B-Spline Basis Function: Questions:_ What is nonzero domain(非零区间)ofB-Splinebasis function Ni(t) ?How many knots does it need?- What is the definition domain(定义区间)of thecurves?4P(t) = ZPNi,4(t)i=0
B-Spline Basis Function Spline Basis Function • Questions: – What is nonzero domain(非零区间) of B-Spline basis function ? ( ) , N t i k – How many knots does it need? – Wh t i th d fi iti d i ( What is the definition domain(定义区间) f th o e curves? 4 ,4 () () P i i t PN t = ∑ i=0
B-Spline Basis FunctionAtake k=4, n=4 as exampleP(t)= ZPN.4(t).i=0to,t,t2,t3, t4,ts, to,tr, ts
B-Spline Basis Function Spline Basis Function • take k=4, n=4 as example 4 ,4 0 () () i i i P t PN t = ∑ 0 1 2 3 4 5 6 7 8 t ,t ,t ,t ,t ,t ,t ,t ,t i=0
B-Spline Basis Function: Properties:- Non-negativity and local support: Ni(t)is non-negative. N,(t)is a non-zero polynomial on [t,ti+]≥0te[t,ti+k]N,k(t)=0otherwise-PartitionofUnity. The sum of all non-zero order k basis functions on [tk-j,tn+1]is 1ZNu (1)=1 te[rk-1, ml]i=0
B-Spline Basis Function Spline Basis Function • Properties: – Non-negati it d l l t tivity and local support • is non-negative is a non ero pol nomial on , ( ) N t i k [ ] t t • N t( )is a non-zero polynomial on ⎨⎧≥ ∈ + t t t N t i i k i k 0 [ , ] ( ) [ , ] i ik t t + , ( ) N t i k – Partition of Unity ⎩⎨= otherwise i k 0 ( ) , • The sum of all non-zero order k basis functions on is 1 ∑ = ∈ n N (t) 1 t [t t ] 1 1 [,] k n t t − + ∑ = = ∈ − + i i k k n N t t t t 0 , 1 1 ( ) 1 [ , ]
B-Spline Basis Function:Properties:- Differential equation of the basis functionk-1k-1N'.(t)ti+kti+k-1-tti+-Please compare withtheBernstein base:Bi,n(t) = n[Bi-1.n-I(t) - Bi.n-i(t)],i = 0,1, ,n;
B-Spline Basis Function Spline Basis Function • Properties: – Diff ti l ti f th b i f ti Differential equation of the basis function: k 1 k 1 , , 1 1, 1 1 1 1 1 () () () ik ik i k ik i ik i k k Nt N t N t t t tt − +− +− + + − − ′ = + − − – Please compare with the Bernstein base: B′ ( ) [B ( ) B ( )] 0,1, , ; ( ) [ ( ) ( )], , 1, 1 , 1 i n B t n B t B t i n i n i n = ⋅⋅⋅ ′ = − − − − , , , ;
B-Spline·Category(分类)of B-Spline- General Curves could be categorized to two groupsby checking if the start point and the endpoint areoverlapped:: Open Curves: Close Curves一 According to the distribution(分布) of the knots inknot vector, B-Spline could be classified to thefollowing four groups:
B-Spline • Category(分类) of B-Spline – General Curves could be categ gp orized to two groups by checking if the start point and the endpoint are overlapped: • Open Curves • Close Curves Close Curves – According to the distribution(分布) of the knots in knot vector B knot vector, B-Spline could be classified to the Spline could be classified to the following four groups: