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
- Similar to Bezier Curve using Bernstein basisfunctions, B-Spline curves uses B-Spline basisfunctions
– 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'sProperties· Formula of B-Spline CurveP(t)=≤P,Bin(t), te[0,1]i=0P(t)= PNi,(t)i=0P,(i = o,l,...,n) are control pointsN,(t) (i=O,1,.,n) are the i-th B-Spline basis functionof order k. B-Spline basis function is a order k(degreek-1)piecewise polynomial （分段多项式）determined by the knot vector, which is a non-decreasing set of numbers
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 nondecreasing set of numbers
·Demo of B-splineThe story of order & degree- G Farin: degree, Computer Aided Geometric Design- Les Piegl: order, Computer Aided Design
• 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· Definition of B-Spline Basis Function- de Boor-Cox recursion formulat,ln+k
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
k=1,i=0Ck=2,i=0C2t, <x<ti+l.OtherwiseCV(t)+k-(t)tti+k-1 -t,ti+kti+
• , 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 + − +− +− + + − − = + − −