19e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com/* 29e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * Copyright 2012 Google Inc. 39e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * 49e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * Use of this source code is governed by a BSD-style license that can be 59e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * found in the LICENSE file. 69e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com */ 7c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com#include "CurveIntersection.h" 827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com#include "Intersections.h" 927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com#include "LineUtilities.h" 1027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com#include "QuadraticUtilities.h" 1127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 12d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com/* 1327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comFind the interection of a line and quadratic by solving for valid t values. 1427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 1527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comFrom http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve 1627accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 17d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com"A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three 18d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.comcontrol points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where 1927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comA, B and C are points and t goes from zero to one. 2027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 2127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comThis will give you two equations: 2227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 2327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com x = a(1 - t)^2 + b(1 - t)t + ct^2 2427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com y = d(1 - t)^2 + e(1 - t)t + ft^2 2527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 26d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.comIf you add for instance the line equation (y = kx + m) to that, you'll end up 2727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comwith three equations and three unknowns (x, y and t)." 2827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 2927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comSimilar to above, the quadratic is represented as 3027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com x = a(1-t)^2 + 2b(1-t)t + ct^2 3127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com y = d(1-t)^2 + 2e(1-t)t + ft^2 3227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comand the line as 3327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com y = g*x + h 3427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 3527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comUsing Mathematica, solve for the values of t where the quadratic intersects the 3627accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comline: 3727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 38d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com (in) t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x, 3927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - g*x - h, x] 40d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com (out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 + 4127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com g (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2) 4227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (in) Solve[t1 == 0, t] 4327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (out) { 4427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com {t -> (-2 d + 2 e + 2 a g - 2 b g - 45d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - 4627accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / 4727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (2 (-d + 2 e - f + a g - 2 b g + c g)) 4827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com }, 4927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com {t -> (-2 d + 2 e + 2 a g - 2 b g + 50d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - 5127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / 5227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (2 (-d + 2 e - f + a g - 2 b g + c g)) 5327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com } 5427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com } 55d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com 5627accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comUsing the results above (when the line tends towards horizontal) 5727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com A = (-(d - 2*e + f) + g*(a - 2*b + c) ) 5827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com B = 2*( (d - e ) - g*(a - b ) ) 5927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com C = (-(d ) + g*(a ) + h ) 6027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 6127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comIf g goes to infinity, we can rewrite the line in terms of x. 6227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com x = g'*y + h' 6327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 6427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comAnd solve accordingly in Mathematica: 6527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 66d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com (in) t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h', 6727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - y, y] 68d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com (out) a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 - 6927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com g' (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2) 7027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (in) Solve[t2 == 0, t] 7127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (out) { 7227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com {t -> (2 a - 2 b - 2 d g' + 2 e g' - 73d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - 7427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) / 7527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (2 (a - 2 b + c - d g' + 2 e g' - f g')) 7627accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com }, 7727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com {t -> (2 a - 2 b - 2 d g' + 2 e g' + 78d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - 7927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/ 8027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com (2 (a - 2 b + c - d g' + 2 e g' - f g')) 8127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com } 8227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com } 8327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 8427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comThus, if the slope of the line tends towards vertical, we use: 8527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com A = ( (a - 2*b + c) - g'*(d - 2*e + f) ) 8627accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com B = 2*(-(a - b ) + g'*(d - e ) ) 8727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com C = ( (a ) - g'*(d ) - h' ) 8827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com */ 89d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com 9027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 91fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comclass LineQuadraticIntersections { 9227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.compublic: 9327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 9427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comLineQuadraticIntersections(const Quadratic& q, const _Line& l, Intersections& i) 9527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com : quad(q) 9627accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com , line(l) 9727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com , intersections(i) { 9827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com} 9927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 100fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comint intersectRay(double roots[2]) { 1013350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com/* 1023350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com solve by rotating line+quad so line is horizontal, then finding the roots 1033350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com set up matrix to rotate quad to x-axis 1043350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com |cos(a) -sin(a)| 1053350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com |sin(a) cos(a)| 1063350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com note that cos(a) = A(djacent) / Hypoteneuse 1073350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com sin(a) = O(pposite) / Hypoteneuse 1083350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com since we are computing Ts, we can ignore hypoteneuse, the scale factor: 1093350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com | A -O | 1103350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com | O A | 1113350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com A = line[1].x - line[0].x (adjacent side of the right triangle) 1123350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com O = line[1].y - line[0].y (opposite side of the right triangle) 1133350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com for each of the three points (e.g. n = 0 to 2) 1143350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com quad[n].y' = (quad[n].y - line[0].y) * A - (quad[n].x - line[0].x) * O 1153350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com*/ 1163350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com double adj = line[1].x - line[0].x; 1173350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com double opp = line[1].y - line[0].y; 1183350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com double r[3]; 1193350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com for (int n = 0; n < 3; ++n) { 1203350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com r[n] = (quad[n].y - line[0].y) * adj - (quad[n].x - line[0].x) * opp; 12127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com } 1223350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com double A = r[2]; 1233350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com double B = r[1]; 1243350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com double C = r[0]; 1253350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com A += C - 2 * B; // A = a - 2*b + c 1263350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com B -= C; // B = -(b - c) 1279f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return quadraticRootsValidT(A, 2 * B, C, roots); 128235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com} 129235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com 130235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.comint intersect() { 131fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com addEndPoints(); 132fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double rootVals[2]; 133fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com int roots = intersectRay(rootVals); 134fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int index = 0; index < roots; ++index) { 135fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double quadT = rootVals[index]; 136fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double lineT = findLineT(quadT); 137fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (pinTs(quadT, lineT)) { 13845a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com _Point pt; 13945a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com xy_at_t(line, lineT, pt.x, pt.y); 14045a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(quadT, lineT, pt); 14124bec79d6f3d71ff97b50db72461a3892bd4f6b5caryclark@google.com } 14224bec79d6f3d71ff97b50db72461a3892bd4f6b5caryclark@google.com } 143fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com return intersections.fUsed; 14427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com} 14527accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 146fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comint horizontalIntersect(double axisIntercept, double roots[2]) { 147198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com double D = quad[2].y; // f 148198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com double E = quad[1].y; // e 149198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com double F = quad[0].y; // d 150198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com D += F - 2 * E; // D = d - 2*e + f 151198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com E -= F; // E = -(d - e) 152198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com F -= axisIntercept; 1539f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return quadraticRootsValidT(D, 2 * E, F, roots); 154198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com} 155198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com 156fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comint horizontalIntersect(double axisIntercept, double left, double right, bool flipped) { 157fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com addHorizontalEndPoints(left, right, axisIntercept); 158fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double rootVals[2]; 159fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com int roots = horizontalIntersect(axisIntercept, rootVals); 160fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int index = 0; index < roots; ++index) { 16145a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com _Point pt; 162fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double quadT = rootVals[index]; 16345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com xy_at_t(quad, quadT, pt.x, pt.y); 16445a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com double lineT = (pt.x - left) / (right - left); 165fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (pinTs(quadT, lineT)) { 16645a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(quadT, lineT, pt); 1673350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com } 1683350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com } 169fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (flipped) { 170fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com flip(); 171fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 172fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com return intersections.fUsed; 1733350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com} 1743350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com 175fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comint verticalIntersect(double axisIntercept, double roots[2]) { 176fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com double D = quad[2].x; // f 177fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com double E = quad[1].x; // e 178fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com double F = quad[0].x; // d 179fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com D += F - 2 * E; // D = d - 2*e + f 180fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com E -= F; // E = -(d - e) 181fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com F -= axisIntercept; 1829f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return quadraticRootsValidT(D, 2 * E, F, roots); 183fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com} 184fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com 185fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comint verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) { 186fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com addVerticalEndPoints(top, bottom, axisIntercept); 187fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double rootVals[2]; 188fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com int roots = verticalIntersect(axisIntercept, rootVals); 189fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int index = 0; index < roots; ++index) { 19045a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com _Point pt; 191fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double quadT = rootVals[index]; 19245a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com xy_at_t(quad, quadT, pt.x, pt.y); 19345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com double lineT = (pt.y - top) / (bottom - top); 194fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (pinTs(quadT, lineT)) { 19545a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(quadT, lineT, pt); 1963350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com } 1973350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com } 198fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (flipped) { 199fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com flip(); 200fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 201fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com return intersections.fUsed; 2023350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com} 2033350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com 20427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comprotected: 205d6176b0dcacb124539e0cfd051e6d93a9782f020rmistry@google.com 206fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com// add endpoints first to get zero and one t values exactly 207fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comvoid addEndPoints() 208fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com{ 209fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int qIndex = 0; qIndex < 3; qIndex += 2) { 210fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int lIndex = 0; lIndex < 2; lIndex++) { 211fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (quad[qIndex] == line[lIndex]) { 21245a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(qIndex >> 1, lIndex, line[lIndex]); 213fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 214fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 215fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 216fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com} 217fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com 218fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comvoid addHorizontalEndPoints(double left, double right, double y) 219fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com{ 220fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int qIndex = 0; qIndex < 3; qIndex += 2) { 221fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (quad[qIndex].y != y) { 222fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com continue; 223fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 224fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (quad[qIndex].x == left) { 22545a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(qIndex >> 1, 0, quad[qIndex]); 226fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 227fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (quad[qIndex].x == right) { 22845a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(qIndex >> 1, 1, quad[qIndex]); 229fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 230fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 231fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com} 232fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com 233fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comvoid addVerticalEndPoints(double top, double bottom, double x) 234fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com{ 235fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int qIndex = 0; qIndex < 3; qIndex += 2) { 236fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (quad[qIndex].x != x) { 237fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com continue; 238fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 239fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (quad[qIndex].y == top) { 24045a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(qIndex >> 1, 0, quad[qIndex]); 241fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 242fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (quad[qIndex].y == bottom) { 24345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com intersections.insert(qIndex >> 1, 1, quad[qIndex]); 244fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 245fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 246fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com} 247fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com 24827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comdouble findLineT(double t) { 249235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com double x, y; 250235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com xy_at_t(quad, t, x, y); 251235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com double dx = line[1].x - line[0].x; 252235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com double dy = line[1].y - line[0].y; 253235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com if (fabs(dx) > fabs(dy)) { 254235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com return (x - line[0].x) / dx; 255235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com } 256235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com return (y - line[0].y) / dy; 25727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com} 25827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 259fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.comvoid flip() { 260fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com // OPTIMIZATION: instead of swapping, pass original line, use [1].y - [0].y 261fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com int roots = intersections.fUsed; 262fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com for (int index = 0; index < roots; ++index) { 263fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com intersections.fT[1][index] = 1 - intersections.fT[1][index]; 264fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 265fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com} 266fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com 2671304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.comstatic bool pinTs(double& quadT, double& lineT) { 268fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (!approximately_one_or_less(lineT)) { 2693350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com return false; 2703350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com } 271fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com if (!approximately_zero_or_more(lineT)) { 272fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com return false; 273fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 2741304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (precisely_less_than_zero(quadT)) { 275fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com quadT = 0; 2761304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } else if (precisely_greater_than_one(quadT)) { 277fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com quadT = 1; 278fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com } 2791304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (precisely_less_than_zero(lineT)) { 2803350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com lineT = 0; 2811304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } else if (precisely_greater_than_one(lineT)) { 2823350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com lineT = 1; 2833350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com } 2843350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com return true; 2853350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com} 2863350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.com 28727accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comprivate: 28827accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com 28927accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comconst Quadratic& quad; 29027accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comconst _Line& line; 29127accef223a27fba437f5e825d99edbae20a045bcaryclark@google.comIntersections& intersections; 29227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com}; 293198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com 294fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com// utility for pairs of coincident quads 295fa0588ff672564af1c235a63589573829035a60bcaryclark@google.comstatic double horizontalIntersect(const Quadratic& quad, const _Point& pt) { 296fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com LineQuadraticIntersections q(quad, *((_Line*) 0), *((Intersections*) 0)); 297fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double rootVals[2]; 298fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com int roots = q.horizontalIntersect(pt.y, rootVals); 29932546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com for (int index = 0; index < roots; ++index) { 30032546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com double x; 301fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double t = rootVals[index]; 30232546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com xy_at_t(quad, t, x, *(double*) 0); 3036d0032a8ec680221c2a704cac2391f2a2d77546fcaryclark@google.com if (AlmostEqualUlps(x, pt.x)) { 30432546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com return t; 30532546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com } 306fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com } 307fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com return -1; 308fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com} 309fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com 310fa0588ff672564af1c235a63589573829035a60bcaryclark@google.comstatic double verticalIntersect(const Quadratic& quad, const _Point& pt) { 311fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com LineQuadraticIntersections q(quad, *((_Line*) 0), *((Intersections*) 0)); 312fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double rootVals[2]; 313fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com int roots = q.verticalIntersect(pt.x, rootVals); 31432546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com for (int index = 0; index < roots; ++index) { 31532546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com double y; 316fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double t = rootVals[index]; 31732546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com xy_at_t(quad, t, *(double*) 0, y); 3186d0032a8ec680221c2a704cac2391f2a2d77546fcaryclark@google.com if (AlmostEqualUlps(y, pt.y)) { 31932546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com return t; 32032546db1494a6c6433a7919844133a6ff5b5c7b2caryclark@google.com } 321fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com } 322fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com return -1; 323fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com} 324fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com 325fa0588ff672564af1c235a63589573829035a60bcaryclark@google.comdouble axialIntersect(const Quadratic& q1, const _Point& p, bool vertical) { 326fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com if (vertical) { 327fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com return verticalIntersect(q1, p); 328fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com } 329fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com return horizontalIntersect(q1, p); 330fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com} 331fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com 332198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.comint horizontalIntersect(const Quadratic& quad, double left, double right, 333198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com double y, double tRange[2]) { 334fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com LineQuadraticIntersections q(quad, *((_Line*) 0), *((Intersections*) 0)); 335fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com double rootVals[2]; 336fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com int result = q.horizontalIntersect(y, rootVals); 337198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com int tCount = 0; 338198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com for (int index = 0; index < result; ++index) { 339198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com double x, y; 340fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com xy_at_t(quad, rootVals[index], x, y); 341198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com if (x < left || x > right) { 342198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com continue; 343198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com } 344fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com tRange[tCount++] = rootVals[index]; 345198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com } 346198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com return tCount; 347198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com} 348198e054b33051a6cd5f606ccbc8d539cefc5631fcaryclark@google.com 349fa0588ff672564af1c235a63589573829035a60bcaryclark@google.comint horizontalIntersect(const Quadratic& quad, double left, double right, double y, 350fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com bool flipped, Intersections& intersections) { 351fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com LineQuadraticIntersections q(quad, *((_Line*) 0), intersections); 352fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com return q.horizontalIntersect(y, left, right, flipped); 353fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com} 354fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com 355fa0588ff672564af1c235a63589573829035a60bcaryclark@google.comint verticalIntersect(const Quadratic& quad, double top, double bottom, double x, 356fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com bool flipped, Intersections& intersections) { 357fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com LineQuadraticIntersections q(quad, *((_Line*) 0), intersections); 358fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com return q.verticalIntersect(x, top, bottom, flipped); 359fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com} 360fa0588ff672564af1c235a63589573829035a60bcaryclark@google.com 3613350c3c68ab75cd08721da3a938b8d2b10096d70caryclark@google.comint intersect(const Quadratic& quad, const _Line& line, Intersections& i) { 36227accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com LineQuadraticIntersections q(quad, line, i); 36327accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com return q.intersect(); 36427accef223a27fba437f5e825d99edbae20a045bcaryclark@google.com} 365235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com 366235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.comint intersectRay(const Quadratic& quad, const _Line& line, Intersections& i) { 367235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com LineQuadraticIntersections q(quad, line, i); 368fb51afb03e76c5701fffaa847584a8b7b2c18a7ecaryclark@google.com return q.intersectRay(i.fT[0]); 369235f56a92f6eb6accbb243e11b3c45e3798f38f2caryclark@google.com} 370