使用postscript画数学插图

CHAPTER 8

Non-linear 2D transformations:

deconstructing paths

Sometimes we want to draw a figure after a non-linear transformation has been applied to it. In image manipula-

tion programs, this is often called morphing. For example, here is a morphed 10 ×10 grid produced by a program

in which the basic drawing commands drew a square grid and these were followed by some transforming code

before stroking.

In order to apply transformations to paths, we just have to understand (a) transformations and (b) paths!

8.1. Two dimensional transformations

transformation may have very different effects in different parts of the plane, and this is the source of much

difficulty in comprehending it. Indeed, the nature of 2D transformations has been in not-so-distant times the

subject of interesting mathematical research. (I am referring to the stability of properties of such transformations

under perturbation, part of the so-called ‘catastrophe theory’.)

I’ll spend some time looking at one which is not too complicated:

f(x, y) = (x

2

− y

2

, 2xy) .

This is not quite a random choice—it is derived from the function of complex numbers that takes z to z

− y

) + i (2xy) .

In other words, it squares r and doubles θ. We can see how this works in the following figures. On the left is the

sector of the unit circle between 0 and π/2, on the right its image with respect to f:

That seems simple enough. And yet, the image that began this chapter is the transformation with respect to this

same f of the square with lower left corner at (1/2, 0), upper right at (3/2, 1), sides aligned with the axes. So

even this simple transformation has some interesting effects. We’ll see others when we look at what it does to

character strings.

A transformation is completely described by the formula for its separate components, i.e. the functions u and v

in the expression

f(x, y) = (u(x, y), v(x, y)) .

This is all we’ll need to know about f in order to apply it. But in order to understand in any deep sense how a





∂u

∂x

∂v

∂x

∂u

∂y

∂v

∂y





.

For the example we have already looked at this is



2x 2y

∆y

f(x + ∆x, y + ∆y) ≈ f (x, y) + [ ∆x ∆y ] Jac

f

(x, y)

Another way to say it:

(x

0

, y

0

)

0

) (y − y

) ] Jac

f

(x, y) .

What this says, roughly, is that locally in small regions the transformation looks as though it were affine.

In our example, we have an exact equation

f(x + ∆x, y + ∆y) = ((x + ∆x)

2

− (y + ∆y)

2

= ((x

2

2

+ 2y∆y + ∆y

which becomes

2

+ 2x∆x) − (y

2

+ 2y∆y ), 2(xy + x∆y + y∆x) ≈ (x

2

− y

2

, 2xy) + [2x∆x − 2y∆y, 2x∆y + y∆x]

= f(x, y) + [ ∆x ∆y ]

approximation. As it always is.

I’ll not use the Jacobian derivative in PostScript procedures, although it could conceivably improve quality at the

cost of complication. But it is important to realize that the reason our transformation procedures will work so

well is precisely because in very small regions the function f looks affine.

In routines to be described later, a 2D transformation f will be given as a procedure with two arguments x and

y, and it will return a pair u v. In our example it would be

Jacobian approximation. In this version the function f would return the Jacobian matrix on the stack as well, in

the form of an array of 4 variables. In practice this doesn’t seem to be important.

As we have seen, the Jacobian matrix of the transform (x, y) 7→ (x

2

− y

2

, 2xy) is



2x 2y

This matrix has the form

interesting properties. One example of such a matrix is the rotation matrix

and another is the scalar matrix

r

−r sin θ r cos θ



.

In fact, if S is any similarity matrix



a b

−b a



We can write