- 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,<x<ti+lN.(t):0Otherwiset-tti+kNi(t)=1,k-1(t)-(t) +ti+k-1 -t,ti+k -1tit!- Knot Vector: a sequence of non-decreasing numberto,ti,", th-,tk,", tn,tn+1Ln+k-l>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 + − +− +− + + − − = + − −