930 likes | 1.21k Vues
Computer Graphics. Representing Curves and Surfaces. Review eq(11.5). Review eq(11.6/11.7). Review eq(11.8). Review eq(11.8). Review eq(11.9). Review eq(11.10). Review. Blending function (also called ‘Basis’ function). Hermite Curves. 以曲線端點 P 1 . P 4 以及端點斜率 R 1 . R 4 求曲線方程式.
E N D
Computer Graphics Representing Curves and Surfaces
Review • Blending function (also called ‘Basis’ function)
Hermite Curves • 以曲線端點P1.P4以及端點斜率R1.R4求曲線方程式
Hermite Curveeq(11.23) • Reduce 6 multiplies and 3 additions to 3 multiplies and 3 additions.
Bezier Curves • 以曲線端點P1.P4以及控制點P2.P3求曲線方程式 • 曲線端點斜率為
Bezier Curves • Define n as the order of Bezier curves. • Define i as control point.
Bezier Curves • De Casteljau iterations
Bezier Curves • Linear Bezier splinesControl points: P0, P1
Bezier Curves • Quadratic Bezier splinesControl points: P0, P1, P2
Bezier Curves • Quadratic Bezier splinesControl points: P0, P1, P2
Bezier Curves • Cubic Bezier splinesControl points: P0, P1, P2, P3
Bezier Curves • Cubic Bezier splinesControl points: P0, P1, P2, P3
Bezier Curves • Cubic Bezier splinesControl points: P0, P1, P2, P3
Bezier Curves • http://www.ibiblio.org/e-notes/Splines/Bezier.htm
Spline • Natural cubic spline • C0, C1, C2 continuous. • Interpolates(passes through) the control points. • Moving any one control point affects the entire curve.
Spline • B-spline • Local control. • Moving a control point affects only a small part of a curve. • Do not interpolate their control points. • Sharing control points between segments.
B-spline • m+1 control points P0, …, Pm, m≥3 • m-2 curve segments Q3, Q4, …, Qm • For each i≥4, there is a join point or knot between Qi-1 and Qi at the parameter value ti.
Uniform Nonrational B-spline • ‘Uniform’ means that the knots are spaced at equal intervals of the parameter t. • ‘Nonrational’ is used to distinguish these splines from rational cubic polynomial curves, see Section 11.2.5 • We assume that t3=0 and the interval ti+1-ti=1