Today's Topics.Why splines?. B-Spline Curves and propertiesB-Spline surfaces. NURBS curves and Surfaces
Today s’ Topics • Why splines? • B-Spline Curves and properties • B-Spline surfaces Spline surfaces • NURBS curves and Surfaces
WhytointroduceB-Spline(B样条）: Bezier curve/surface has many advantages,but they have two main shortcomings:- Bezier curve/surface cannot be modified locally(局部修改)- It is very complex to satisfy geometric continuityconditions for Bezier curves or surfaces joining
Why to introduce B Why to introduce B -Spline (B Spline (B样条 ) • Bezier curve/surface has many advantages, but the y have two main shortcomin gs: – B e e cu ve/su ace ca ot be od ed oca y zi er cu rve/surface cannot be modified locall y (局部修改). – It i l t ti f t i ti it It is very comp lex to sati s fy geome t r ic continuity conditions for Bezier curves or surfaces joining
. History of B-splines- In 1946, Schoenberg proposed a spline-basedmethod to approximate curves.- It's motivated by runge-kutta problem ininterpolation: high degree polynomial may surgeupper and down_ Why not use lower degree piecewise polynomialwith continuous joining?- that's Spline
• History of B-splines – I 1946 n, Sh b c oen erg proposed li a sp ne-b d ase method to approximate curves. – It’s motivated by runge-kutta problem in interp g g py y g olation: high degree polynomial may surge upper and down – Wh t l d i i l i l Why not use lower degree piecewise polynomial with continuous joining? – that’s Spline
- But people thought it's impossible to use Splinein shape design, because complicatedcomputation- In 1972, based on Schoenberg's work, Gordonand Riesenfeld introduced “B-Spline" and lotsof corresponding geometric algorithms.- B-Spline retains all advantages of Bezier curves,and overcomes the shortcomings ofBeziercurves
– But people thought it’s impossible to use Spline in shape design because complicated in shape design, because complicated computation – I 1972 b d S h b ’ k G d In 1972, base d on S c hoen berg’s wor k, Gor don and Riesenfeld introduced “B-Spline” and lots of di i l i h f corresponding geometr ic a lgor i t hms. – B-Spline retains all advantages of Bezier curves, and overcomes the shortcomings of Bezier curves
. Tips for understanding B-Spline?- Spline function interpolation is well known, it canbe calculated by solving a tridiagonalequations(三对角方程)- For a given partition of an interval, we cancompute Spline curve interpolation similarly- All splines over a given partition will form alinear space. The basis function of this linearspace is called B-Spline basis function
• Tips for understanding B-Spline? – Spline function interpolation is well known it can Spline function interpolation is well known, it can be calculated by solving a tridiagonal equat ions (三对角方程). – For a given partition of an interval, we can compute Spline curve interpolation similarly. – All splines over a given partition will form a All splines over a given partition will form a linear space. The basis function of this linear space i ll d B is call e d B -S li b i f i S pline bas is funct ion