triangle-centroid.md

Centroid of Triangle Rendered with Lines $f(x)$, $g(x)$ and $h(x)$

$$ f(x) = ax + b $$

$$ g(x) = cx + d $$

$$ h(x) = ex + f $$

$$ \implies $$

$$ P = \xi([a,b], [e,f]) $$

$$ Q = \xi([a,b], [c,d]) $$

$$ R = \xi([c,d], [e,f]) $$

$$ p = \Xi(R, Q) $$

$$ r = \Xi(P, Q) $$

$$ i = \lambda(P, p) $$

$$ j = \lambda(R, r) $$

$$ C = \xi([i_a, i_b], [j_a, j_b]) $$

$$ \iff $$

$$ \xi([A,B], [C,D]) = \left( \frac{D-B}{A-C}, \frac{A \cdot (D-B)}{A-C} + B \right) $$

$$ \Xi([x_1,y_1], [x_2,y_2]) = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) $$

$$ \lambda([x_1,y_1], [x_2,y_2]) = \begin{cases} a = \frac{y_2-y_1}{x_2-x_1} \\ b = \frac{x_2y_1 - x_1y_2}{x_2-x_1} \end{cases} $$

Implementation

function triangleCentroid([a,b], [c,d], [e,f]) {
    const P = intersection([a,b], [e,f]);
    const Q = intersection([a,b], [c,d]);
    const R = intersection([c,d], [e,f]);

    const p = half(R,Q);
    const r = half(P,Q);

    const i = functionByPoints(P, p);
    const j = functionByPoints(R, r);

    const C = intersection(i, j);
    return C;

    function intersection([A,B], [C,D]) {
        return [
            (D-B)/(A-C),
            (A*(D-B))/(A-C)+B
        ];
    }
    function half([x1,y1], [x2,y2]) {
        return [
            (x1+x2)/2,
            (y1+y2)/2
        ];
    }
    function functionByPoints([x1,y1], [x2,y2]) {
        const A = (y2-y1)/(x2-x1);
        const B = (x2*y1 - x1*y2)/(x2-x1);
        return [A, B];
    }
}