├── .gitignore
├── .travis.yml
├── Readme.md
├── dist
    └── bezier-intersect.js
├── index.js
├── package-lock.json
├── package.json
├── rollup.config.js
├── src
    ├── cubic.js
    ├── polynomial.js
    └── quadratic.js
└── test
    └── test.js


/.gitignore:
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1 | node_modules
2 | 


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/.travis.yml:
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 1 | language: node_js
 2 | node_js:
 3 |   - "8"
 4 | 
 5 | env:
 6 |   global:
 7 |     - BUILD_TIMEOUT=10000
 8 | 
 9 | install:
10 |   - npm install
11 | 


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/Readme.md:
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 1 | # bezier-intersect [![npm version](https://badge.fury.io/js/bezier-intersect.svg)](https://badge.fury.io/js/bezier-intersect) [![build](https://travis-ci.org/w8r/bezier-intersect.svg?branch=master)](https://travis-ci.org/w8r/bezier-intersect)
 2 | 
 3 | Set of functions to find intersections between lines and rectangles and Bezier curves of order 2 and 3. Based on [thelonious/js-intersections](https://github.com/thelonious/js-intersections/), but with the abstractions removed and some performance tweaking.
 4 | 
 5 | ## Install
 6 | 
 7 | ```
 8 | npm i -S bezier-intersect
 9 | ```
10 | 
11 | ```js
12 | import {
13 |   quadBezierLine,
14 |   cubicBezierLine,
15 |   quadBezierAABB,
16 |   cubicBezierAABB
17 | } from 'bezier-intersect';
18 | ```
19 | 
20 | ```html
21 | <script src="https://unpkg.com/bezier-intersect"></script>
22 | <script>
23 | var res = [];
24 | bezierIntersect.quadBezierLine(..., res);
25 | console.log(res);
26 | </script>
27 | ```
28 | 
29 | ## API
30 | 
31 | ### `quadBezierLine(ax, ay, cx, cy, bx, by, l1x, l1y, l2x, l2y, [result:Array<number>]):number`
32 | 
33 | Calculates the intersection points between the quadratic Bezier curve and line segment. If `result` is passed, returns the exact number of intersections, and stores them in `result` as `[x, y, x, y]`. If not - stops at the first intersection and returns `1` or `0` if there are no intersections.
34 | 
35 | ### `quadBezierAABB(ax, ay, cx, cy, bx, by, minx, miny, maxx, maxy, [result:Array<number>]):number`
36 | 
37 | Calculates the intersection points between the quadratic Bezier curve and axis-aligned box. If `result` is passed, returns the exact number of intersections, and stores them in `result` as `[x, y, x, y]`. If not - stops at the first intersection and returns `1` or `0` if there are no intersections.
38 | 
39 | ### `cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, l1x, l1y, l2x, l2y, [result:Array<number>]):number`
40 | 
41 | Calculates the intersection points between the cubic Bezier curve and line segment. If `result` is passed, returns the exact number of intersections, and stores them in `result` as `[x, y, x, y]`. If not - stops at the first intersection and returns `1` or `0` if there are no intersections.
42 | 
43 | ### `cubicBezierAABB(ax, ay, c1x, c1y, c2x, c2y, bx, by, minx, miny, maxx, maxy, [result:Array<number>]):number`
44 | 
45 | Calculates the intersection points between the cubic Bezier curve and axis-aligned box. If `result` is passed, returns the exact number of intersections, and stores them in `result` as `[x, y, x, y]`. If not - stops at the first intersection and returns `1` or `0` if there are no intersections.
46 | 
47 | ## TODO
48 | 
49 | - [ ] More tests
50 | - [ ] Bezier/Polygon
51 | - [ ] Bezier/Ellipse/Circle
52 | 
53 | ## License
54 | 
55 | MIT
56 | 


--------------------------------------------------------------------------------
/dist/bezier-intersect.js:
--------------------------------------------------------------------------------
  1 | (function (global, factory) {
  2 | 	typeof exports === 'object' && typeof module !== 'undefined' ? factory(exports) :
  3 | 	typeof define === 'function' && define.amd ? define(['exports'], factory) :
  4 | 	(factory((global.bezierIntersect = {})));
  5 | }(this, (function (exports) { 'use strict';
  6 | 
  7 | var POLYNOMIAL_TOLERANCE = 1e-6;
  8 | var TOLERANCE            = 1e-12;
  9 | 
 10 | function getPolynomialRoots() {
 11 |   var C  = arguments;
 12 |   var degree  = C.length - 1;
 13 |   var n       = degree;
 14 |   var results = [];
 15 |   for (var i = 0; i <= degree; i++) {
 16 |     if (Math.abs(C[i]) <= TOLERANCE) { degree--; } else { break; }
 17 |   }
 18 | 
 19 |   switch (degree) {
 20 |     case 1: getLinearRoots(C[n], C[n - 1], results); break;
 21 |     case 2: getQuadraticRoots(C[n], C[n - 1], C[n - 2], results); break;
 22 |     case 3: getCubicRoots(C[n], C[n - 1], C[n - 2], C[n - 3], results); break;
 23 |     default: break;
 24 |   }
 25 | 
 26 |   return results;
 27 | }
 28 | 
 29 | function getLinearRoots(C0, C1, results) {
 30 |   if ( results === void 0 ) results = [];
 31 | 
 32 |   if (C1 !== 0) { results.push(-C0 / C1); }
 33 |   return results;
 34 | }
 35 | 
 36 | function getQuadraticRoots(C0, C1, C2, results) {
 37 |   if ( results === void 0 ) results = [];
 38 | 
 39 |   var a = C2;
 40 |   var b = C1 / a;
 41 |   var c = C0 / a;
 42 |   var d = b * b - 4 * c;
 43 | 
 44 |   if (d > 0) {
 45 |     var e = Math.sqrt(d);
 46 | 
 47 |     results.push(0.5 * (-b + e));
 48 |     results.push(0.5 * (-b - e));
 49 |   } else if (d === 0) {
 50 |     results.push( 0.5 * -b);
 51 |   }
 52 | 
 53 |   return results;
 54 | }
 55 | 
 56 | 
 57 | function getCubicRoots(C0, C1, C2, C3, results) {
 58 |   if ( results === void 0 ) results = [];
 59 | 
 60 |   var c3 = C3;
 61 |   var c2 = C2 / c3;
 62 |   var c1 = C1 / c3;
 63 |   var c0 = C0 / c3;
 64 | 
 65 |   var a       = (3 * c1 - c2 * c2) / 3;
 66 |   var b       = (2 * c2 * c2 * c2 - 9 * c1 * c2 + 27 * c0) / 27;
 67 |   var offset  = c2 / 3;
 68 |   var discrim = b * b / 4 + a * a * a / 27;
 69 |   var halfB   = b / 2;
 70 |   var tmp, root;
 71 | 
 72 |   if (Math.abs(discrim) <= POLYNOMIAL_TOLERANCE) { discrim = 0; }
 73 | 
 74 |   if (discrim > 0) {
 75 |     var e = Math.sqrt(discrim);
 76 | 
 77 |     tmp = -halfB + e;
 78 |     if ( tmp >= 0 ) { root = Math.pow(  tmp, 1/3); }
 79 |     else            { root = -Math.pow(-tmp, 1/3); }
 80 | 
 81 |     tmp = -halfB - e;
 82 |     if ( tmp >= 0 ) { root += Math.pow( tmp, 1/3); }
 83 |     else            { root -= Math.pow(-tmp, 1/3); }
 84 | 
 85 |     results.push(root - offset);
 86 |   } else if (discrim < 0) {
 87 |     var distance = Math.sqrt(-a/3);
 88 |     var angle    = Math.atan2(Math.sqrt(-discrim), -halfB) / 3;
 89 |     var cos      = Math.cos(angle);
 90 |     var sin      = Math.sin(angle);
 91 |     var sqrt3    = Math.sqrt(3);
 92 | 
 93 |     results.push( 2 * distance * cos - offset);
 94 |     results.push(-distance * (cos + sqrt3 * sin) - offset);
 95 |     results.push(-distance * (cos - sqrt3 * sin) - offset);
 96 |   } else {
 97 |     if (halfB >= 0) { tmp = -Math.pow(halfB, 1/3); }
 98 |     else            { tmp =  Math.pow(-halfB, 1/3); }
 99 | 
100 |     results.push(2 * tmp - offset);
101 |     // really should return next root twice, but we return only one
102 |     results.push(-tmp - offset);
103 |   }
104 | 
105 |   return results;
106 | }
107 | 
108 | function quadBezierLine(
109 |   p1x, p1y, p2x, p2y, p3x, p3y,
110 |   a1x, a1y, a2x, a2y, result) {
111 |   var ax, ay, bx, by;                // temporary variables
112 |   var c2x, c2y, c1x, c1y, c0x, c0y;  // coefficients of quadratic
113 |   var cl;               // c coefficient for normal form of line
114 |   var nx, ny;           // normal for normal form of line
115 |   // used to determine if point is on line segment
116 |   var minx = Math.min(a1x, a2x),
117 |       miny = Math.min(a1y, a2y),
118 |       maxx = Math.max(a1x, a2x),
119 |       maxy = Math.max(a1y, a2y);
120 | 
121 |   ax = p2x * -2; ay = p2y * -2;
122 |   c2x = p1x + ax + p3x;
123 |   c2y = p1y + ay + p3y;
124 | 
125 |   ax = p1x * -2; ay = p1y * -2;
126 |   bx = p2x * 2;  by = p2y * 2;
127 |   c1x = ax + bx;
128 |   c1y = ay + by;
129 | 
130 |   c0x = p1x; c0y = p1y; // vec
131 | 
132 |   // Convert line to normal form: ax + by + c = 0
133 |   // Find normal to line: negative inverse of original line's slope
134 |   nx = a1y - a2y; ny = a2x - a1x;
135 | 
136 |   // Determine new c coefficient
137 |   cl = a1x * a2y - a2x * a1y;
138 | 
139 |   // Transform cubic coefficients to line's coordinate system
140 |   // and find roots of cubic
141 |   var roots = getPolynomialRoots(
142 |     // dot products => x * x + y * y
143 |     nx * c2x + ny * c2y,
144 |     nx * c1x + ny * c1y,
145 |     nx * c0x + ny * c0y + cl
146 |   );
147 | 
148 |   // Any roots in closed interval [0,1] are intersections on Bezier, but
149 |   // might not be on the line segment.
150 |   // Find intersections and calculate point coordinates
151 |   for (var i = 0; i < roots.length; i++) {
152 |     var t = roots[i];
153 |     if ( 0 <= t && t <= 1 ) { // We're within the Bezier curve
154 |       // Find point on Bezier
155 |       // lerp: x1 + (x2 - x1) * t
156 |       var p4x = p1x + (p2x - p1x) * t;
157 |       var p4y = p1y + (p2y - p1y) * t;
158 | 
159 |       var p5x = p2x + (p3x - p2x) * t;
160 |       var p5y = p2y + (p3y - p2y) * t;
161 | 
162 |       // candidate
163 |       var p6x = p4x + (p5x - p4x) * t;
164 |       var p6y = p4y + (p5y - p4y) * t;
165 | 
166 |       // See if point is on line segment
167 |       // Had to make special cases for vertical and horizontal lines due
168 |       // to slight errors in calculation of p6
169 |       if (a1x === a2x) {
170 |         if (miny <= p6y && p6y <= maxy) {
171 |           if (result) { result.push(p6x, p6y); }
172 |           else        { return 1; }
173 |         }
174 |       } else if (a1y === a2y) {
175 |         if (minx <= p6x && p6x <= maxx) {
176 |           if (result) { result.push(p6x, p6y); }
177 |           else        { return 1; }
178 |         }
179 | 
180 |       // gte: (x1 >= x2 && y1 >= y2)
181 |       // lte: (x1 <= x2 && y1 <= y2)
182 |       } else if (p6x >= minx && p6y >= miny && p6x <= maxx && p6y <= maxy) {
183 |         if (result) { result.push(p6x, p6y); }
184 |         else        { return 1; }
185 |       }
186 |     }
187 |   }
188 |   return result ? result.length / 2 : 0;
189 | }
190 | 
191 | 
192 | function quadBezierAABB(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmax, ymax, result) {
193 |   if (result) { // all intersections
194 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmax, ymin, result);
195 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmax, ymin, xmax, ymax, result);
196 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymax, xmax, ymax, result);
197 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmin, ymax, result);
198 |     return result.length / 2;
199 |   } else {      // any intersections
200 |     // trivial cases
201 |     if (xmin <= ax && xmax >= ax && ymin <= ay && ymax >= ay) { return 1; }
202 |     if (xmin <= bx && xmax >= bx && ymin <= by && ymax >= by) { return 1; }
203 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmax, ymin)) { return 1; }
204 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmax, ymin, xmax, ymax)) { return 1; }
205 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymax, xmax, ymax)) { return 1; }
206 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmin, ymax)) { return 1; }
207 |     return 0;
208 |   }
209 | }
210 | 
211 | function cubicBezierLine(
212 |   p1x, p1y, p2x, p2y, p3x, p3y, p4x, p4y,
213 |   a1x, a1y, a2x, a2y, result) {
214 |   var ax, ay, bx, by, cx, cy, dx, dy;         // temporary variables
215 |   var c3x, c3y, c2x, c2y, c1x, c1y, c0x, c0y; // coefficients of cubic
216 |   var cl;         // c coefficient for normal form of line
217 |   var nx, ny;     // normal for normal form of line
218 | 
219 |   // used to determine if point is on line segment
220 |   var minx = Math.min(a1x, a2x),
221 |       miny = Math.min(a1y, a2y),
222 |       maxx = Math.max(a1x, a2x),
223 |       maxy = Math.max(a1y, a2y);
224 | 
225 |   // Start with Bezier using Bernstein polynomials for weighting functions:
226 |   //     (1-t^3)P1 + 3t(1-t)^2P2 + 3t^2(1-t)P3 + t^3P4
227 |   //
228 |   // Expand and collect terms to form linear combinations of original Bezier
229 |   // controls.  This ends up with a vector cubic in t:
230 |   //     (-P1+3P2-3P3+P4)t^3 + (3P1-6P2+3P3)t^2 + (-3P1+3P2)t + P1
231 |   //             /\                  /\                /\       /\
232 |   //             ||                  ||                ||       ||
233 |   //             c3                  c2                c1       c0
234 | 
235 |   // Calculate the coefficients
236 |   ax = p1x * -1; ay = p1y * -1;
237 |   bx = p2x * 3;  by = p2y * 3;
238 |   cx = p3x * -3; cy = p3y * -3;
239 |   dx = ax + bx + cx + p4x;
240 |   dy = ay + by + cy + p4y;
241 |   c3x = dx; c3y = dy; // vec
242 | 
243 |   ax = p1x * 3;  ay = p1y * 3;
244 |   bx = p2x * -6; by = p2y * -6;
245 |   cx = p3x * 3;  cy = p3y * 3;
246 |   dx = ax + bx + cx;
247 |   dy = ay + by + cy;
248 |   c2x = dx; c2y = dy; // vec
249 | 
250 |   ax = p1x * -3; ay = p1y * -3;
251 |   bx = p2x * 3;  by = p2y * 3;
252 |   cx = ax + bx;
253 |   cy = ay + by;
254 |   c1x = cx;
255 |   c1y = cy; // vec
256 | 
257 |   c0x = p1x;
258 |   c0y = p1y;
259 | 
260 |   // Convert line to normal form: ax + by + c = 0
261 |   // Find normal to line: negative inverse of original line's slope
262 |   nx = a1y - a2y;
263 |   ny = a2x - a1x;
264 | 
265 |   // Determine new c coefficient
266 |   cl = a1x * a2y - a2x * a1y;
267 | 
268 |   // ?Rotate each cubic coefficient using line for new coordinate system?
269 |   // Find roots of rotated cubic
270 |   var roots = getPolynomialRoots(
271 |     // dot products => x * x + y * y
272 |     nx * c3x + ny * c3y,
273 |     nx * c2x + ny * c2y,
274 |     nx * c1x + ny * c1y,
275 |     nx * c0x + ny * c0y + cl
276 |   );
277 | 
278 |   // Any roots in closed interval [0,1] are intersections on Bezier, but
279 |   // might not be on the line segment.
280 |   // Find intersections and calculate point coordinates
281 |   for (var i = 0; i < roots.length; i++) {
282 |     var t = roots[i];
283 | 
284 |     if (0 <= t && t <= 1) { // We're within the Bezier curve
285 |       // Find point on Bezier
286 |       // lerp: x1 + (x2 - x1) * t
287 |       var p5x = p1x + (p2x - p1x) * t;
288 |       var p5y = p1y + (p2y - p1y) * t; // lerp(p1, p2, t);
289 | 
290 |       var p6x = p2x + (p3x - p2x) * t;
291 |       var p6y = p2y + (p3y - p2y) * t;
292 | 
293 |       var p7x = p3x + (p4x - p3x) * t;
294 |       var p7y = p3y + (p4y - p3y) * t;
295 | 
296 |       var p8x = p5x + (p6x - p5x) * t;
297 |       var p8y = p5y + (p6y - p5y) * t;
298 | 
299 |       var p9x = p6x + (p7x - p6x) * t;
300 |       var p9y = p6y + (p7y - p6y) * t;
301 | 
302 |       // candidate
303 |       var p10x = p8x + (p9x - p8x) * t;
304 |       var p10y = p8y + (p9y - p8y) * t;
305 | 
306 |       // See if point is on line segment
307 |       if (a1x === a2x) {                       // vertical
308 |         if (miny <= p10y && p10y <= maxy) {
309 |           if (result) { result.push(p10x, p10y); }
310 |           else        { return 1; }
311 |         }
312 |       } else if (a1y === a2y) {               // horizontal
313 |         if (minx <= p10x && p10x <= maxx) {
314 |           if (result) { result.push(p10x, p10y); }
315 |           else        { return 1; }
316 |         }
317 |       } else if (p10x >= minx && p10y >= miny && p10x <= maxx && p10y <= maxy) {
318 |         if (result) { result.push(p10x, p10y); }
319 |         else        { return 1; }
320 |       }
321 |     }
322 |   }
323 |   return result ? result.length / 2 : 0;
324 | }
325 | 
326 | 
327 | function cubicBezierAABB(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmax, ymax, result) {
328 |   if (result) { // all intersections
329 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmax, ymin, result);
330 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmax, ymin, xmax, ymax, result);
331 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymax, xmax, ymax, result);
332 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmin, ymax, result);
333 |     return result.length / 2;
334 |   } else {     // any intersections
335 |     // trivial cases
336 |     if (xmin <= ax && xmax >= ax && ymin <= ay && ymax >= ay) { return 1; }
337 |     if (xmin <= bx && xmax >= bx && ymin <= by && ymax >= by) { return 1; }
338 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmax, ymin)) { return 1; }
339 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmax, ymin, xmax, ymax)) { return 1; }
340 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymax, xmax, ymax)) { return 1; }
341 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmin, ymax)) { return 1; }
342 |     return 0;
343 |   }
344 | }
345 | 
346 | exports.quadBezierLine = quadBezierLine;
347 | exports.quadBezierAABB = quadBezierAABB;
348 | exports.cubicBezierLine = cubicBezierLine;
349 | exports.cubicBezierAABB = cubicBezierAABB;
350 | 
351 | Object.defineProperty(exports, '__esModule', { value: true });
352 | 
353 | })));
354 | 


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/index.js:
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1 | export * from './src/quadratic';
2 | export * from './src/cubic';
3 | 


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/package-lock.json:
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  1 | {
  2 |   "name": "bezier-intersect",
  3 |   "version": "0.0.3",
  4 |   "lockfileVersion": 1,
  5 |   "requires": true,
  6 |   "dependencies": {
  7 |     "acorn": {
  8 |       "version": "3.3.0",
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 10 |       "integrity": "sha1-ReN/s56No/JbruP/U2niu18iAXo=",
 11 |       "dev": true
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 17 |       "dev": true,
 18 |       "requires": {
 19 |         "acorn": "3.3.0"
 20 |       }
 21 |     },
 22 |     "acorn-object-spread": {
 23 |       "version": "1.0.0",
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 26 |       "dev": true,
 27 |       "requires": {
 28 |         "acorn": "3.3.0"
 29 |       }
 30 |     },
 31 |     "ansi-regex": {
 32 |       "version": "2.1.1",
 33 |       "resolved": "https://registry.npmjs.org/ansi-regex/-/ansi-regex-2.1.1.tgz",
 34 |       "integrity": "sha1-w7M6te42DYbg5ijwRorn7yfWVN8=",
 35 |       "dev": true
 36 |     },
 37 |     "ansi-styles": {
 38 |       "version": "2.2.1",
 39 |       "resolved": "https://registry.npmjs.org/ansi-styles/-/ansi-styles-2.2.1.tgz",
 40 |       "integrity": "sha1-tDLdM1i2NM914eRmQ2gkBTPB3b4=",
 41 |       "dev": true
 42 |     },
 43 |     "arr-diff": {
 44 |       "version": "2.0.0",
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 46 |       "integrity": "sha1-jzuCf5Vai9ZpaX5KQlasPOrjVs8=",
 47 |       "dev": true,
 48 |       "requires": {
 49 |         "arr-flatten": "1.1.0"
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513 |       "dev": true,
514 |       "requires": {
515 |         "browser-stdout": "1.3.0",
516 |         "commander": "2.11.0",
517 |         "debug": "3.1.0",
518 |         "diff": "3.3.1",
519 |         "escape-string-regexp": "1.0.5",
520 |         "glob": "7.1.2",
521 |         "growl": "1.10.3",
522 |         "he": "1.1.1",
523 |         "mkdirp": "0.5.1",
524 |         "supports-color": "4.4.0"
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531 |           "dev": true,
532 |           "requires": {
533 |             "has-flag": "2.0.0"
534 |           }
535 |         }
536 |       }
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541 |       "integrity": "sha1-VgiurfwAvmwpAd9fmGF4jeDVl8g=",
542 |       "dev": true
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547 |       "integrity": "sha1-GrKLVW4Zg2Oowab35vogE3/mrtk=",
548 |       "dev": true,
549 |       "requires": {
550 |         "remove-trailing-separator": "1.1.0"
551 |       }
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556 |       "integrity": "sha1-Gpx0SCnznbuFjHbKNXmuKlTr0fo=",
557 |       "dev": true,
558 |       "requires": {
559 |         "for-own": "0.1.5",
560 |         "is-extendable": "0.1.1"
561 |       }
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566 |       "integrity": "sha1-WDsap3WWHUsROsF9nFC6753Xa9E=",
567 |       "dev": true,
568 |       "requires": {
569 |         "wrappy": "1.0.2"
570 |       }
571 |     },
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573 |       "version": "1.0.2",
574 |       "resolved": "https://registry.npmjs.org/os-homedir/-/os-homedir-1.0.2.tgz",
575 |       "integrity": "sha1-/7xJiDNuDoM94MFox+8VISGqf7M=",
576 |       "dev": true
577 |     },
578 |     "parse-glob": {
579 |       "version": "3.0.4",
580 |       "resolved": "https://registry.npmjs.org/parse-glob/-/parse-glob-3.0.4.tgz",
581 |       "integrity": "sha1-ssN2z7EfNVE7rdFz7wu246OIORw=",
582 |       "dev": true,
583 |       "requires": {
584 |         "glob-base": "0.3.0",
585 |         "is-dotfile": "1.0.3",
586 |         "is-extglob": "1.0.0",
587 |         "is-glob": "2.0.1"
588 |       }
589 |     },
590 |     "path-is-absolute": {
591 |       "version": "1.0.1",
592 |       "resolved": "https://registry.npmjs.org/path-is-absolute/-/path-is-absolute-1.0.1.tgz",
593 |       "integrity": "sha1-F0uSaHNVNP+8es5r9TpanhtcX18=",
594 |       "dev": true
595 |     },
596 |     "path-parse": {
597 |       "version": "1.0.5",
598 |       "resolved": "https://registry.npmjs.org/path-parse/-/path-parse-1.0.5.tgz",
599 |       "integrity": "sha1-PBrfhx6pzWyUMbbqK9dKD/BVxME=",
600 |       "dev": true
601 |     },
602 |     "pathval": {
603 |       "version": "1.1.0",
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605 |       "integrity": "sha1-uULm1L3mUwBe9rcTYd74cn0GReA=",
606 |       "dev": true
607 |     },
608 |     "preserve": {
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611 |       "integrity": "sha1-gV7R9uvGWSb4ZbMQwHE7yzMVzks=",
612 |       "dev": true
613 |     },
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618 |       "dev": true,
619 |       "requires": {
620 |         "is-number": "3.0.0",
621 |         "kind-of": "4.0.0"
622 |       },
623 |       "dependencies": {
624 |         "is-number": {
625 |           "version": "3.0.0",
626 |           "resolved": "https://registry.npmjs.org/is-number/-/is-number-3.0.0.tgz",
627 |           "integrity": "sha1-JP1iAaR4LPUFYcgQJ2r8fRLXEZU=",
628 |           "dev": true,
629 |           "requires": {
630 |             "kind-of": "3.2.2"
631 |           },
632 |           "dependencies": {
633 |             "kind-of": {
634 |               "version": "3.2.2",
635 |               "resolved": "https://registry.npmjs.org/kind-of/-/kind-of-3.2.2.tgz",
636 |               "integrity": "sha1-MeohpzS6ubuw8yRm2JOupR5KPGQ=",
637 |               "dev": true,
638 |               "requires": {
639 |                 "is-buffer": "1.1.6"
640 |               }
641 |             }
642 |           }
643 |         },
644 |         "kind-of": {
645 |           "version": "4.0.0",
646 |           "resolved": "https://registry.npmjs.org/kind-of/-/kind-of-4.0.0.tgz",
647 |           "integrity": "sha1-IIE989cSkosgc3hpGkUGb65y3Vc=",
648 |           "dev": true,
649 |           "requires": {
650 |             "is-buffer": "1.1.6"
651 |           }
652 |         }
653 |       }
654 |     },
655 |     "regex-cache": {
656 |       "version": "0.4.4",
657 |       "resolved": "https://registry.npmjs.org/regex-cache/-/regex-cache-0.4.4.tgz",
658 |       "integrity": "sha512-nVIZwtCjkC9YgvWkpM55B5rBhBYRZhAaJbgcFYXXsHnbZ9UZI9nnVWYZpBlCqv9ho2eZryPnWrZGsOdPwVWXWQ==",
659 |       "dev": true,
660 |       "requires": {
661 |         "is-equal-shallow": "0.1.3"
662 |       }
663 |     },
664 |     "remove-trailing-separator": {
665 |       "version": "1.1.0",
666 |       "resolved": "https://registry.npmjs.org/remove-trailing-separator/-/remove-trailing-separator-1.1.0.tgz",
667 |       "integrity": "sha1-wkvOKig62tW8P1jg1IJJuSN52O8=",
668 |       "dev": true
669 |     },
670 |     "repeat-element": {
671 |       "version": "1.1.2",
672 |       "resolved": "https://registry.npmjs.org/repeat-element/-/repeat-element-1.1.2.tgz",
673 |       "integrity": "sha1-7wiaF40Ug7quTZPrmLT55OEdmQo=",
674 |       "dev": true
675 |     },
676 |     "repeat-string": {
677 |       "version": "1.6.1",
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679 |       "integrity": "sha1-jcrkcOHIirwtYA//Sndihtp15jc=",
680 |       "dev": true
681 |     },
682 |     "resolve": {
683 |       "version": "1.5.0",
684 |       "resolved": "https://registry.npmjs.org/resolve/-/resolve-1.5.0.tgz",
685 |       "integrity": "sha512-hgoSGrc3pjzAPHNBg+KnFcK2HwlHTs/YrAGUr6qgTVUZmXv1UEXXl0bZNBKMA9fud6lRYFdPGz0xXxycPzmmiw==",
686 |       "dev": true,
687 |       "requires": {
688 |         "path-parse": "1.0.5"
689 |       }
690 |     },
691 |     "rollup": {
692 |       "version": "0.46.3",
693 |       "resolved": "https://registry.npmjs.org/rollup/-/rollup-0.46.3.tgz",
694 |       "integrity": "sha512-ygbpqQczZTLr7ca2tVgi/AUPIRiI2sSp+3xKQRHgovM489GJPaH5mYUeRDb43Sq1td3zq6GUGovLi9vzmbQmVw==",
695 |       "dev": true
696 |     },
697 |     "rollup-plugin-buble": {
698 |       "version": "0.15.0",
699 |       "resolved": "https://registry.npmjs.org/rollup-plugin-buble/-/rollup-plugin-buble-0.15.0.tgz",
700 |       "integrity": "sha1-g8PonH/SJmx5GPQbo5gDE1Gcf9A=",
701 |       "dev": true,
702 |       "requires": {
703 |         "buble": "0.15.2",
704 |         "rollup-pluginutils": "1.5.2"
705 |       }
706 |     },
707 |     "rollup-plugin-commonjs": {
708 |       "version": "8.2.6",
709 |       "resolved": "https://registry.npmjs.org/rollup-plugin-commonjs/-/rollup-plugin-commonjs-8.2.6.tgz",
710 |       "integrity": "sha512-qK0+uhktmnAgZkHkqFuajNmPw93fjrO7+CysDaxWE5jrUR9XSlSvuao5ZJP+XizxA8weakhgYYBtbVz9SGBpjA==",
711 |       "dev": true,
712 |       "requires": {
713 |         "acorn": "5.3.0",
714 |         "estree-walker": "0.5.1",
715 |         "magic-string": "0.22.4",
716 |         "resolve": "1.5.0",
717 |         "rollup-pluginutils": "2.0.1"
718 |       },
719 |       "dependencies": {
720 |         "acorn": {
721 |           "version": "5.3.0",
722 |           "resolved": "https://registry.npmjs.org/acorn/-/acorn-5.3.0.tgz",
723 |           "integrity": "sha512-Yej+zOJ1Dm/IMZzzj78OntP/r3zHEaKcyNoU2lAaxPtrseM6rF0xwqoz5Q5ysAiED9hTjI2hgtvLXitlCN1/Ug==",
724 |           "dev": true
725 |         },
726 |         "estree-walker": {
727 |           "version": "0.5.1",
728 |           "resolved": "https://registry.npmjs.org/estree-walker/-/estree-walker-0.5.1.tgz",
729 |           "integrity": "sha512-7HgCgz1axW7w5aOvgOQkoR1RMBkllygJrssU3BvymKQ95lxXYv6Pon17fBRDm9qhkvXZGijOULoSF9ShOk/ZLg==",
730 |           "dev": true
731 |         },
732 |         "magic-string": {
733 |           "version": "0.22.4",
734 |           "resolved": "https://registry.npmjs.org/magic-string/-/magic-string-0.22.4.tgz",
735 |           "integrity": "sha512-kxBL06p6iO2qPBHsqGK2b3cRwiRGpnmSuVWNhwHcMX7qJOUr1HvricYP1LZOCdkQBUp0jiWg2d6WJwR3vYgByw==",
736 |           "dev": true,
737 |           "requires": {
738 |             "vlq": "0.2.3"
739 |           }
740 |         },
741 |         "rollup-pluginutils": {
742 |           "version": "2.0.1",
743 |           "resolved": "https://registry.npmjs.org/rollup-pluginutils/-/rollup-pluginutils-2.0.1.tgz",
744 |           "integrity": "sha1-fslbNXP2VDpGpkYb2afFRFJdD8A=",
745 |           "dev": true,
746 |           "requires": {
747 |             "estree-walker": "0.3.1",
748 |             "micromatch": "2.3.11"
749 |           },
750 |           "dependencies": {
751 |             "estree-walker": {
752 |               "version": "0.3.1",
753 |               "resolved": "https://registry.npmjs.org/estree-walker/-/estree-walker-0.3.1.tgz",
754 |               "integrity": "sha1-5rGlHPcpJSTnI3wxLl/mZgwc4ao=",
755 |               "dev": true
756 |             }
757 |           }
758 |         }
759 |       }
760 |     },
761 |     "rollup-plugin-node-resolve": {
762 |       "version": "3.0.2",
763 |       "resolved": "https://registry.npmjs.org/rollup-plugin-node-resolve/-/rollup-plugin-node-resolve-3.0.2.tgz",
764 |       "integrity": "sha512-ZwmMip/yqw6cmDQJuCQJ1G7gw2z11iGUtQNFYrFZHmqadRHU+OZGC3nOXwXu+UTvcm5lzDspB1EYWrkTgPWybw==",
765 |       "dev": true,
766 |       "requires": {
767 |         "builtin-modules": "1.1.1",
768 |         "is-module": "1.0.0",
769 |         "resolve": "1.5.0"
770 |       }
771 |     },
772 |     "rollup-pluginutils": {
773 |       "version": "1.5.2",
774 |       "resolved": "https://registry.npmjs.org/rollup-pluginutils/-/rollup-pluginutils-1.5.2.tgz",
775 |       "integrity": "sha1-HhVud4+UtyVb+hs9AXi+j1xVJAg=",
776 |       "dev": true,
777 |       "requires": {
778 |         "estree-walker": "0.2.1",
779 |         "minimatch": "3.0.4"
780 |       }
781 |     },
782 |     "strip-ansi": {
783 |       "version": "3.0.1",
784 |       "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-3.0.1.tgz",
785 |       "integrity": "sha1-ajhfuIU9lS1f8F0Oiq+UJ43GPc8=",
786 |       "dev": true,
787 |       "requires": {
788 |         "ansi-regex": "2.1.1"
789 |       }
790 |     },
791 |     "supports-color": {
792 |       "version": "2.0.0",
793 |       "resolved": "https://registry.npmjs.org/supports-color/-/supports-color-2.0.0.tgz",
794 |       "integrity": "sha1-U10EXOa2Nj+kARcIRimZXp3zJMc=",
795 |       "dev": true
796 |     },
797 |     "type-detect": {
798 |       "version": "4.0.6",
799 |       "resolved": "https://registry.npmjs.org/type-detect/-/type-detect-4.0.6.tgz",
800 |       "integrity": "sha512-qZ3bAurt2IXGPR3c57PyaSYEnQiLRwPeS60G9TahElBZsdOABo+iKYch/PhRjSTZJ5/DF08x43XMt9qec2g3ig==",
801 |       "dev": true
802 |     },
803 |     "vlq": {
804 |       "version": "0.2.3",
805 |       "resolved": "https://registry.npmjs.org/vlq/-/vlq-0.2.3.tgz",
806 |       "integrity": "sha512-DRibZL6DsNhIgYQ+wNdWDL2SL3bKPlVrRiBqV5yuMm++op8W4kGFtaQfCs4KEJn0wBZcHVHJ3eoywX8983k1ow==",
807 |       "dev": true
808 |     },
809 |     "wrappy": {
810 |       "version": "1.0.2",
811 |       "resolved": "https://registry.npmjs.org/wrappy/-/wrappy-1.0.2.tgz",
812 |       "integrity": "sha1-tSQ9jz7BqjXxNkYFvA0QNuMKtp8=",
813 |       "dev": true
814 |     }
815 |   }
816 | }
817 | 


--------------------------------------------------------------------------------
/package.json:
--------------------------------------------------------------------------------
 1 | {
 2 |   "name": "bezier-intersect",
 3 |   "version": "0.0.3",
 4 |   "description": "intersections between bezier curves of order 2, 3 and lines and rectangles",
 5 |   "main": "dist/bezier-intersect.js",
 6 |   "module": "index",
 7 |   "jsnext:main": "index",
 8 |   "scripts": {
 9 |     "build": "rollup -c",
10 |     "dev": "rollup -c -w",
11 |     "test": "mocha test/test.js",
12 |     "pretest": "npm run build",
13 |     "prepublish": "npm run build"
14 |   },
15 |   "devDependencies": {
16 |     "chai": "^4.1.2",
17 |     "mocha": "^5.0.0",
18 |     "rollup": "^0.46.0",
19 |     "rollup-plugin-buble": "^0.15.0",
20 |     "rollup-plugin-commonjs": "^8.0.2",
21 |     "rollup-plugin-node-resolve": "^3.0.0"
22 |   },
23 |   "keywords": [
24 |     "bezier",
25 |     "curve",
26 |     "intersection",
27 |     "quadratic",
28 |     "cubic"
29 |   ],
30 |   "files": [
31 |     "dist",
32 |     "src",
33 |     "index.js"
34 |   ],
35 |   "author": "Alexander Milevski <info@w8r.name>",
36 |   "license": "MIT",
37 |   "directories": {
38 |     "test": "test"
39 |   },
40 |   "dependencies": {},
41 |   "repository": {
42 |     "type": "git",
43 |     "url": "git+https://github.com/w8r/bezier-intersect.git"
44 |   },
45 |   "bugs": {
46 |     "url": "https://github.com/w8r/bezier-intersect/issues"
47 |   },
48 |   "homepage": "https://github.com/w8r/bezier-intersect#readme"
49 | }
50 | 


--------------------------------------------------------------------------------
/rollup.config.js:
--------------------------------------------------------------------------------
 1 | import resolve  from 'rollup-plugin-node-resolve';
 2 | import commonjs from 'rollup-plugin-commonjs';
 3 | import buble    from 'rollup-plugin-buble';
 4 | import pkg      from './package.json';
 5 | 
 6 | export default [
 7 |   // browser-friendly UMD build
 8 |   {
 9 |     entry: 'index.js',
10 |     dest: pkg.main,
11 |     format: 'umd',
12 |     moduleName: 'bezierIntersect',
13 |     plugins: [
14 |       resolve(), // so Rollup can find `ms`
15 |       commonjs(), // so Rollup can convert `ms` to an ES module
16 |       buble({  // transpile ES2015+ to ES5
17 |         exclude: ['node_modules/**']
18 |       })
19 |     ]
20 |   }
21 | 
22 |   // CommonJS (for Node) and ES module (for bundlers) build.
23 |   // (We could have three entries in the configuration array
24 |   // instead of two, but it's quicker to generate multiple
25 |   // builds from a single configuration where possible, using
26 |   // the `targets` option which can specify `dest` and `format`)
27 |   // {
28 |   //   entry: 'src/main.js',
29 |   //   external: ['ms'],
30 |   //   targets: [
31 |   //     { dest: pkg.main, format: 'cjs' },
32 |   //     { dest: pkg.module, format: 'es' }
33 |   //   ],
34 |   //   plugins: [
35 |   //     buble({
36 |   //       exclude: ['node_modules/**']
37 |   //     })
38 |   //   ]
39 |   // }
40 | ];
41 | 


--------------------------------------------------------------------------------
/src/cubic.js:
--------------------------------------------------------------------------------
  1 | import { getPolynomialRoots } from './polynomial';
  2 | 
  3 | export function cubicBezierLine(
  4 |   p1x, p1y, p2x, p2y, p3x, p3y, p4x, p4y,
  5 |   a1x, a1y, a2x, a2y, result) {
  6 |   var ax, ay, bx, by, cx, cy, dx, dy;         // temporary variables
  7 |   var c3x, c3y, c2x, c2y, c1x, c1y, c0x, c0y; // coefficients of cubic
  8 |   var cl;         // c coefficient for normal form of line
  9 |   var nx, ny;     // normal for normal form of line
 10 | 
 11 |   // used to determine if point is on line segment
 12 |   var minx = Math.min(a1x, a2x),
 13 |       miny = Math.min(a1y, a2y),
 14 |       maxx = Math.max(a1x, a2x),
 15 |       maxy = Math.max(a1y, a2y);
 16 | 
 17 |   // Start with Bezier using Bernstein polynomials for weighting functions:
 18 |   //     (1-t^3)P1 + 3t(1-t)^2P2 + 3t^2(1-t)P3 + t^3P4
 19 |   //
 20 |   // Expand and collect terms to form linear combinations of original Bezier
 21 |   // controls.  This ends up with a vector cubic in t:
 22 |   //     (-P1+3P2-3P3+P4)t^3 + (3P1-6P2+3P3)t^2 + (-3P1+3P2)t + P1
 23 |   //             /\                  /\                /\       /\
 24 |   //             ||                  ||                ||       ||
 25 |   //             c3                  c2                c1       c0
 26 | 
 27 |   // Calculate the coefficients
 28 |   ax = p1x * -1; ay = p1y * -1;
 29 |   bx = p2x * 3;  by = p2y * 3;
 30 |   cx = p3x * -3; cy = p3y * -3;
 31 |   dx = ax + bx + cx + p4x;
 32 |   dy = ay + by + cy + p4y;
 33 |   c3x = dx; c3y = dy; // vec
 34 | 
 35 |   ax = p1x * 3;  ay = p1y * 3;
 36 |   bx = p2x * -6; by = p2y * -6;
 37 |   cx = p3x * 3;  cy = p3y * 3;
 38 |   dx = ax + bx + cx;
 39 |   dy = ay + by + cy;
 40 |   c2x = dx; c2y = dy; // vec
 41 | 
 42 |   ax = p1x * -3; ay = p1y * -3;
 43 |   bx = p2x * 3;  by = p2y * 3;
 44 |   cx = ax + bx;
 45 |   cy = ay + by;
 46 |   c1x = cx;
 47 |   c1y = cy; // vec
 48 | 
 49 |   c0x = p1x;
 50 |   c0y = p1y;
 51 | 
 52 |   // Convert line to normal form: ax + by + c = 0
 53 |   // Find normal to line: negative inverse of original line's slope
 54 |   nx = a1y - a2y;
 55 |   ny = a2x - a1x;
 56 | 
 57 |   // Determine new c coefficient
 58 |   cl = a1x * a2y - a2x * a1y;
 59 | 
 60 |   // ?Rotate each cubic coefficient using line for new coordinate system?
 61 |   // Find roots of rotated cubic
 62 |   var roots = getPolynomialRoots(
 63 |     // dot products => x * x + y * y
 64 |     nx * c3x + ny * c3y,
 65 |     nx * c2x + ny * c2y,
 66 |     nx * c1x + ny * c1y,
 67 |     nx * c0x + ny * c0y + cl
 68 |   );
 69 | 
 70 |   // Any roots in closed interval [0,1] are intersections on Bezier, but
 71 |   // might not be on the line segment.
 72 |   // Find intersections and calculate point coordinates
 73 |   for (var i = 0; i < roots.length; i++) {
 74 |     var t = roots[i];
 75 | 
 76 |     if (0 <= t && t <= 1) { // We're within the Bezier curve
 77 |       // Find point on Bezier
 78 |       // lerp: x1 + (x2 - x1) * t
 79 |       var p5x = p1x + (p2x - p1x) * t;
 80 |       var p5y = p1y + (p2y - p1y) * t; // lerp(p1, p2, t);
 81 | 
 82 |       var p6x = p2x + (p3x - p2x) * t;
 83 |       var p6y = p2y + (p3y - p2y) * t;
 84 | 
 85 |       var p7x = p3x + (p4x - p3x) * t;
 86 |       var p7y = p3y + (p4y - p3y) * t;
 87 | 
 88 |       var p8x = p5x + (p6x - p5x) * t;
 89 |       var p8y = p5y + (p6y - p5y) * t;
 90 | 
 91 |       var p9x = p6x + (p7x - p6x) * t;
 92 |       var p9y = p6y + (p7y - p6y) * t;
 93 | 
 94 |       // candidate
 95 |       var p10x = p8x + (p9x - p8x) * t;
 96 |       var p10y = p8y + (p9y - p8y) * t;
 97 | 
 98 |       // See if point is on line segment
 99 |       if (a1x === a2x) {                       // vertical
100 |         if (miny <= p10y && p10y <= maxy) {
101 |           if (result) result.push(p10x, p10y);
102 |           else        return 1;
103 |         }
104 |       } else if (a1y === a2y) {               // horizontal
105 |         if (minx <= p10x && p10x <= maxx) {
106 |           if (result) result.push(p10x, p10y);
107 |           else        return 1;
108 |         }
109 |       } else if (p10x >= minx && p10y >= miny && p10x <= maxx && p10y <= maxy) {
110 |         if (result) result.push(p10x, p10y);
111 |         else        return 1;
112 |       }
113 |     }
114 |   }
115 |   return result ? result.length / 2 : 0;
116 | }
117 | 
118 | 
119 | export function cubicBezierAABB(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmax, ymax, result) {
120 |   if (result) { // all intersections
121 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmax, ymin, result);
122 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmax, ymin, xmax, ymax, result);
123 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymax, xmax, ymax, result);
124 |     cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmin, ymax, result);
125 |     return result.length / 2;
126 |   } else {     // any intersections
127 |     // trivial cases
128 |     if (xmin <= ax && xmax >= ax && ymin <= ay && ymax >= ay) return 1;
129 |     if (xmin <= bx && xmax >= bx && ymin <= by && ymax >= by) return 1;
130 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmax, ymin)) return 1;
131 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmax, ymin, xmax, ymax)) return 1;
132 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymax, xmax, ymax)) return 1;
133 |     if (cubicBezierLine(ax, ay, c1x, c1y, c2x, c2y, bx, by, xmin, ymin, xmin, ymax)) return 1;
134 |     return 0;
135 |   }
136 | }
137 | 


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/src/polynomial.js:
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 1 | const POLYNOMIAL_TOLERANCE = 1e-6;
 2 | const TOLERANCE            = 1e-12;
 3 | 
 4 | export function getPolynomialRoots() {
 5 |   var C  = arguments;
 6 |   var degree  = C.length - 1;
 7 |   var n       = degree;
 8 |   var results = [];
 9 |   for (var i = 0; i <= degree; i++) {
10 |     if (Math.abs(C[i]) <= TOLERANCE) degree--; else break;
11 |   }
12 | 
13 |   switch (degree) {
14 |     case 1: getLinearRoots(C[n], C[n - 1], results); break;
15 |     case 2: getQuadraticRoots(C[n], C[n - 1], C[n - 2], results); break;
16 |     case 3: getCubicRoots(C[n], C[n - 1], C[n - 2], C[n - 3], results); break;
17 |     default: break;
18 |   }
19 | 
20 |   return results;
21 | }
22 | 
23 | export function getLinearRoots(C0, C1, results = []) {
24 |   if (C1 !== 0) results.push(-C0 / C1);
25 |   return results;
26 | }
27 | 
28 | export function getQuadraticRoots(C0, C1, C2, results = []) {
29 |   var a = C2;
30 |   var b = C1 / a;
31 |   var c = C0 / a;
32 |   var d = b * b - 4 * c;
33 | 
34 |   if (d > 0) {
35 |     var e = Math.sqrt(d);
36 | 
37 |     results.push(0.5 * (-b + e));
38 |     results.push(0.5 * (-b - e));
39 |   } else if (d === 0) {
40 |     results.push( 0.5 * -b);
41 |   }
42 | 
43 |   return results;
44 | }
45 | 
46 | 
47 | export function getCubicRoots(C0, C1, C2, C3, results = []) {
48 |   var c3 = C3;
49 |   var c2 = C2 / c3;
50 |   var c1 = C1 / c3;
51 |   var c0 = C0 / c3;
52 | 
53 |   var a       = (3 * c1 - c2 * c2) / 3;
54 |   var b       = (2 * c2 * c2 * c2 - 9 * c1 * c2 + 27 * c0) / 27;
55 |   var offset  = c2 / 3;
56 |   var discrim = b * b / 4 + a * a * a / 27;
57 |   var halfB   = b / 2;
58 |   var tmp, root;
59 | 
60 |   if (Math.abs(discrim) <= POLYNOMIAL_TOLERANCE) discrim = 0;
61 | 
62 |   if (discrim > 0) {
63 |     var e = Math.sqrt(discrim);
64 | 
65 |     tmp = -halfB + e;
66 |     if ( tmp >= 0 ) root = Math.pow(  tmp, 1/3);
67 |     else            root = -Math.pow(-tmp, 1/3);
68 | 
69 |     tmp = -halfB - e;
70 |     if ( tmp >= 0 ) root += Math.pow( tmp, 1/3);
71 |     else            root -= Math.pow(-tmp, 1/3);
72 | 
73 |     results.push(root - offset);
74 |   } else if (discrim < 0) {
75 |     var distance = Math.sqrt(-a/3);
76 |     var angle    = Math.atan2(Math.sqrt(-discrim), -halfB) / 3;
77 |     var cos      = Math.cos(angle);
78 |     var sin      = Math.sin(angle);
79 |     var sqrt3    = Math.sqrt(3);
80 | 
81 |     results.push( 2 * distance * cos - offset);
82 |     results.push(-distance * (cos + sqrt3 * sin) - offset);
83 |     results.push(-distance * (cos - sqrt3 * sin) - offset);
84 |   } else {
85 |     if (halfB >= 0) tmp = -Math.pow(halfB, 1/3);
86 |     else            tmp =  Math.pow(-halfB, 1/3);
87 | 
88 |     results.push(2 * tmp - offset);
89 |     // really should return next root twice, but we return only one
90 |     results.push(-tmp - offset);
91 |   }
92 | 
93 |   return results;
94 | }
95 | 


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/src/quadratic.js:
--------------------------------------------------------------------------------
  1 | import { getPolynomialRoots } from './polynomial';
  2 | 
  3 | export function quadBezierLine(
  4 |   p1x, p1y, p2x, p2y, p3x, p3y,
  5 |   a1x, a1y, a2x, a2y, result) {
  6 |   var ax, ay, bx, by;                // temporary variables
  7 |   var c2x, c2y, c1x, c1y, c0x, c0y;  // coefficients of quadratic
  8 |   var cl;               // c coefficient for normal form of line
  9 |   var nx, ny;           // normal for normal form of line
 10 |   // used to determine if point is on line segment
 11 |   var minx = Math.min(a1x, a2x),
 12 |       miny = Math.min(a1y, a2y),
 13 |       maxx = Math.max(a1x, a2x),
 14 |       maxy = Math.max(a1y, a2y);
 15 | 
 16 |   ax = p2x * -2; ay = p2y * -2;
 17 |   c2x = p1x + ax + p3x;
 18 |   c2y = p1y + ay + p3y;
 19 | 
 20 |   ax = p1x * -2; ay = p1y * -2;
 21 |   bx = p2x * 2;  by = p2y * 2;
 22 |   c1x = ax + bx;
 23 |   c1y = ay + by;
 24 | 
 25 |   c0x = p1x; c0y = p1y; // vec
 26 | 
 27 |   // Convert line to normal form: ax + by + c = 0
 28 |   // Find normal to line: negative inverse of original line's slope
 29 |   nx = a1y - a2y; ny = a2x - a1x;
 30 | 
 31 |   // Determine new c coefficient
 32 |   cl = a1x * a2y - a2x * a1y;
 33 | 
 34 |   // Transform cubic coefficients to line's coordinate system
 35 |   // and find roots of cubic
 36 |   var roots = getPolynomialRoots(
 37 |     // dot products => x * x + y * y
 38 |     nx * c2x + ny * c2y,
 39 |     nx * c1x + ny * c1y,
 40 |     nx * c0x + ny * c0y + cl
 41 |   );
 42 | 
 43 |   // Any roots in closed interval [0,1] are intersections on Bezier, but
 44 |   // might not be on the line segment.
 45 |   // Find intersections and calculate point coordinates
 46 |   for (var i = 0; i < roots.length; i++) {
 47 |     var t = roots[i];
 48 |     if ( 0 <= t && t <= 1 ) { // We're within the Bezier curve
 49 |       // Find point on Bezier
 50 |       // lerp: x1 + (x2 - x1) * t
 51 |       var p4x = p1x + (p2x - p1x) * t;
 52 |       var p4y = p1y + (p2y - p1y) * t;
 53 | 
 54 |       var p5x = p2x + (p3x - p2x) * t;
 55 |       var p5y = p2y + (p3y - p2y) * t;
 56 | 
 57 |       // candidate
 58 |       var p6x = p4x + (p5x - p4x) * t;
 59 |       var p6y = p4y + (p5y - p4y) * t;
 60 | 
 61 |       // See if point is on line segment
 62 |       // Had to make special cases for vertical and horizontal lines due
 63 |       // to slight errors in calculation of p6
 64 |       if (a1x === a2x) {
 65 |         if (miny <= p6y && p6y <= maxy) {
 66 |           if (result) result.push(p6x, p6y);
 67 |           else        return 1;
 68 |         }
 69 |       } else if (a1y === a2y) {
 70 |         if (minx <= p6x && p6x <= maxx) {
 71 |           if (result) result.push(p6x, p6y);
 72 |           else        return 1;
 73 |         }
 74 | 
 75 |       // gte: (x1 >= x2 && y1 >= y2)
 76 |       // lte: (x1 <= x2 && y1 <= y2)
 77 |       } else if (p6x >= minx && p6y >= miny && p6x <= maxx && p6y <= maxy) {
 78 |         if (result) result.push(p6x, p6y);
 79 |         else        return 1;
 80 |       }
 81 |     }
 82 |   }
 83 |   return result ? result.length / 2 : 0;
 84 | }
 85 | 
 86 | 
 87 | export function quadBezierAABB(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmax, ymax, result) {
 88 |   if (result) { // all intersections
 89 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmax, ymin, result);
 90 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmax, ymin, xmax, ymax, result);
 91 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymax, xmax, ymax, result);
 92 |     quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmin, ymax, result);
 93 |     return result.length / 2;
 94 |   } else {      // any intersections
 95 |     // trivial cases
 96 |     if (xmin <= ax && xmax >= ax && ymin <= ay && ymax >= ay) return 1;
 97 |     if (xmin <= bx && xmax >= bx && ymin <= by && ymax >= by) return 1;
 98 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmax, ymin)) return 1;
 99 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmax, ymin, xmax, ymax)) return 1;
100 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymax, xmax, ymax)) return 1;
101 |     if (quadBezierLine(ax, ay, c1x, c1y, bx, by, xmin, ymin, xmin, ymax)) return 1;
102 |     return 0;
103 |   }
104 | }
105 | 


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/test/test.js:
--------------------------------------------------------------------------------
 1 | const {
 2 |   cubicBezierLine,
 3 |   quadBezierAABB,
 4 |   cubicBezierAABB
 5 | } = require('../dist/bezier-intersect');
 6 | 
 7 | const { assert } = require('chai');
 8 | 
 9 | describe('intersectCubicBezierLine', () => {
10 |   it ('vertical line crosses', () => {
11 |     const result = [];
12 |     assert.isTrue(!!cubicBezierLine(0,0, 100, 100, 200, 100, 300, 0, 100, 0, 100, 100, result))
13 |     assert.deepEqual(result, [99.99999999999997, 66.66666666666666]);
14 |   });
15 | 
16 |   it ('diagonal line crosses', () => {
17 |     const result = [];
18 |     assert.isTrue(!!cubicBezierLine(0,0, 100, 100, 200, 100, 300, 0, 100, 0, 200, 100, result))
19 |     assert.deepEqual(result, [173.20508075688772, 73.20508075688774]);
20 |   });
21 | });
22 | 
23 | describe('intersectCubicBezierAABB', () => {
24 | 
25 |   it ('intersects', () => {
26 |     const x = 981.7516776530498, y = 1202.2380271093887;
27 |     const dx = 860.2542723247064, dy = 1323.735432437732;
28 |     const res = [];
29 |     const q = [843, 1228, 943, 1328];
30 |     assert.equal(cubicBezierAABB(x, y, dx, y, x, dy, x, y, q[0], q[1], q[2], q[3], res), 2);
31 |     assert.deepEqual(res, [ 927.7910501404327, 1228, 942.9999999999999, 1252.974906993395 ]);
32 |   });
33 | 
34 |   it ('does not intersect', () => {
35 |     const x = 981.7516776530498, y = 1202.2380271093887;
36 |     const dx = 860.2542723247064, dy = 1323.735432437732;
37 |     const res = [];
38 |     const q = [843 - 100, 1228 - 100, 943 -100, 1328 - 100]
39 |     assert.equal(cubicBezierAABB(x, y, dx, y, x, dy, x, y, q[0], q[1], q[2], q[3], res), 0);
40 |     assert.deepEqual(res, []);
41 |   });
42 | 
43 | });
44 | 
45 | describe('quad bezier rectangle', () => {
46 |   it ('crosses the rectangle at 2 points', () => {
47 |     const result = [];
48 |     assert.equal(quadBezierAABB(0, 0, 150, 100, 300, 0, 100, 0, 200, 100, result), 2);
49 |     assert.deepEqual(result, [200, 44.44444444444445, 100, 44.44444444444444]);
50 |   });
51 | });
52 | 


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