Maths: Binormial Co-efficient

Binormial co-efficient

(ⁿ_i) = n! / i!(n-i)!

Binormial theorem

(x + y)ⁿ = ∑ⁿ_i=0 (ⁿ_i) xⁱ y^n-i

in relation to FFD:

conventional polynomial basis: 1 t t² t³
Bernstein polynomial basis: (1-t)³ 3(1-t)²t 3(1-t)t² t³


1-D example (real line)

given an arbitrary point, v, on a line of P0, P1, P2, P3.

P₀=0
P₁=1
P₂=2
P₃=3

v is given as x. t is given as the length of v to P0 divided by P3 to P0.

in general mathematical terms:

v = ∑ ³_i=0 P _i W _i (t)


W = weight = scalar = constant

in this context:

t= ||v P₀|| / ||P₃ P₀|| = x/3

∴ using dot product:

v = 1 • W₁(t) + 2 • W₂(t) + 3 • W₃(t)

(P0 is ignored because P0=0; where the multiplication will results in 0)

= (³₁)t¹ (1-t²) + 2 • (³₂)t²(1-t) + 3 • (³₃)t³

= 3t (1-t²) + 6 t²(1-t) + 3 t³

=3t [ (1-t)² + 2t(1-t) + t² ]

=3t [ (1-t) + t ] ² = 3t = x