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Use better epsilon values in Numerical.solveQuadratic() and solveCubic()

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To finally satisfy both #541 and #708.

With this change in place, https://github.com/paperjs/boolean-test is also finally back to run with 0 errors. Woop!
This commit is contained in:
Jürg Lehni 2015-08-18 23:47:28 +02:00
parent 098ddda3bc
commit e476672748
2 changed files with 28 additions and 35 deletions
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8
src/path/Curve.js
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@ -617,13 +617,7 @@ statics: {
p2 = v[coord + 6], p2 = v[coord + 6],
c = 3 * (c1 - p1), c = 3 * (c1 - p1),
b = 3 * (c2 - c1) - c, b = 3 * (c2 - c1) - c,
a = p2 - p1 - c - b, a = p2 - p1 - c - b;
isZero = Numerical.isZero;
// If both a and b are near zero, we should treat the curve as a line in
// order to find the right solutions in some edge-cases in
// Curve.getParameterOf()
if (isZero(a) && isZero(b))
a = b = 0;
return Numerical.solveCubic(a, b, c, p1 - val, roots, min, max); return Numerical.solveCubic(a, b, c, p1 - val, roots, min, max);
}, },

55
src/util/Numerical.js
View file

@ -161,10 +161,12 @@ var Numerical = new function() {
* http://www.cs.berkeley.edu/~wkahan/Math128/Cubic.pdf * http://www.cs.berkeley.edu/~wkahan/Math128/Cubic.pdf
* Blinn J. - "How to solve a Quadratic Equation" * Blinn J. - "How to solve a Quadratic Equation"
* *
* @param {Number} a The quadratic term * @param {Number} a the quadratic term
* @param {Number} b The linear term * @param {Number} b the linear term
* @param {Number} c The constant term * @param {Number} c the constant term
* @param {Number[]} roots The array to store the roots in * @param {Number[]} roots the array to store the roots in
* @param {Number} [min] the lower bound of the allowed roots
* @param {Number} [max] the upper bound of the allowed roots
* @return {Number} The number of real roots found, or -1 if there are * @return {Number} The number of real roots found, or -1 if there are
* infinite solutions * infinite solutions
* *
@ -177,21 +179,15 @@ var Numerical = new function() {
D; D;
b /= 2; b /= 2;
D = b * b - a * c; // Discriminant D = b * b - a * c; // Discriminant
/* // If the discriminant is very small, we can try to pre-condition
* If the discriminant is very small, we can try to pre-condition // the coefficients, so that we may get better accuracy
* the coefficients, so that we may get better accuracy
*/
if (D !== 0 && abs(D) < MACHINE_EPSILON) { if (D !== 0 && abs(D) < MACHINE_EPSILON) {
// If the geometric mean of the coefficients is small enough // If the geometric mean of the coefficients is small enough
var pow = Math.pow, var gmC = pow(abs(a * b * c), 1 / 3);
gmC = pow(abs(a*b*c), 1/3);
if (gmC < 1e-8) { if (gmC < 1e-8) {
/* // We multiply with a factor to normalize the coefficients.
* we multiply with a factor to normalize the // The factor is just the nearest exponent of 10, big enough
* coefficients. The factor is just the nearest exponent // to raise all the coefficients to nearly [-1, +1] range.
* of 10, big enough to raise all the coefficients to
* nearly [-1, +1] range.
*/
var mult = pow(10, abs( var mult = pow(10, abs(
Math.floor(Math.log(gmC) * Math.LOG10E))); Math.floor(Math.log(gmC) * Math.LOG10E)));
if (!isFinite(mult)) if (!isFinite(mult))
@ -203,10 +199,10 @@ var Numerical = new function() {
D = b * b - a * c; D = b * b - a * c;
} }
} }
if (abs(a) < MACHINE_EPSILON) { if (abs(a) < EPSILON) {
// This could just be a linear equation // This could just be a linear equation
if (abs(B) < MACHINE_EPSILON) if (abs(B) < EPSILON)
return abs(c) < MACHINE_EPSILON ? -1 : 0; return abs(c) < EPSILON ? -1 : 0;
x1 = -c / B; x1 = -c / B;
} else { } else {
// No real roots if D < 0 // No real roots if D < 0
@ -251,26 +247,29 @@ var Numerical = new function() {
* W. Kahan's paper contains inferences on accuracy of cubic * W. Kahan's paper contains inferences on accuracy of cubic
* zero-finding methods. Also testing methods for robustness. * zero-finding methods. Also testing methods for robustness.
* *
* @param {Number} a The cubic term (x³ term). * @param {Number} a the cubic term (x³ term)
* @param {Number} b The quadratic term (x² term). * @param {Number} b the quadratic term (x² term)
* @param {Number} c The linear term (x term). * @param {Number} c the linear term (x term)
* @param {Number} d The constant term * @param {Number} d the constant term
* @param {Number[]} roots The array to store the roots in * @param {Number[]} roots the array to store the roots in
* @return {Number} The number of real roots found, or -1 if there are * @param {Number} [min] the lower bound of the allowed roots
* @param {Number} [max] the upper bound of the allowed roots
* @return {Number} the number of real roots found, or -1 if there are
* infinite solutions * infinite solutions
* *
* @author Harikrishnan Gopalakrishnan * @author Harikrishnan Gopalakrishnan
*/ */
solveCubic: function(a, b, c, d, roots, min, max) { solveCubic: function(a, b, c, d, roots, min, max) {
var x, b1, c2, count = 0; var count = 0,
x, b1, c2;
// If a or d is zero, we only need to solve a quadratic, so we set // If a or d is zero, we only need to solve a quadratic, so we set
// the coefficients appropriately. // the coefficients appropriately.
if (a === 0) { if (abs(a) < EPSILON) {
a = b; a = b;
b1 = c; b1 = c;
c2 = d; c2 = d;
x = Infinity; x = Infinity;
} else if (d === 0) { } else if (abs(d) < EPSILON) {
b1 = b; b1 = b;
c2 = c; c2 = c;
x = 0; x = 0;