Abstract
This chapter explores the antecedents of NonUniform Rational B-Spline (NURBS) curves, specifically Bézier curves. A Bézier curve, which is a special case of a NURBS curve, is determined by a control polygon. The chapter also illustrates the shapes of the curve generated by a Bézier polygon. The curve exhibits the variation-diminishing property. This means that the curve does not oscillate about any straight line more often than the control polygon. Although it is not necessary to numerically specify the tangent vectors at the ends of an individual Bézier curve, maintaining slope and curvature continuity when joining Bézier curves, determining surface normals for lighting or numerical control tool path calculation, or local curvature for smoothness or fairness calculations requires a knowledge of both the first and second derivatives of a Bézier curve.
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