OLD | NEW |
| (Empty) |
1 /* | |
2 * Copyright 2012 Google Inc. | |
3 * | |
4 * Use of this source code is governed by a BSD-style license that can be | |
5 * found in the LICENSE file. | |
6 */ | |
7 #include "CurveIntersection.h" | |
8 #include "CubicUtilities.h" | |
9 #include "Intersections.h" | |
10 #include "LineUtilities.h" | |
11 | |
12 /* | |
13 Find the interection of a line and cubic by solving for valid t values. | |
14 | |
15 Analogous to line-quadratic intersection, solve line-cubic intersection by | |
16 representing the cubic as: | |
17 x = a(1-t)^3 + 2b(1-t)^2t + c(1-t)t^2 + dt^3 | |
18 y = e(1-t)^3 + 2f(1-t)^2t + g(1-t)t^2 + ht^3 | |
19 and the line as: | |
20 y = i*x + j (if the line is more horizontal) | |
21 or: | |
22 x = i*y + j (if the line is more vertical) | |
23 | |
24 Then using Mathematica, solve for the values of t where the cubic intersects the | |
25 line: | |
26 | |
27 (in) Resultant[ | |
28 a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - x, | |
29 e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - i*x - j, x] | |
30 (out) -e + j + | |
31 3 e t - 3 f t - | |
32 3 e t^2 + 6 f t^2 - 3 g t^2 + | |
33 e t^3 - 3 f t^3 + 3 g t^3 - h t^3 + | |
34 i ( a - | |
35 3 a t + 3 b t + | |
36 3 a t^2 - 6 b t^2 + 3 c t^2 - | |
37 a t^3 + 3 b t^3 - 3 c t^3 + d t^3 ) | |
38 | |
39 if i goes to infinity, we can rewrite the line in terms of x. Mathematica: | |
40 | |
41 (in) Resultant[ | |
42 a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - i*y - j, | |
43 e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y] | |
44 (out) a - j - | |
45 3 a t + 3 b t + | |
46 3 a t^2 - 6 b t^2 + 3 c t^2 - | |
47 a t^3 + 3 b t^3 - 3 c t^3 + d t^3 - | |
48 i ( e - | |
49 3 e t + 3 f t + | |
50 3 e t^2 - 6 f t^2 + 3 g t^2 - | |
51 e t^3 + 3 f t^3 - 3 g t^3 + h t^3 ) | |
52 | |
53 Solving this with Mathematica produces an expression with hundreds of terms; | |
54 instead, use Numeric Solutions recipe to solve the cubic. | |
55 | |
56 The near-horizontal case, in terms of: Ax^3 + Bx^2 + Cx + D == 0 | |
57 A = (-(-e + 3*f - 3*g + h) + i*(-a + 3*b - 3*c + d) ) | |
58 B = 3*(-( e - 2*f + g ) + i*( a - 2*b + c ) ) | |
59 C = 3*(-(-e + f ) + i*(-a + b ) ) | |
60 D = (-( e ) + i*( a ) + j ) | |
61 | |
62 The near-vertical case, in terms of: Ax^3 + Bx^2 + Cx + D == 0 | |
63 A = ( (-a + 3*b - 3*c + d) - i*(-e + 3*f - 3*g + h) ) | |
64 B = 3*( ( a - 2*b + c ) - i*( e - 2*f + g ) ) | |
65 C = 3*( (-a + b ) - i*(-e + f ) ) | |
66 D = ( ( a ) - i*( e ) - j ) | |
67 | |
68 For horizontal lines: | |
69 (in) Resultant[ | |
70 a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - j, | |
71 e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y] | |
72 (out) e - j - | |
73 3 e t + 3 f t + | |
74 3 e t^2 - 6 f t^2 + 3 g t^2 - | |
75 e t^3 + 3 f t^3 - 3 g t^3 + h t^3 | |
76 So the cubic coefficients are: | |
77 | |
78 */ | |
79 | |
80 class LineCubicIntersections { | |
81 public: | |
82 | |
83 LineCubicIntersections(const Cubic& c, const _Line& l, Intersections& i) | |
84 : cubic(c) | |
85 , line(l) | |
86 , intersections(i) { | |
87 } | |
88 | |
89 // see parallel routine in line quadratic intersections | |
90 int intersectRay(double roots[3]) { | |
91 double adj = line[1].x - line[0].x; | |
92 double opp = line[1].y - line[0].y; | |
93 Cubic r; | |
94 for (int n = 0; n < 4; ++n) { | |
95 r[n].x = (cubic[n].y - line[0].y) * adj - (cubic[n].x - line[0].x) * opp
; | |
96 } | |
97 double A, B, C, D; | |
98 coefficients(&r[0].x, A, B, C, D); | |
99 return cubicRootsValidT(A, B, C, D, roots); | |
100 } | |
101 | |
102 int intersect() { | |
103 addEndPoints(); | |
104 double rootVals[3]; | |
105 int roots = intersectRay(rootVals); | |
106 for (int index = 0; index < roots; ++index) { | |
107 double cubicT = rootVals[index]; | |
108 double lineT = findLineT(cubicT); | |
109 if (pinTs(cubicT, lineT)) { | |
110 _Point pt; | |
111 xy_at_t(line, lineT, pt.x, pt.y); | |
112 intersections.insert(cubicT, lineT, pt); | |
113 } | |
114 } | |
115 return intersections.fUsed; | |
116 } | |
117 | |
118 int horizontalIntersect(double axisIntercept, double roots[3]) { | |
119 double A, B, C, D; | |
120 coefficients(&cubic[0].y, A, B, C, D); | |
121 D -= axisIntercept; | |
122 return cubicRootsValidT(A, B, C, D, roots); | |
123 } | |
124 | |
125 int horizontalIntersect(double axisIntercept, double left, double right, bool fl
ipped) { | |
126 addHorizontalEndPoints(left, right, axisIntercept); | |
127 double rootVals[3]; | |
128 int roots = horizontalIntersect(axisIntercept, rootVals); | |
129 for (int index = 0; index < roots; ++index) { | |
130 _Point pt; | |
131 double cubicT = rootVals[index]; | |
132 xy_at_t(cubic, cubicT, pt.x, pt.y); | |
133 double lineT = (pt.x - left) / (right - left); | |
134 if (pinTs(cubicT, lineT)) { | |
135 intersections.insert(cubicT, lineT, pt); | |
136 } | |
137 } | |
138 if (flipped) { | |
139 flip(); | |
140 } | |
141 return intersections.fUsed; | |
142 } | |
143 | |
144 int verticalIntersect(double axisIntercept, double roots[3]) { | |
145 double A, B, C, D; | |
146 coefficients(&cubic[0].x, A, B, C, D); | |
147 D -= axisIntercept; | |
148 return cubicRootsValidT(A, B, C, D, roots); | |
149 } | |
150 | |
151 int verticalIntersect(double axisIntercept, double top, double bottom, bool flip
ped) { | |
152 addVerticalEndPoints(top, bottom, axisIntercept); | |
153 double rootVals[3]; | |
154 int roots = verticalIntersect(axisIntercept, rootVals); | |
155 for (int index = 0; index < roots; ++index) { | |
156 _Point pt; | |
157 double cubicT = rootVals[index]; | |
158 xy_at_t(cubic, cubicT, pt.x, pt.y); | |
159 double lineT = (pt.y - top) / (bottom - top); | |
160 if (pinTs(cubicT, lineT)) { | |
161 intersections.insert(cubicT, lineT, pt); | |
162 } | |
163 } | |
164 if (flipped) { | |
165 flip(); | |
166 } | |
167 return intersections.fUsed; | |
168 } | |
169 | |
170 protected: | |
171 | |
172 void addEndPoints() | |
173 { | |
174 for (int cIndex = 0; cIndex < 4; cIndex += 3) { | |
175 for (int lIndex = 0; lIndex < 2; lIndex++) { | |
176 if (cubic[cIndex] == line[lIndex]) { | |
177 intersections.insert(cIndex >> 1, lIndex, line[lIndex]); | |
178 } | |
179 } | |
180 } | |
181 } | |
182 | |
183 void addHorizontalEndPoints(double left, double right, double y) | |
184 { | |
185 for (int cIndex = 0; cIndex < 4; cIndex += 3) { | |
186 if (cubic[cIndex].y != y) { | |
187 continue; | |
188 } | |
189 if (cubic[cIndex].x == left) { | |
190 intersections.insert(cIndex >> 1, 0, cubic[cIndex]); | |
191 } | |
192 if (cubic[cIndex].x == right) { | |
193 intersections.insert(cIndex >> 1, 1, cubic[cIndex]); | |
194 } | |
195 } | |
196 } | |
197 | |
198 void addVerticalEndPoints(double top, double bottom, double x) | |
199 { | |
200 for (int cIndex = 0; cIndex < 4; cIndex += 3) { | |
201 if (cubic[cIndex].x != x) { | |
202 continue; | |
203 } | |
204 if (cubic[cIndex].y == top) { | |
205 intersections.insert(cIndex >> 1, 0, cubic[cIndex]); | |
206 } | |
207 if (cubic[cIndex].y == bottom) { | |
208 intersections.insert(cIndex >> 1, 1, cubic[cIndex]); | |
209 } | |
210 } | |
211 } | |
212 | |
213 double findLineT(double t) { | |
214 double x, y; | |
215 xy_at_t(cubic, t, x, y); | |
216 double dx = line[1].x - line[0].x; | |
217 double dy = line[1].y - line[0].y; | |
218 if (fabs(dx) > fabs(dy)) { | |
219 return (x - line[0].x) / dx; | |
220 } | |
221 return (y - line[0].y) / dy; | |
222 } | |
223 | |
224 void flip() { | |
225 // OPTIMIZATION: instead of swapping, pass original line, use [1].y - [0].y | |
226 int roots = intersections.fUsed; | |
227 for (int index = 0; index < roots; ++index) { | |
228 intersections.fT[1][index] = 1 - intersections.fT[1][index]; | |
229 } | |
230 } | |
231 | |
232 static bool pinTs(double& cubicT, double& lineT) { | |
233 if (!approximately_one_or_less(lineT)) { | |
234 return false; | |
235 } | |
236 if (!approximately_zero_or_more(lineT)) { | |
237 return false; | |
238 } | |
239 if (precisely_less_than_zero(cubicT)) { | |
240 cubicT = 0; | |
241 } else if (precisely_greater_than_one(cubicT)) { | |
242 cubicT = 1; | |
243 } | |
244 if (precisely_less_than_zero(lineT)) { | |
245 lineT = 0; | |
246 } else if (precisely_greater_than_one(lineT)) { | |
247 lineT = 1; | |
248 } | |
249 return true; | |
250 } | |
251 | |
252 private: | |
253 | |
254 const Cubic& cubic; | |
255 const _Line& line; | |
256 Intersections& intersections; | |
257 }; | |
258 | |
259 int horizontalIntersect(const Cubic& cubic, double left, double right, double y, | |
260 double tRange[3]) { | |
261 LineCubicIntersections c(cubic, *((_Line*) 0), *((Intersections*) 0)); | |
262 double rootVals[3]; | |
263 int result = c.horizontalIntersect(y, rootVals); | |
264 int tCount = 0; | |
265 for (int index = 0; index < result; ++index) { | |
266 double x, y; | |
267 xy_at_t(cubic, rootVals[index], x, y); | |
268 if (x < left || x > right) { | |
269 continue; | |
270 } | |
271 tRange[tCount++] = rootVals[index]; | |
272 } | |
273 return result; | |
274 } | |
275 | |
276 int horizontalIntersect(const Cubic& cubic, double left, double right, double y, | |
277 bool flipped, Intersections& intersections) { | |
278 LineCubicIntersections c(cubic, *((_Line*) 0), intersections); | |
279 return c.horizontalIntersect(y, left, right, flipped); | |
280 } | |
281 | |
282 int verticalIntersect(const Cubic& cubic, double top, double bottom, double x, | |
283 bool flipped, Intersections& intersections) { | |
284 LineCubicIntersections c(cubic, *((_Line*) 0), intersections); | |
285 return c.verticalIntersect(x, top, bottom, flipped); | |
286 } | |
287 | |
288 int intersect(const Cubic& cubic, const _Line& line, Intersections& i) { | |
289 LineCubicIntersections c(cubic, line, i); | |
290 return c.intersect(); | |
291 } | |
292 | |
293 int intersectRay(const Cubic& cubic, const _Line& line, Intersections& i) { | |
294 LineCubicIntersections c(cubic, line, i); | |
295 return c.intersectRay(i.fT[0]); | |
296 } | |
OLD | NEW |