onefinity-firmware/src/avr/BezierMath.md

Cubic Bezier

f(x) = A(1 - x)^3 + 3B(1 - x)^2 x + 3C(1 - x) x^2 + Dx^3

 -Ax^3 + 3Ax^2 - 3Ax + A
 3Bx^3 - 6Bx^2 + 3Bx
-3Cx^3 + 3Cx^2
  Dx^3


f(x) = (-A + 3B -3C + D)x^3 + (3A - 6B + 3C)x^2 + (-3A + 3B)x + A

  a =  -A + 3B - 3C + D
  b =  3A - 6B + 3C
  c = -3A + 3B
  d =   A

f(x) = ax^3 + bx^2 + cx + d

integral f(x) dx = a/4 x^4 + b/3 x^3 + c/2 x^2 + dx + E

= (-A + 3B - 3C + D)/4 x^4 + (A - 2B + B) x^3 + 3/2 (B - A) x^2 + Ax + E

Quintic Bezier

A(1 - x)^5 + 5A(1 - x)^4 x + 10A(1 - x)^3 x^2 + 10B(1 - x)^2 x^3 +
5B(1 - x)x^4 + Bx^5

(-6A + 6B)x^5 + (15A - 15B)x^4 + (-10A + 10B)x^3 + A

6(B - A)x^5 + 15(A - B)x^4 + 10(B - A)x^3 + A

x^3 (6(B - A)x^2 + 15(A - B)x + 10(B - A)) + A

a =   6(B - A)
b = -15(B - A)
c =  10(B - A)
d = A

f(x) = ax^5 + bx^4 + cx^3 + d

f(x) = (ax^2 + bx + c)x^3 + d


integral f(x) = a/6 x^6 + b/5 x^5 + c/4 x^4 + dx + e

= (B - A)x^6 - 3(B - A)x^5 + 5/2(B - A)x^4 + Ax + e

= (B - A)x^4 (x^2 - 3x + 5/2) + Ax + e

A = 0
B = 1
e = 0

f(x) = 6x^5 -15x^4 + 10x^3
int f(x) dx = x^6 - 3x^5 + 5/2x^4 + C