quadratic bezier curve length in javascript

quadratic-bezier.js

/*
 * http://en.wikipedia.org/wiki/B%C3%A9zier_curve
 * http://www.malczak.linuxpl.com/blog/quadratic-bezier-curve-length/g
 */

function Point(x, y) {
    this.x = x;
    this.y = y;
}

function quadraticBezierLength(p0, p1, p2) {
    var a = new Point(
        p0.x - 2 * p1.x + p2.x,
        p0.y - 2 * p1.y + p2.y
    );
    var b = new Point(
        2 * p1.x - 2 * p0.x,
        2 * p1.y - 2 * p0.y
    );
    var A = 4 * (a.x * a.x + a.y * a.y);
    var B = 4 * (a.x * b.x + a.y * b.y);
    var C = b.x * b.x + b.y * b.y;

    var Sabc = 2 * sqrt(A+B+C);
    var A_2 = sqrt(A);
    var A_32 = 2 * A * A_2;
    var C_2 = 2 * sqrt(C);
    var BA = B / A_2;

    return (A_32 * Sabc + A_2 * B * (Sabc - C_2) + (4 * C * A - B * B) * log((2 * A_2 + BA + Sabc) / (BA + C_2))) / (4 * A_32);
}

How to adapt this in 3D space ?

@robindricks This is the worst type of feedback, but I believe your code has a bug in it—though I can't find it. The point seems to be behind where it should be, compared to other functions.

This works for me:

function quadraticBezierLength(
	x1: number,
	y1: number,
	x2: number,
	y2: number,
	x3: number,
	y3: number
) {
	let a: number, b: number, c: number, u: number

	let v1x = x2 * 2
	let v1y = y2 * 2
	let d = x1 - v1x + x3
	let d1 = y1 - v1y + y3
	let e = v1x - 2 * x1
	let e1 = v1y - 2 * y1
	let c1 = (a = 4 * (d * d + d1 * d1))
	c1 += b = 4 * (d * e + d1 * e1)
	c1 += c = e * e + e1 * e1
	c1 = 2 * Math.sqrt(c1)
	let a1 = 2 * a * (u = Math.sqrt(a))
	let u1 = b / u
	a = 4 * c * a - b * b
	c = 2 * Math.sqrt(c)
	return (
		(a1 * c1 + u * b * (c1 - c) + a * Math.log((2 * u + u1 + c1) / (u1 + c))) /
		(4 * a1)
	)
}

Thank you for this code, it works beautifully. I stumbled upon two edge cases that I had to handle:

1. A "flat" bezier curve,
   as in both control points are on the same point along a straight line. In this case, (BA + C_2), which is used as a divisor, is zero, resulting in a division by zero. To handle this:

    // […]
    let BA = B / A_2;

    let Y = (BA + C_2) > 0 ? Math.log((2 * A_2 + BA + Sabc) / (BA + C_2)) : 0;
    
    return (A_32 * Sabc + A_2 * B * (Sabc - C_2) + (4 * C * A - B * B) * y) / (4 * A_32);

2. {0,0}{0,0}{0,0}
   If all three points are passed as {0,0}, A_32 ends up as zero, resulting in a division by zero. To handle this:

    return A_32 === 0 ? 0 : (A_32 * Sabc + A_2 * B * (Sabc - C_2) + (4 * C * A - B * B) * Y) / (4 * A_32);

Note that JavaScript does not throw errors or warnings because it evaluates divisions by zero to "Infinity", which then results in NaN when used in further calculations. So what you end up with is the function returning "NaN" if the edge cases are not handled.

how do you calculate the length for Bezier Curve in three dimensions? Is this the right approach? @steveruizok @chknoflach

    function quadraticBezierLength(x1, y1, z1, x2, y2, z2, x3, y3, z3) {
          let a, b, c, u
          let v1x = x2 * 2
          let v1y = y2 * 2
          let v1z = z2 * 2
          let d = x1 - v1x + x3
          let d1 = y1 - v1y + y3
          let d2 = z1 - v1z + z3
          let e = v1x - 2 * x1
          let e1 = v1y - 2 * y1
          let e2 = v1z - 2 * z1
          let c1 = (a = 4 * (d * d + d1 * d1 + d2 * d2))
          c1 += b = 4 * (d * e + d1 * e1 + d2 * e2)
          c1 += c = e * e + e1 * e1 + e2 * e2
          c1 = 2 * Math.sqrt(c1)
          let a1 = 2 * a * (u = Math.sqrt(a))
          let u1 = b / u
          a = 4 * c * a - b * b
          c = 2 * Math.sqrt(c)
          return (
            (a1 * c1 + u * b * (c1 - c) + a * Math.log((2 * u + u1 + c1) / (u1 + c))) /
            (4 * a1)
          )
    }

@chknoflach Yes but in the first case you mentioned, if P0 is (10, 10), P1 is (10.00001, 10.00012), P2 is (20, 20), the divisor will be so small and the division result will be so big, makes a poor precision as a double number. It's annoying for me to process these points data from users drawing on a GUI.

As commented before, we should also check for actually linear commands:

/**
* p0: previous command's final point
* cp1: quadratic bézier control point
* p: final point
*/

function quadraticBezierLength(p0, cp1, p, t = 1) {
    if (t === 0) {
        return 0;
    }

    const interpolate = (p1, p2, t) => {
        let pt = { x: (p2.x - p1.x) * t + p1.x, y: (p2.y - p1.y) * t + p1.y };
        return pt;
    }
    const getLineLength = (p1, p2) => {
        return Math.sqrt(
            (p2.x - p1.x) * (p2.x - p1.x) + (p2.y - p1.y) * (p2.y - p1.y)
        );
    }

    // is flat/linear 
    let l1 = getLineLength(p0, cp1) + getLineLength(cp1, p);
    let l2 = getLineLength(p0, p);
    if (l1 === l2) {
        let m1 = interpolate(p0, cp1, t);
        let m2 = interpolate(cp1, p, t);
        p = interpolate(m1, m2, t);
        let lengthL;
        lengthL = Math.sqrt((p.x - p0.x) * (p.x - p0.x) + (p.y - p0.y) * (p.y - p0.y));
        return lengthL;
    }

    let a, b, c, d, e, e1, d1, v1x, v1y;

    v1x = cp1.x * 2;
    v1y = cp1.y * 2;
    d = p0.x - v1x + p.x;
    d1 = p0.y - v1y + p.y;
    e = v1x - 2 * p0.x;
    e1 = v1y - 2 * p0.y;
    a = 4 * (d * d + d1 * d1);
    b = 4 * (d * e + d1 * e1);
    c = e * e + e1 * e1;

    const bt = b / (2 * a),
        ct = c / a,
        ut = t + bt,
        k = ct - bt ** 2;

    return (
        (Math.sqrt(a) / 2) *
        (ut * Math.sqrt(ut ** 2 + k) -
            bt * Math.sqrt(bt ** 2 + k) +
            k *
            Math.log((ut + Math.sqrt(ut ** 2 + k)) / (bt + Math.sqrt(bt ** 2 + k))))
    );
}