Game Development Reference
In-Depth Information
Chapter 13
Curves in 3D
I didn't discover curves; I only uncovered them.
— Mae West (1892-1980)
This chapter talks about how to represent curves mathematically in 3D.
Recreating a curve from its mathematical definition is relatively easy; the
tricky part is obtaining a curve with desired properties, or alternatively,
making a tool that designers can use to draw such curves. Our goal in this
chapter is to provide a graceful and intuitive introduction to the mathemat-
ics of curves. In comparison with most of the other topics on the subject,
our aim is to hit the most important points, without stopping every other
paragraph to prove that what we are saying is true. (We will, however, stop
periodically to discuss correct pronunciation, which is probably appropriate
considering that most of the people who developed the math we'll be using
in this chapter were French.) Curves and splines are very useful for all sorts
of reasons. There are obvious applications such as moving objects around
on curved trajectories. But then the coordinates of our curve need not have
a spatial interpretation; essentially, any time we wish to fit a function for a
color, intensity, or other property to given data points, we have a potential
application for curves and splines.
The chapter is divided roughly into two parts. The first part is about
simple, “short” curves that can be described by one equation.
Section 13.1 introduces the specific type of curve we focus on almost
exclusively: the parametric polynomial curve. (It pays special atten-
tion to cubic polynomials.)
Section 13.2 describes polynomial interpolation, whereby a curve is
threaded through specified control points.
Section 13.3 discusses Hermite form, which describes a curve in terms
of its endpoints and the derivatives at those endpoints.
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