Uniform B-Spline(均匀B样条)- (1) Uniform B-Spline: The knots are uniformed distributed, like 0,1,2,3,4,5,6,7.Thiskind ofknot vector defines uniform B-Spline basisfunctionuniformB-SplineofDegree3
U if B n orm B-S li p ne(均匀B样条) – (1) Uniform B-Spline • The knots are uniformed distributed, , like 0,1,2,3,4,5,6,7 • This kind of knot vector defines uniform B-Spline basis function unif B orm-S li f D 3 Spline of Degree 3
Quasi-UniformB-Spline(准均匀B样条)- (2) Quasi-Uniform B-Spline. Different from uniform B-Spline, it has:-thestart-knotandend-knothaverepetitiveness(重复度)ofk-Uniform B-Spline does not retain the“end point"property ofBezierCurve, whichmeans thestartpointand end point ofuniform B-Spline are no-longer the same as the start point andend point of the control points. However, quasi-Uniform B-Spline retains this“end point"propertyQuasi-uniformB-Splinecurveofdegree3
Quasi-Uniform B Uniform B-Spline( Spline(准均匀B样条) – (2) Quasi-Uniform B-Spline • Different from uniform B-Spline, it has: – the start-knot and end-knot have repetitiveness(重复度) of k – Uniform B-Spline does not retain the “end point” property of Bezier Curve, which means the start point and end point of uniform B-Spline are no-longer the same as the start point and end point of the control points However quasi end point of the control points. However, quasi-Uniform B Uniform BSpline retains this “end point” property Quasi-uniform B-Spline curve of degree 3
Piecewise BezierCurve(分段Bezier曲线(3)PiecewiseBezierCurve·the start-knot and end-knot have repetitiveness(重复度)ofk: all other knots have repetitiveness of k-1. Then each curve segment will be Bezier curvesPiecewiseB-SplineCurveof degree3
Piecewise Bezier Curve( Piecewise Bezier Curve(分段Bezier曲线 ) – (3) Piecewise Bezier Curve • the start-knot and en d-knot have re p ( etitiveness (重复度 ) of k • all other knots have re petitiveness of k-1 • Then each curve segment will be Bezier curves Piecewise B-Spline Curve of degree 3
PiecewiseBezierCurve(分段Bezier曲线:For piecewise Bezier curve, the different pieces ofthe curve are relatively independent. Moving thecontrol point will only influence the correspondingpiece of curve, while other pieces of curves will benot change. Furthermore, the algorithms for Beziercould also be used for piecewise Bezier Curve. But this method need more data to define the curve(more control points, more knots)
Piecewise Bezier Curve( Piecewise Bezier Curve(分段Bezier曲线 ) • For piecewise Bezier curve, the different pieces of th l ti l i d d t M i th the curve are re l ative ly i n depen den t. Mov ing the control point will only influence the corresponding piece of curve, while other pieces of curves will be not change. Furthermore, the algorithms for Bezier not change. Furthermore, the algorithms for Bezier could also be used for piecewise Bezier Curve. • But this method need more data to define the curve (more control points, more knots)
Non-uniform B-Spline (非均匀B样条)- (4) Non-uniform B-Spline. The knot vectors T =[to,t,...,tn+k]satisfyconditions that the sequence of knots is non-decreasing(非递减)-repetitivenessoftwoendknots,<=k-repetitiveness of other knots, <= k-1 This kind of knot vector defines the non-uniformB-Spline
Non-uniform B-S p ( line (非均匀 B 样条 ) – (4) Non-uniform B-Spline • Th e o vec o s sa s y kn ot vec t o r s T tt t = [, , ] 0 1 L n k + sati s fy conditions that the sequence of knots is nondecreasing(非递减 ) 0 1 [, , ] n k + decreasing(非递减 ). – repetitiveness of two end knots, <= k – repetitiveness of other knots, <= k-1 • This kind of knot vector defines the non-uniform B-Spline