清华大学：《计算机图形学基础》课程教学资源（授课教案）B样条曲线曲面

Today s’ Topics • Why splines? • B-Spline Curves and properties • B-Spline surfaces Spline surfaces • NURBS curves and Surfaces

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 – 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 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 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

– Similar to Bezier Curve using Bernstein basis f nctions B functions, B -Spline c r es ses B Spline c u r ves uses B -Spline basis Spline basis functions

B-Spline curves and it’s Properties • Formula of B-Spline Curve. ∑ n ( ) ( ), [0,1] , 0 =Σ ∈ = P t P B t t i i n n i are control points ∑ = = i i i k P t PN t 0 , ( ) ( ) – P (i 0 1 ) are control points. – (i=0,1,.,n) are the i-th B-Spline basis function P (i 0,1, , n) i = L ( ) , N t i k of order k. B-Spline basis function is a order k (degree k -1) piecewise polynomial (分段多项式) determined by the knot vector, which is a non￾decreasing set of numbers

• Demo of B-spline • The story of order & degree The story of order & degree – G Farin: degree, Computer Aided Geometric Design – L Pi l d C Aid d D i Les Pieg l: or der, Computer Aid e d Des ign

B-Spline Basis Function Spline Basis Function • Definition of B-Spline Basis Function – de Boor-Cox recursion formula: ⎧ t < x < t 1 i i 1 ⎩⎨⎧ < < = + Otherwise t x t N t i i i 01 ( ) 1 ,1 tt t t − − , , 1 1, 1 1 1 () () () i ik ik ik i k ik i ik i tt t t Nt N t N t t t tt + − +− +− + + = + − − – Knot Vector: a sequence of non-decreasing number t t L t t L t t L t t k k n n n k n k t t t t t t t t − + + − + , , , , , , , , , , 0 1 L 1 L 1 L 1

• , i = 0 k =1 • , i = 0 k = 2 ⎧1 t < x < t ⎩⎨⎧ < < = + Otherwise t x t N t i i i 01 ( ) 1 ,1 , , 1 1, 1 1 1 () () () i ik ik ik i k ik i ik i tt t t Nt N t N t t t tt + − +− +− + + − − = + − −