14 typedef TRect<BReal> BRect;
16 extern int Factorial(int n);
17 extern int BinomialCoeff(int n, int k);
18 extern BReal Bernstein(int k, int n, const BReal & u);
19 extern std::pair<BReal,BReal> BezierTurningPoints(const BReal & p0, const BReal & p1, const BReal & p2, const BReal & p3);
21 extern std::vector<BReal> SolveQuadratic(const BReal & a, const BReal & b, const BReal & c, const BReal & min = 0, const BReal & max = 1);
23 extern std::vector<BReal> SolveCubic(const BReal & a, const BReal & b, const BReal & c, const BReal & d, const BReal & min = 0, const BReal & max = 1, const BReal & delta = 1e-9);
25 /** A _cubic_ bezier. **/
33 typedef enum {UNKNOWN, LINE, QUADRATIC, CUSP, LOOP, SERPENTINE} Type;
36 //Bezier() = default; // Needed so we can fread/fwrite this struct... for now.
37 Bezier(BReal _x0=0, BReal _y0=0, BReal _x1=0, BReal _y1=0, BReal _x2=0, BReal _y2=0, BReal _x3=0, BReal _y3=0) : x0(_x0), y0(_y0), x1(_x1), y1(_y1), x2(_x2), y2(_y2), x3(_x3), y3(_y3), type(UNKNOWN)
44 if (type != Bezier::UNKNOWN)
46 // From Loop-Blinn 2005, with w0 == w1 == w2 == w3 = 1
47 // Transformed control points: (a0 = x0, b0 = y0)
48 BReal a1 = (x1-x0)*BReal(3);
49 BReal a2 = (x0- x1*BReal(2) +x2)*BReal(3);
50 BReal a3 = (x3 - x0 + (x1 - x2)*BReal(3));
52 BReal b1 = (y1-y0)*BReal(3);
53 BReal b2 = (y0- y1*BReal(2) +y2)*BReal(3);
54 BReal b3 = (y3 - y0 + (y1 - y2)*BReal(3));
56 // d vector (d0 = 0 since all w = 1)
57 BReal d1 = a2*b3 - a3*b2;
58 BReal d2 = a3*b1 - a1*b3;
59 BReal d3 = a1*b2 - a2*b1;
61 if (Abs(d1+d2+d3) < BReal(1e-6))
64 //Debug("LINE %s", Str().c_str());
68 BReal delta1 = -(d1*d1);
70 BReal delta3 = d1*d3 -(d2*d2);
71 if (Abs(delta1+delta2+delta3) < BReal(1e-6))
75 //Debug("QUADRATIC %s", Str().c_str());
79 BReal discriminant = d1*d3*BReal(4) -d2*d2;
80 if (Abs(discriminant) < BReal(1e-6))
83 //Debug("CUSP %s", Str().c_str());
85 else if (discriminant > BReal(0))
88 //Debug("SERPENTINE %s", Str().c_str());
93 //Debug("LOOP %s", Str().c_str());
95 //Debug("disc %.30f", discriminant);
100 std::string Str() const
103 s << "Bezier{" << Float(x0) << "," << Float(y0) << " -> " << Float(x1) << "," << Float(y1) << " -> " << Float(x2) << "," << Float(y2) << " -> " << Float(x3) << "," << Float(y3) << "}";
108 * Construct absolute control points using relative control points to a bounding rectangle
109 * ie: If cpy is relative to bounds rectangle, this will be absolute
111 Bezier(const Bezier & cpy, const BRect & t = BRect(0,0,1,1)) : x0(cpy.x0), y0(cpy.y0), x1(cpy.x1), y1(cpy.y1), x2(cpy.x2),y2(cpy.y2), x3(cpy.x3), y3(cpy.y3), type(cpy.type)
131 BRect SolveBounds() const;
133 std::pair<BReal,BReal> GetTop() const;
134 std::pair<BReal,BReal> GetBottom() const;
135 std::pair<BReal,BReal> GetLeft() const;
136 std::pair<BReal,BReal> GetRight() const;
138 Bezier ToAbsolute(const BRect & bounds) const
140 return Bezier(*this, bounds);
143 /** Convert absolute control points to control points relative to bounds
144 * (This basically does the opposite of the Copy constructor)
145 * ie: If this is absolute, the returned Bezier will be relative to the bounds rectangle
147 Bezier ToRelative(const BRect & bounds) const
149 // x' <- (x - x0)/w etc
150 // special cases when w or h = 0
151 // (So can't just use the Copy constructor on the inverse of bounds)
152 // BRect inverse = {-bounds.x/bounds.w, -bounds.y/bounds.h, BReal(1)/bounds.w, BReal(1)/bounds.h};
163 result.x0 = (x0 - bounds.x)/bounds.w;
164 result.x1 = (x1 - bounds.x)/bounds.w;
165 result.x2 = (x2 - bounds.x)/bounds.w;
166 result.x3 = (x3 - bounds.x)/bounds.w;
178 result.y0 = (y0 - bounds.y)/bounds.h;
179 result.y1 = (y1 - bounds.y)/bounds.h;
180 result.y2 = (y2 - bounds.y)/bounds.h;
181 result.y3 = (y3 - bounds.y)/bounds.h;
186 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
187 Bezier DeCasteljauSubdivideLeft(const BReal& t)
189 BReal one_minus_t = BReal(1) - t;
192 BReal x01 = x1*t + x0*one_minus_t;
193 BReal x12 = x2*t + x1*one_minus_t;
194 BReal x23 = x3*t + x2*one_minus_t;
196 BReal x012 = x12*t + x01*one_minus_t;
197 BReal x123 = x23*t + x12*one_minus_t;
199 BReal x0123 = x123*t + x012*one_minus_t;
202 BReal y01 = y1*t + y0*one_minus_t;
203 BReal y12 = y2*t + y1*one_minus_t;
204 BReal y23 = y3*t + y2*one_minus_t;
206 BReal y012 = y12*t + y01*one_minus_t;
207 BReal y123 = y23*t + y12*one_minus_t;
209 BReal y0123 = y123*t + y012*one_minus_t;
211 return Bezier(x0, y0, x01, y01, x012, y012, x0123, y0123);
213 // Performs one round of De Casteljau subdivision and returns the [t,1] part.
214 Bezier DeCasteljauSubdivideRight(const BReal& t)
216 BReal one_minus_t = BReal(1) - t;
219 BReal x01 = x1*t + x0*one_minus_t;
220 BReal x12 = x2*t + x1*one_minus_t;
221 BReal x23 = x3*t + x2*one_minus_t;
223 BReal x012 = x12*t + x01*one_minus_t;
224 BReal x123 = x23*t + x12*one_minus_t;
226 BReal x0123 = x123*t + x012*one_minus_t;
229 BReal y01 = y1*t + y0*one_minus_t;
230 BReal y12 = y2*t + y1*one_minus_t;
231 BReal y23 = y3*t + y2*one_minus_t;
233 BReal y012 = y12*t + y01*one_minus_t;
234 BReal y123 = y23*t + y12*one_minus_t;
236 BReal y0123 = y123*t + y012*one_minus_t;
238 return Bezier(x0123, y0123, x123, y123, x23, y23, x3, y3);
241 Bezier ReParametrise(const BReal& t0, const BReal& t1)
243 //Debug("Reparametrise: %f -> %f",Double(t0),Double(t1));
245 // Subdivide to get from [0,t1]
246 new_bezier = DeCasteljauSubdivideLeft(t1);
247 // Convert t0 from [0,1] range to [0, t1]
248 BReal new_t0 = t0 / t1;
249 //Debug("New t0 = %f", Double(new_t0));
250 new_bezier = new_bezier.DeCasteljauSubdivideRight(new_t0);
252 //Debug("%s becomes %s", this->Str().c_str(), new_bezier.Str().c_str());
256 std::vector<Bezier> ClipToRectangle(const BRect & r)
258 // Find points of intersection with the rectangle.
259 Debug("Clipping Bezier to BRect %s", r.Str().c_str());
263 std::vector<BReal> x_intersection = SolveXParam(r.x);
264 //Debug("Found %d intersections on left edge", x_intersection.size());
266 // And for the other side.
268 std::vector<BReal> x_intersection_pt2 = SolveXParam(r.x + r.w);
269 x_intersection.insert(x_intersection.end(), x_intersection_pt2.begin(), x_intersection_pt2.end());
270 //Debug("Found %d intersections on right edge (total x: %d)", x_intersection_pt2.size(), x_intersection.size());
273 std::vector<BReal> y_intersection = SolveYParam(r.y);
274 //Debug("Found %d intersections on top edge", y_intersection.size());
276 std::vector<BReal> y_intersection_pt2 = SolveYParam(r.y+r.h);
277 y_intersection.insert(y_intersection.end(), y_intersection_pt2.begin(), y_intersection_pt2.end());
278 //Debug("Found %d intersections on bottom edge (total y: %d)", y_intersection_pt2.size(), y_intersection.size());
281 x_intersection.insert(x_intersection.end(), y_intersection.begin(), y_intersection.end());
282 x_intersection.push_back(BReal(0));
283 x_intersection.push_back(BReal(1));
284 std::sort(x_intersection.begin(), x_intersection.end());
286 //Debug("Found %d intersections.\n", x_intersection.size());
287 /*for(auto t : x_intersection)
290 Evaluate(ptx, pty, t);
291 Debug("Root: t = %f, (%f,%f)", Double(t), Double(ptx), Double(pty));
294 std::vector<Bezier> all_beziers;
295 if (x_intersection.size() <= 2)
297 all_beziers.push_back(*this);
300 BReal t0 = *(x_intersection.begin());
301 for (auto it = x_intersection.begin()+1; it != x_intersection.end(); ++it)
304 if (t1 == t0) continue;
305 //Debug(" -- t0: %f to t1: %f: %f", Double(t0), Double(t1), Double((t1 + t0)/BReal(2)));
307 Evaluate(ptx, pty, ((t1 + t0) / BReal(2)));
308 if (r.PointIn(ptx, pty))
310 //Debug("Adding segment: (point at %f, %f)", Double(ptx), Double(pty));
311 all_beziers.push_back(this->ReParametrise(t0, t1));
315 //Debug("Segment removed (point at %f, %f)", Double(ptx), Double(pty));
322 /** Evaluate the Bezier at parametric parameter u, puts resultant point in (x,y) **/
323 void Evaluate(BReal & x, BReal & y, const BReal & u) const
326 for (unsigned i = 0; i < 4; ++i)
327 coeff[i] = Bernstein(i,3,u);
328 x = x0*coeff[0] + x1*coeff[1] + x2*coeff[2] + x3*coeff[3];
329 y = y0*coeff[0] + y1*coeff[1] + y2*coeff[2] + y3*coeff[3];
331 std::vector<Vec2> Evaluate(const std::vector<BReal> & u) const;
333 std::vector<BReal> SolveXParam(const BReal & x) const;
334 std::vector<BReal> SolveYParam(const BReal & x) const;
336 // Get points with same X
337 inline std::vector<Vec2> SolveX(const BReal & x) const
339 return Evaluate(SolveXParam(x));
341 // Get points with same Y
342 inline std::vector<Vec2> SolveY(const BReal & y) const
344 return Evaluate(SolveYParam(y));
347 bool operator==(const Bezier & equ) const
349 return (x0 == equ.x0 && y0 == equ.y0
350 && x1 == equ.x1 && y1 == equ.y1
351 && x2 == equ.x2 && y2 == equ.y2
352 && x3 == equ.x3 && y3 == equ.y3);
354 bool operator!=(const Bezier & equ) const {return !this->operator==(equ);}