of 2=3 and translates them

1=3 unit to the right. Let's summarize these results by saying what f and g do to subsets

of R2:

(i) f rotates sets 45_ counterclockwise about the origin, and shrinks them toward the

origin by a factor of p2=2 _ 0:7;

(ii) g shrinks sets toward the origin by a factor of 2=3, and then translates the resulting

sets 1=3 unit to the right.

It would be nice to see an example of f and g doing these things to a subset of R2, and

for our purposes, the best example is the set S itself! This is illustrated in Figure 5, where

S is the entire red-and-blue set, f(S) is the red set, and g(S) is the blue set. That is, if we

apply f to every point x of S, and color the resulting points f(x) red, we get the set f(S) in

Figure 5, and similarly for g(S), which is blue. In terms of the descriptions (i) and (ii), the

set f(S) is what we get when we rotate the set S through an angle of 45_ counterclockwise

48

about the origin, and shrink it toward the origin by a factor of p2=2 _ 0:7. The set g(S)

is what we get when we shrink the set S toward the origin by a factor of 2=3, and then

translate the resulting set 1=3 unit to the right. Apparently, a very unusual thing happens:

the set S is precisely the union of f(S) and g(S). That is, apparently,

S = f(S) [ g(S):

Figure 5. The set S has the property that S = f(S) [ g(S).

We'll show that this is indeed the case. In fact, this situation is related to one of

the most important properties of fractals, sometimes referred to as \self-similarity." We'll

say more about self-similarity later, but basically it means that \the parts look like the

whole."

Before going into that, let's take care of some other business. In order to be good at

drawing pictures with iterated function systems, let's precisely

I. De_ne what we mean by an iterated function system (IFS);

II. De_ne the attractor S of an IFS;

III. De_ne what it means for the attractor to \attract" points, and show that it does so;

IV. Show that in the example above, S = f(S) [ g(S), and generalize the result to other

IFS's.

Notice that the a_ne maps f and g were contraction mappings on R2. Speci_cally,

from what we have said it follows that kA1k = p2=2 and kA2k = 2=3. These are the

49

kinds of mappings which we will use in our iterated function systems. Since we used a

coin to choose f or g, each map had a 1=2 probability of being chosen. More generally, we

may use more than just two mappings, and we will allow our maps to have any nonzero

probability of being chosen, the only restriction being that the probabilities of all the maps

must add up to 1, so that it's absolutely certain that some map will be chosen on each

turn. Although more general types of iterated function systems exist, the IFS's we will

use here are capable of generating a fantastically rich variety of fractals. Our de_nition is

as follows.

I. An iterated function system (IFS) is a collection of a_ne contraction

mappings

f1(x) = A1x+b1; f2(x) = A2x+b2; . . .; fn(x) = Anx+bn

on R2, with corresponding nonzero probabilities

p1; p2; . . . ; pn such that p1 + p2 + _ _ _+pn = 1:

We described the attractor S in Figure 4 as being \in_nitely complex," and we'll see

that this is true in a sense. It may seem impossible to precisely de_ne such a set, but in

fact, it's not hard to do. Before giving the de_nition, however, let's discuss our example

with f and g a little further, so that we can see where the idea for the de_nition comes

from. We claimed that the sequence of points x0; x1; x2; . . . was being \attracted" by the

set S. If we look at this sequence in terms of the matrices A1, A2 and the vectors b1, b2,

we see that

x1 = f(x0) = A1x0 +b1

= b1 +A1x0

x2 = g(x1) = A2(b1 +A1x0) + b2

= b2 +A2b1 +A2A1x0

x3 = g(x2) = A2(b2 +A2b1 +A2A1x0) + b2

= b2 +A2b2 +A2A2b1 +A2A2A1x0

x4 = f(x3) = A1(b2 +A2b2 +A2A2b1 +A2A2A1x0) + b1

= b1 +A1b2 +A1A2b2 +A1A2A2b1 +A1A2A2A1x0

...

We said earlier that the point x4 seemed to be very close to the set S, so let's look at the

expression for x4 more closely to see if we can guess what the formula for a \typical point"

of S should look like. Equation (2) gives the expression for x4 in color-coded form:

50

x4 = b1 + A1b2 + A1A2b2 + A1A2A2b1 + A1A2A2A1x0 .

(2)

The magenta symbols came from the _rst application of f to x0; the blue symbols came

from the subsequent application of g to the point x1 = f(x0); the green symbols came

from the application of g to the point x2 = g(x1); and the red symbols came from the

_nal application of f to the point x3 = g(x2). If we continue this process, we'll run out of

colors, so let's replace colors with subscripts. If the red symbols are given new subscripts

k1, the green symbols given new subscripts k2, the blue symbols given new subscripts k3,

and the magenta symbols given subscripts k4, then Equation (2) becomes

x4 = bk1 + Ak1bk2 + Ak1Ak2bk3 + Ak1Ak2Ak3bk4 + Ak1Ak2Ak3Ak4x0: (3)

Notice that the subscript k1 is associated with the last function used, not the _rst one, so

if we applied another function to x4, we'd have to re-number all the subscripts. However,

if we keep things in this more general form, it's easy to write down the form of a generic

point xN:

xN =bk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _+Ak1Ak2 _ _ _AkN􀀀1bkN

+Ak1Ak2 _ _ _AkNx0:

If S really attracts points as we have said, then for large N, the point xN should be

close to some point of S. This suggests that a generic point y of S should be de_ned by a

formula that looks something like the formula for xN. However, we implied earlier that the

\attraction" takes place regardless of the starting point x0, so we shouldn't use x0 in the

formula for y. Moreover, there is no logical number of terms to stop with, so let's not stop;

let's try writing y as an in_nite series of vectors bkj , all except the _rst one multiplied by

matrices Akl :

y = bk1 + Ak1bk2 + Ak1Ak2bk3 + Ak1Ak2Ak3bk4 + _ _ _ ; (4)

where each subscript kj is either a 1 or a 2; that is, each vector bkj is either b1 or

b2, and each matrix Akj is either A1 or A2. It's important to realize that if, say, Ak5

represents the matrix A2 (that is, if k5 = 2), then Ak5 represents A2 everywhere it appears,

and furthermore, bk5 must then represent b2. We have achieved what we were after|a

reasonable guess at a formula that describes a \typical" element of S. Of course, the series

in (4) will not be meaningful unless it converges, but we can see that it does, by looking

at the corresponding series of norms:

kbk1k + kAk1bk2k + kAk1Ak2bk3k + kAk1Ak2Ak3bk4k + _ _ _ : (5)

In this series, the Nth term, for N _ 2, is kAk1Ak2 _ _ _AkN􀀀1bkNk. If we set _ =

maxfkA1k; kA2kg, _ = maxfkb1k; kb2kg, we see that the Nth term satis_es

kAk1Ak2 _ _ _AkN􀀀1bkNk _ _N􀀀1_;

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so the series in (5) is dominated by the geometric series

_ + __ + _2_ + _ _ _ ;

which converges, since 0 < _ = p2=2 < 1. It follows that the series in (4) converges also,

and we are now ready to give our de_nition of the attractor of an IFS.

II. Let an iterated function system be given, with a_ne contraction map-

pings

f1(x) = A1x+b1; f2(x) = A2x+b2; . . .; fn(x) = Anx+bn

on R2. The attractor S of the system is the set of all points of the form

bk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _, where each matrix Akj , and each vector

bkj , is one of those listed above. That is,

S = fbk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _ j1 _ kj _ n for all jg:

Now let's show that the attractor S really does \attract" points. Suppose we have an

iterated function system like that given in Item I, and suppose we have a starting point

x0 2 R2. We begin creating a sequence of points x0; x1; x2; . . . by applying the functions

fk, choosing them at random according to their probabilities pk. At the Nth step of the

process, we have a point

xN =bk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _+Ak1Ak2 _ _ _AkN􀀀1bkN

+Ak1Ak2 _ _ _AkNx0:

We would like to show that such points eventually stay very close to points of the attractor

S. Speci_cally, if someone picks a small distance " > 0 and says, \After some step M, I

want each point xN, for N > M, to be less than " units away from some point on the

attractor S." Can we do it? Yes, and here's how. Following our pervious argument, we

de_ne

_ = maxfkA1k; kA2k; . . . ;kAnkg and _ = maxfkb1k; kb2k; . . . ;kbnkg;

and notice that since 0 < _ < 1, the geometric series

_ + __ + _2_ + _ _ _

converges. Thus we can choose a positive integer M large enough so that

_N_ + _N+1_ + _ _ _ < "=2

52

for every N >M. We can also insure that M is large enough so that

_Nkx0k < "=2

for every N >M.

Now suppose that N >M, and consider the point

xN = bk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _+Ak1Ak2 _ _ _AkN􀀀1bkN

+Ak1Ak2 _ _ _AkNx0

and the point y in S de_ned by

y = bk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _+Ak1Ak2 _ _ _AkN􀀀1bkN

+Ak1Ak2 _ _ _AkNbkN+1 + Ak1Ak2 _ _ _AkN+1bkN+2 _ _ _

in S. Notice that the _rst N terms in the expansions of xN and y are the same. Thus

they cancel when we subtract y 􀀀 xN, so the distance between y and xN is

ky 􀀀 xNk = k 􀀀Ak1Ak2 _ _ _AkNx0

+Ak1Ak2 _ _ _AkNbkN+1 + Ak1Ak2 _ _ _AkN+1bkN+2 + _ _ _ k

_ k􀀀Ak1Ak2 _ _ _AkNx0k

+kAk1Ak2 _ _ _AkNbkN+1k + kAk1Ak2 _ _ _AkN+1bkN+2k + _ _ _

< "=2 + (_N_ +_N+1_ + _ _ _)

< "=2 + "=2 = ";

which is what we wanted to show. Therefore, we can say that

III. For any starting point x0 and any " > 0, there exists a positive integer

M, such that for all N > M, each of the points xN is less than " units

from some point of S. This is the sense in which the attractor S \attracts"

the points xN.

Thus, if N >M, then xN is within " units of some point y in S, and xN+1 is within

some point z in S, and so on. However, y and z may be far apart from each other! That

is, xN+1 may jump clear to the other side of the attractor from xN, but nevertheless, xN

and xN+1 will each be close to some point of S. This is very important from an artistic

point of view, since it means that the points xN will eventually be indistinguishable from

points of S, the set we want to draw.

However, there is another artistic/mathematical issue we have not taken care of. What

we have said so far is like saying that if we send a group of bees (the points xN for large

53

enough N) into a flower garden (the attractor S), then every bee will be close to a flower (a

point of S). But will every flower be close to a bee? Maybe not: even if we have lots of bees,

maybe some bees will huddle around the same flower, and not distribute themselves nicely

around the garden. Analogously, maybe the \dots" xN will not \_ll out" the attractor

S and give us a complete picture. However, each point y of S eventually will have some

point xN close to it|in fact, as close as we want. To see why, suppose that

y = bk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _+Ak1Ak2 _ _ _AkN􀀀1bkN

+Ak1Ak2 _ _ _AkNbkN+1 + Ak1Ak2 _ _ _AkN+1bkN+2 _ _ _ ;

is a point of S, and suppose we want the series expansion of some point xN to agree with

the _rst three terms of y. Since the functions fk are being chosen at random, eventually

three of them will be chosen in the order fk3 , fk2 , fk1 . We're assuming that our iterated

function game is already in progress, so we already have gone through Q steps, and we are

working on a point xQ. Going through three more steps, we can write N = Q+3, and we

have

xN = fk1 (fk2 (fk3 (xQ)))

= bk1 + Ak1bk2 + Ak1Ak2bk3 + Ak1Ak2Ak3xQ;

where the _rst three terms are the same as those in the expansion of y. A similar argument

can be made to show that we will eventually get a point which agrees with y in the _rst

thousand terms, or the _rst million terms, etc. Then, by using an argument like the one

we used to verify Item III, we can show that such points will get arbitrarily close to y.

Finally, let's take care of Item IV.

IV. The attractor S of the iterated function system with a_ne contraction

mappings

f1(x) = A1x+b1; f2(x) = A2x+b2; . . .; fn(x) = Anx+bn

satis_es

S = f1(S) [ f2(S) [ _ _ _ [ fn(S):

To show that S = f1(S)[f2(S) [ _ _ _ [fn(S), we must show that each element of S is

an element of f1(S)[f2(S)[_ _ _[fn(S), and that each element of f1(S)[f2(S)[_ _ _[fn(S)

is an element of S. To begin, let

y = bk1 + Ak1bk2 + Ak1Ak2bk3 + _ _ _

be an element of S. By our de_nition of S, the point

z = bk2 + Ak2bk3 + Ak2Ak3bk4 + _ _ _

54

is also an element of S, and since we can apply the matrix A1 term-by-term to z, we have

fk1 (z) = Ak1z+bk1 = bk1 +Ak1bk2 +Ak1Ak2bk3 +Ak1Ak2Ak3bk4 +_ _ _ = y;

so y belongs to fk1 (S), and therefore it belongs to f1(S) [ f2(S) [ _ _ _ [ fn(S).

Going the other way, suppose y belongs to f1(S)[f2(S)[_ _ _[fn(S). Then y belongs

to fk(S) for some k. We can label this k as k1, so that y 2 fk1 (S), which means that

y = fk1 (z) for some z 2 S. It follows that z can be written in the form

z = bk2 + Ak2bk3 + Ak2Ak3bk4 + _ _ _ ;

so

y = fk1 (z) = Ak1z+bk1 = bk1 +Ak1bk2 +Ak1Ak2bk3 +Ak1Ak2Ak3bk4 +_ _ _ ;

which is an element of S, by the de_nition of S. This _nishes the proof.

For our purposes, Item IV is perhaps the most important item of all. Items I{III

are important, too, because they guarantee that we have a dependable way of drawing

fractals; the artistic equivalent would be making sure that we have reliable paints and

brushes. However, Item IV is what allows us to understand the artwork itself. To see

how, recall that the interpretation of Item IV that is appropriate to the set S in Figure 5

(reproduced in Figure 6 below) is the statement

S = f(S) [ g(S):

Figure 6.

55

Now think about the set f(S). Since S = f(S)[g(S), we can write f(S) = f(f(S)[g(S)).

Thus we get f(S) by applying f to every point of f(S) itself, [thereby obtaining f(f(S))],

and by applying f to every point in g(S) [thereby obtaining f(g(S))]. It follows that f(S)

is the union of the two sets f(f(S)) and f(g(S)):

f(S) = f(f(S)) [ f(g(S)):

This situation is illustrated in Figure 7, in which f(f(S)) is shaded in red and f(g(S)) is

shaded in green.

Figure 7.

In a similar manner, we can deduce that the set g(S) is also a union of two sets:

g(S) = g(f(S)) [ g(g(S)):

The sets g(f(S)) and g(g(S)) have been shaded in gold and blue, respectively. We can

keep going like this, breaking the set S down into smaller and smaller parts. For instance,

can you take a pencil and circle the part of S which is f(f(f(S)))?

All of this subdividing of S is more than just a mathematical exercise. It can help us

answer questions such as

_ Why does the set S look so complicated?

_ Why does the set S look like it is made of trees?

The set S looks complicated because it consists of smaller and smaller copies of itself.

If such a set has any interesting detail at all, then that detail will be repeated at smaller

56

and smaller scales, without end. This is why we were justi_ed in referring to Figure 4 as

being \in_nitely complex."

The reason that the set S looks like it is made of trees is related to the same property:

S consists of smaller and smaller copies of itself! If you cut a branch o_ a tree (remove

a subset of the tree) and stick it in the ground, it looks like a tree. If you cut a smaller

branch o_ that branch and stick it in the ground, it also looks like a tree. Of course, this

process cannot go on forever, because, for example, the tree does not consist of little trees

at the microscopic scale; it consists of cells, which don't look anything like trees. However,

to a good approximation, a tree consists of smaller copies of itself, so it is not too surprising

that a fractal can look like a tree. In fact, you can make all kinds of fractal trees. As an

example, Figure 8 features a fractal tree S which is the attractor of a _ve-map IFS (the

origin is at the base of the tree).

Figure 8.

Below are the a_ne maps which make the tree. We use the usual notation fk(x) =

Akx+bk , and denote the corresponding probability of being chosen by pk.

f1 : A1 = _:45 􀀀:23

:23 :45 _ b1 = _ 0

:28 _ p1 = :2

f2 : A2 = _:52 􀀀:13

:13 :52 _ b2 = _ 0

:40 _ p2 = :2

f3 : A3 = _􀀀:57 :05

:05 :57 _ b3 = _ 0

:40 _ p3 = :2

f4 : A4 = _􀀀:50 :23

:23 :50 _ b4 = _ 0

:32 _ p4 = :2

f5 : A5 = _0 0

0 :45 _ b4 = _0

0 _ p4 = :2

57

In Figure 8, we see that S = f1(S) [ f2(S) [ f3(S) [ f4(S) [ f5(S). However, it wouldn't

be quite right to say that the \tree" S is the union of _ve \copies" of itself, because the

"trunk" f5(S) is just a line segment, not a little tree. In order to partially classify the

di_erent ways in which a set may be the union of \copies of itself," terms like \self-similar"

and \self-a_ne" have been introduced. We won't be too picky about the use of these terms,

because the exact de_nitions are too restrictive for artistic (and even scienti_c) purposes.

However, \self-similar" has a nice ring to it. It captures the essence of most fractals and

many natural phenomena, so we will use it freely. For instance, it seems appropriate to

call the set S in Figure 8 self-similar, despite the presence of the skinny set f5(S). It also

seems _tting to describe the set S in Figure 5 as self-similar, even though the sets f(S)

and g(S) overlap a bit (thereby disqualifying S as being self-similar, according to some

technical de_nitions).

(a) (b) (c)

Figure 9. Self-similarity in grass: a part looks like the whole.

Another example of self-similarity in plants can be seen in Figure 9 (a) and (b). The

picture in (a) was made by scanning a stem of grass in a flatbed scanner. A small sub-stem,

indicated by an arrow, was broken o_, then scanned at a higher resolution and magni_ed,

resulting in (b). Notice how much the image in (b) resembles the whole stem in (a). Part of

this resemblance is due to some fortuitous squashing in the scanner, but it's fairly obvious

that some self-similarity is present. It therefore seems likely that the grass stem could be

58

modeled reasonably well by fractal geometry. A _rst approximation (an artist would say

a rough sketch) of the grass stem is the IFS attractor in Figure 9(c).

You shouldn't conclude from our discussion so far that the self-similarity of IFS at-

tractors makes them suitable only for modeling plants. As far as natural forms go, there

are many shapes in nature which are approximately self-similar. For instance, consider the

IFS

f1 : A1 = _:28 :06

:06 􀀀:28 _ b1 = _0

0 _ p1 = :33

f2 : A2 = _ :25 :06

􀀀:06 :25 _ b2 = _:28

:06 _ p2 = :33

f3 : A3 = _:47 0

0 􀀀:47 _ b3 = _:53

0 _ p3 = :34

whose attractor S looks like Figure 10(a).

(a) (b) (c)

Figure 10. Fractal lightning and real lightning. (Photo courtesy of NASA.)

Jagged fractal curves like this make nice models of things like rivers, coastlines, pro-

_les of mountain ranges or clouds, etc. For example, if we turn the attractor sideways

(Figure 10(b)) it compares nicely with a lightning bolt (Figure 10(c)). Later, we'll learn a

quantitative method for making such comparisons when we discuss a concept called fractal

dimension. As another example, we can use the curve as the outline of a mountain range,

as in Figure 11. In this image, we reversed the curve (reected it in a vertical line) to

make the pro_le of a second mountain range. The clouds in Figure 11 are also an IFS

attractor which we'll discuss later. The mountains and clouds were put together in Adobe

Photoshop, and some digital \airbrushing" was added to make the atmospheric haze.

59

Figure 11. A landscape made from IFS attractors.

60