import { toPowerBasis2 } from '../../to-power-basis/to-power-basis/double/to-power-basis.js';


/**
 * Returns the implicit form of the given quadratic bezier curve.
 * 
 * * returned coefficients are subscripted to match their monomial's variables,
 * e.g. `vₓᵧ` is the coefficient of the monomial `vₓᵧxy`
 * * the implicit form is given by: `vₓₓx² +vₓᵧxy + vᵧᵧy² + vₓx + vᵧy + v = 0`
 * * intermediate calculations are done in **double** precision
 * * adapted from [Indrek Mandre](http://www.mare.ee/indrek/misc/2d.pdf)
 * 
 * @param ps a quadratic bezier curve given as an array of its control points, 
 * e.g. `[[1,2],[3,4],[5,7]]`
 * 
 * @doc mdx
 */
function getImplicitForm2(
        ps: number[][]): { 
            vₓₓ: number; vₓᵧ: number; vᵧᵧ: number;
            vₓ: number; vᵧ: number;
            v: number;
        } {

    // The implicit form is given by:
    // vₓₓx² +vₓᵧxy + vᵧᵧy² + vₓx + vᵧy + v = 0
    
    const [[a2,a1,a0],[b2,b1,b0]] = toPowerBasis2(ps);

    const q1 = a2*b1 - a1*b2;
    const q2 = a2*b0 - a0*b2;

    const vₓₓ = -b2*b2;
    const vₓᵧ = 2*a2*b2;
    const vᵧᵧ = -a2*a2;
    const vₓ = b1*q1 - 2*b2*q2;
    const vᵧ = 2*a2*q2 - a1*q1;
    const v = q1*(a1*b0 - a0*b1) - q2*q2;


    return { vₓₓ, vₓᵧ, vᵧᵧ, vₓ, vᵧ, v };
}


export { getImplicitForm2 }