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Real life methods for constructing curves and surfaces often start with points and vectors, which is why we start with a short discussion of the properties of these mathematical enti- ties. The material in this section applies to both two-dimensional and three-dimensional points and vectors, while the examples are given in two-dimensions.

In order to achieve realism, the many algorithms and techniques employed in computer graphics have to construct mathematical models of curved surfaces, models that are based on curves. It seems that straight line segments and flat surface patches, which are simple geometric figures, cannot play an important role in achieving realism, yet they turn out to be useful in many instances. A smooth curve can be approximated by a set of short straight segments. A smooth, curved surface can similarly be approximated by a set of surface patches, each a small, flat polygon. Thus, this chapter discusses straight lines and flat surfaces that are defined by points. The application of these simple geometric figures to computer graphics is referred to as . The chapter also presents two types of surfaces, bilinear and lofted, that are curved, but are partly based on straight lines.

Definition: A polynomial of degree n in is the function where a i are the coefficients of the polynomial (in our case, they are real numbers). Note that there are + 1 coefficients.

The curve and surface methods of the preceding chapters are based on points. Using polynomials, it is easy to construct a parametric curve segment (or surface patch) that passes through a given one-dimensional array or two-dimensional grid of points.

Given a set of points, it is easy to compute a polynomial that passes through the points. The LP of Section 3.2 is an example of such a polynomial. However, as the discussion in Section 1.5 (especially exercise 1.20) illustrates, a curve based on a high-degree poly- nomial may wiggle wildly and its shape may be far from what the user has in mind. In practical work we are normally interested in a smooth, tight curve that proceeds from point to point such that each segment between two points is a smooth arc. The spline approach to curve design, discussed in this chapter, constructs such a curve from indi- vidual segments, each a simple curve, generally a parametric cubic (PC). This chapter illustrates spline interpolation with three examples, cubic splines (Section 5.1), cardinal splines (Section 5.4), and Kochanek-Bartels splines (Section 5.6). Another important type, the B-spline, is the topic of Chapter 7. Other types of splines are known and are discussed in the scientific literature. A short history of splines can be found in [Schumaker 81] and [Farin 04].

Bézier methods for curves and surfaces are popular, are commonly used in practical work, and are described here in detail. Two approaches to the design of a Bézier curve are described, one using Bernstein polynomials and the other using the mediation operator. Both rectangular and triangular Bézier surface patches are discussed, with examples.

B-spline methods for curves and surfaces were first proposed in the 1940s but were seriously developed only in the 1970s, by several researchers, most notably R. Riesenfeld. They have been studied extensively, have been considerably extended since the 1970s, and much is currently known about them. The designation „B“ stands for Basis, so the full name of this approach to curve and surface design is the basis spline. This chapter discusses the important types of B-spline curves and surfaces, including the most versatile one, the nonuniform rational B-spline (NURBS, Section 7.14)

The Bézier curve can be constructed either as a weighted sum of control points or by the process of scaffolding. These are two very different approaches that lead to the same result. A third approach to curve and surface design, employing the process of (also known as or ), is the topic of this chapter. Refinement is a general approach that can produce Bézier curves, B-spline curves, and other types of curves. Its main advantage is that it can easily be extended to surfaces.

The surfaces described in this chapter are obtained by transforming a curve. They are not generated as interpolations or approximations of points or vectors and are con- sequently different from the surfaces described in previous chapters. A reader who wishes a full understanding of this chapter should be familiar with the important three- dimensional transformations (rotation, translation, scaling, reflection, and shearing)and how they are described mathematically by a 4 × 4 transformation matrix. This mate- rial is available in most texts on computer graphics, but the next paragraph is a short summary, for those who only need a refresher.