Splines -- 3/3/2020

Curves (cubic) that appeared to be smooth/continuous

Circle

x = rθ + cx -> x = rcos(2πt) + cx
y = rθ + cy -> y = rsin(2πt) + cy
0 <= θ <= π
0 <= T <= 1

Bezier

Defined By: 2 Endpoints (P0, P3) && 2 Influence Points(P1, P2)
Influence points pull the curve
Quadratic curve would only have on influence point

Line

Pt = (1 - t)P0 + tP1

Quadratric

Q0 = (1 - t)Q0 + tQ1
Q0t = (1 - t)P0 + tP1
Q1t = (1 - t)P1 + tP2
Qt = (1 - t)[(1 - t)P0 + tP1] + t[(1 - t)P1 + tP2]
Qt = (1 - t)²P0 + 2t(1 - t)P1 + t²P2

Cubic

O = (1 - t)S + tT
S = (1 - t)²A + 2t(1 - t)B + t²C
T = (1 - t)²B + 2t(1 - t)C + t²D
O = (1 - t)[(1 - t)²A + 2t(1 - t)B + t²C] + t[(1 - t)²B + 2t(1 - t)C + t²D]
O = (1 - t)³A + 3t(1 - t)²B + 3t²(1 - t)C + t³D
(-A + 3B - 3C + D)t³ + (3A - 6B + 3C)t² + (-3A + 3B)t + A
Form: at³ + bt² + ct + d

Hermite

Defined By: 2 endpoints: P0, P1 && Rates of change at each endpoint: R0, R1
Hermite curves are better when you're typing instructions rather than clicking and drawing things
f(t) = at³ + bt² + ct + d Points on curve
f'(t) = 3at² + 2bt + c Rates of change
f(0) = d = P0
f(1) = a + b + c + d = P1
f'(0) = C = R0
f'(1) = 3a + 2b + c = R1

[0 0 0 1] * [a] = [      d      ] => [P0]
[1 1 1 1] * [b] = [a + b + c + d] => [P1]
[0 0 1 0] * [c] = [      c      ] => [R0]
[1 2 1 0] * [d] = [ 3a + 2b + c ] => [R1]

[ 2 -2  1  1] * [P0] = [ 2P0 - 2P1 + R0 + R1 ] = a
[-3  3 -2 -1] * [P0] = [-3P0 + 3p1 - 2R0 - R1] = b
[ 0  0  1  0] * [P0] = [          R0         ] = c
[ 1  0  0  0] * [P0] = [          P0         ] = d