get-medial-point-coeffs-bez3-same-curve

function getMedialPointCoeffsBez3_SameCurve(t: number, v: number[], ps: number[][]): { A: number[]; B: number[]; C: number[]; D: number[]; H: number[]; }

Defined in get-medial-points/double/get-medial-point-coeffs-bez3-same-curve.ts:26

Returns polynomial coefficients for the same-curve cubic medial-point case, where p = b(t) on the same cubic bezier ps and v is normal to the curve at that parameter t.

The returned coefficients encode:

• A, B: E2(s, τ) = (s - t)^2⋅(A(s)⋅τ + B(s))
• C, D: E1(s, τ) = (s - t)⋅(C(s)⋅τ + D(s))
• H: the reduced eliminant in s

Here τ is the ray parameter in q(τ) = p + τ⋅v. For a candidate root s of H, recover τ from A(s)⋅τ + B(s) = 0, i.e. τ = -B(s)/A(s) when A(s) ≠ 0.

In this same-curve setup, the full eliminant has the form H_full(s) = (s - t)^4*H(s). This function returns only the reduced factor H(s) (the repeated (s - t)^4 factor is omitted).

Parameters:

Name	Type	Description
t	number	parameter on ps where p = b(t)
v	number[]	ray direction from p; assumed normal to ps at t
ps	number[][]	order 3 bezier control points, e.g. [[0,0],[1,1],[2,1],[3,0]]