TABLE OF CONTENTS Chapter 3 developed a parametric rational quadratic representation for conics and in Example 3.16, a constructive algorithm was developed for evaluating a special case conic section. Later in Section 3.9.1, the algorithm is generalized to evaluate all conic sections constructively. Recall that attractive properties of the geometric approach for conics included:
the curve interpolated the first point and the last point;
the curve was tangent to the sides of the polygon formed by the three points that determine it;
the rational parametric form used the defining three points for coefficients in the rational parametric formulation derived.
Property 3 means that no computationally costly process is needed to derive the analytical coefficients from the geometric solution.
In this chapter, that constructive process is extended to create curves from ordered sets of n + 1 points, for arbitrary n. We also investigate which of the above three characteristics are retained by curves constructed with the generalized formulation, and also if any other shape characteristics can be determined.
This leads us to the formulation of the B zier curve, presentation of its properties, and corresponding proofs. We also show its relationship to the Bernstein curve approximation method. This relationship allows us to use approximation and convergence properties from the Bernstein method to understand the behavior of the B zier curve.
An arbitrary collection of n + 1 points in 
, 
, is used to define the constructive process and the resulting curve ?. Without loss of generality,...
