Chapter 10: More NURBS Surfaces

What Is a Curve?

I should get this out of the way as soon as possible. This is a book about curves (and surfaces), so it would be quite improper to go too much farther without first defining exactly what a curve is.

Definition

A curve is defined as a plot of the results of a function in space.

 

Now, keep in mind that this definition isn't entirely correct from a strict mathematical perspective. Because of this, I will probably never be invited to swanky mathematician dinner parties. We all must make sacrifices. A strict mathematical definition could have filled an entire chapter and this is a fairly short book. Instead, my definition of a curve will be good enough for our purposes. In short, I am defining a curve as a plot and vice versa.

Having said that, keep in mind that the idea of a "plot" can encompass much more than something that you'd draw on a screen or a piece of paper. Imagine a car; obviously, you can think of the outline of the body as a curve, but don't forget that the motion of the car follows a curve, the fuel consumption follows a curve, and much more.

Most of the examples in this book are focused on curves and surfaces as geometric and visible graphs because visual examples are very helpful and because curves can be quite useful in the visual space. Don't let that lull you into a single mindset. Remember that curves can be used to represent so much more. This is one of the reasons I am starting out with polynomials . Polynomial functions are less valuable for geometry and more valuable for ideas like motion or consumption. For example, you can easily describe the motion of a car as a polynomial function of time. However, the rich curves of the curves of the car require something more flexible. In fact, Pierre Bezier invented Bezier curves while working for an automobile designer. I'll discuss this more in Chapter 3, "Paremetric Equations and Bezier Curves."

In the meantime, polynomial curves provide an easy framework in which to discuss some of the properties of curves.