计算机图形学课件Chapter08.ppt

Curves and Surfaces Contents The transformation matrix for conversion from a periodic, cubic B-spline representation to a cubic, Bézier spline representation. pn-2 pn-1 pn p0 p1 p2 p3 4. Cubic Bézier curves n = 3. Four control points. Bézier matrix 8-11 Bézier Surfaces pj,k : (m + 1) by (n + 1) control points m = 2, n = 2 m = 3, n = 3 Connecting two Bézier surfaces 8-12 B-Spline Curves Advantages of B-Spline over Bézier curves: 1. B-Spline Curves Control points: Blending function Bk, d (u) has degree d – 1 Polynomial degree is independent of control point number. Local control of curve shape. Cox-deBoor recursive formulation: A sequence of (n+d+1) non-decreasing values Any term with denominator evaluated as zero is assigned the value zero. knot vector: Properties of blending functions: Bk, d has degree d – 1 and Cd-2 continuity Bk, d is non-zero over d subintervals uk ? u ? uk+d ? For n + 1 control points, the curve is described with n + 1 blending functions u0 u1 un un+1 . . . u2 un-1 u0 u1 un un+1 . . . u2 u3 un-1 un+2 B0,1 B1,1 Bn-1,1 Bn,1 B0,2 B1,2 B2,2 Bn-1,2 Bn,2 Bn-2,2 d = 1 d = 2 The resulting B-Spline curve is defined only in the interval ud-1 ? u ? un+1 Each section of the spline curve (between two successive knot values) is influenced by d control points, and lies within the convex hull of these control points. Each control point affects the shape of at most d sections. Properties of B-Spline curve: ? The range of parameter u is divided into n + d subintervals by the n + d + 1 values specified in the knot vector. Example of local control: 2. Uniform, Periodic B-Spline Curves Blending functions are periodic Knot values have constant spacing: {-1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0} {0.0, 0.2, 0.4, 0.6, 0.8, 1.0} {0, 1, 2, 3, 4, 5, 6, 7} Example: knot vector: {0, 1, 2, 3, 4, 5, 6} B0,3 B0,2 B1,2 B0,1 B1,1 B2,1 (uk = k) n = d = 3. p0 p1 p2 p3 3. Cubic, Periodic B-Spline Curves d = 4 For n = 3: (1) Blending functions. knot vector {0, 1, 2, 3