Finding the Derivative of NURBS Curves

In Conclusion

This chapter has set the stage for most of the curve concepts you'll see throughout the rest of this book. Bezier curves have their limitations, but they are a little bit easier to deal with than the more general B-splines. Before you move on to the next chapter, make sure you understand this chapter as much as possible. Here is a list of some of the things you should remember.

• Parametric equations are more flexible than the forms seen in earlier chapters, in part because they allow you to define a shape with control points rather than an abstract mathematical equation. They are the first example of parametric curve equations found in this and later chapters.

• Slopes of parametric curves can be computed by first finding the derivatives with respect to t.

• Bezier curves are defined in parametric terms as functions of a set of control points and basis functions.

• The endpoints of a Bezier curve are defined by the endpoints of the control polygon. All other points on the curve are affected by all of the control points.

• Bezier curves are limited in that the number of control points determines the degree of the curve and they do not give you any local control.

• Joined Bezier curves are continuous if the endpoints match. They are C1 continuous if the last and first line segments of the control polygons are collinear.

• You can find the derivative of any point on a Bezier curve as a function of the derivatives of the basis functions.