Fundamentals of Computer Graphics, 3rd Edition

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15

Michael Gleicher

Curves

15.1 Curves

Intuitively, think of a curve as something you can draw with a pen. The curve is

the set of points that the pen traces over an interval of time. While we usually

think of a pen writing on paper (e.g., a curve that is in a 2D space), the pen could

move in 3D to generate a space curve, or you could imagine the pen moving in

some other kind of space.

Mathematically, definitions of curve can be seen in at least two ways:

1. The continuous image of some interval in an n-dimensional space.

2. A continuousmap from a one-dimensionalspace to an n-dimensionalspace.

Both of these definitions start with the idea of an interval range (the time over

which the pen traces the curve). However, there is a significant difference: in

the first definition, the curve is the set of points the pen traces (the image), while

in the second definition, the curve is the mapping between time and that set of

points. For this chapter, we use the first definition.

A curve is an infinitely large set of points. The points in a curve have the

property that any point has two neighbors, except for a small number of points

that have one neighbor (these are the endpoints). Some curves have no endpoints,

either because they are infinite (like a line) or they are closed (loop around and

connect to themselves).

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340 15. Curves

Because the “pen” of the curve is thin (infinitesimally), it is difficult to create

filled regions. While space-filling curves are possible (by having them fold over

themselves infinitely many times), we do not consider such mathematical oddities

here. Generally, we think of curves as the outlines of things, not the “insides.”

The problem that we need to address is how to specify a curve—to give a

name or representation to a curve so that we can represent it on a computer. For

some curves, the problem of naming them is easy since they have known shapes:

line segments, circles, elliptical arcs, etc. A general curve that does not have a

“named” shape is sometimes called a free-form curve. Because a free-form curve

can take on just about any shape, they are much harder to specify.

There are three main ways to specify curves mathematically:

1. Im plicit curve representations define the set of points on a curve by giving a

procedure that can test to see if a point in on the curve. Usually, an implicit

curve representation is defined by an implicit function of the form

f(x, y)=0,

so that the curve is the set of points for which this equation is true. Note that

the implicit function f is a scalar function (it returns a single real number).

2. Parametric curve representations provide a mapping from a free parameter

to the set of points on the curve. That is, this free parameter provides an

index to the points on the curve. The parametric form of a curve is a func-

tion that assigns positions to values of the free parameter. Intuitively, if you

think of a curve as something you can draw with a pen on a piece of paper,

the free parameter is time, ranging over the interval from the time that we

began drawing the curve to the time that we finish. The parametric function

of this curve tells us where the pen is at any instant in time:

(x, y)=f(t).

Note that the parametric function is a vector-valued function. This example

is a 2D curve, so the output of the function is a 2-vector; in 3D it would be

a 3-vector.

3. Generative or procedural curve representations provide procedures that can

generate the points on the curve that do not fall into the first two categories.

Examples of generative curve descriptions include subdivision schemes and

fractals.

Remember that a curve is a set of points. These representations give us ways

to specify those sets. Any curve has many possible representations. For this