608 lines
18 KiB
C++
608 lines
18 KiB
C++
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#include "tcurveutil.h"
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#include "tcurves.h"
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#include "tmathutil.h"
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#include "tbezier.h"
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//=============================================================================
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/*
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This function returns a vector of
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pairs of double (DoublePair) which identifies the parameters
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of the points of intersection.
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The integer returned is the number of intersections which
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have been identified (for two segments).
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If the segments are parallel to the parameter it is set to -1.
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*/
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int intersect(const TSegment &first, const TSegment &second,
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std::vector<DoublePair> &intersections) {
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return intersect(first.getP0(), first.getP1(), second.getP0(), second.getP1(),
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intersections);
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}
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int intersect(const TPointD &p1, const TPointD &p2, const TPointD &p3,
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const TPointD &p4, std::vector<DoublePair> &intersections) {
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// This algorithm is presented in Graphics Geems III pag 199
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static double Ax, Bx, Ay, By, Cx, Cy, d, f, e;
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static double x1lo, x1hi, y1lo, y1hi;
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Ax = p2.x - p1.x;
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Bx = p3.x - p4.x;
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// test delle BBox
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if (Ax < 0.0) {
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x1lo = p2.x;
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x1hi = p1.x;
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} else {
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x1lo = p1.x;
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x1hi = p2.x;
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}
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if (Bx > 0.0) {
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if (x1hi < p4.x || x1lo > p3.x) return 0;
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} else if (x1hi < p3.x || x1lo > p4.x)
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return 0;
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Ay = p2.y - p1.y;
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By = p3.y - p4.y;
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if (Ay < 0) {
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y1lo = p2.y;
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y1hi = p1.y;
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} else {
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y1lo = p1.y;
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y1hi = p2.y;
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}
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if (By > 0) {
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if (y1hi < p4.y || y1lo > p3.y) return 0;
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} else if (y1hi < p3.y || y1lo > p4.y)
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return 0;
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Cx = p1.x - p3.x;
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Cy = p1.y - p3.y;
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d = By * Cx - Bx * Cy;
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f = Ay * Bx - Ax * By;
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e = Ax * Cy - Ay * Cx;
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if (f > 0) {
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if (d < 0) return 0;
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if (!areAlmostEqual(d, f))
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if (d > f) return 0;
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if (e < 0) return 0;
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if (!areAlmostEqual(e, f))
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if (e > f) return 0;
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} else if (f < 0) {
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if (d > 0) return 0;
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if (!areAlmostEqual(d, f))
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if (d < f) return 0;
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if (e > 0) return 0;
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if (!areAlmostEqual(e, f))
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if (e < f) return 0;
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} else {
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if (d < 0 || d > 1 || e < 0 || e > 1) return 0;
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if (p1 == p2 && p3 == p4) {
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intersections.push_back(DoublePair(0, 0));
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return 1;
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}
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// Check that the segments are not on the same line.
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if (!cross(p2 - p1, p4 - p1)) {
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// Calculation of Barycentric combinations.
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double distp2p1 = norm2(p2 - p1);
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double distp3p4 = norm2(p3 - p4);
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double dist2_p3p1 = norm2(p3 - p1);
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double dist2_p4p1 = norm2(p4 - p1);
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double dist2_p3p2 = norm2(p3 - p2);
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double dist2_p4p2 = norm2(p4 - p2);
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int intersection = 0;
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// Calculation of the first two solutions.
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double vol1;
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if (distp3p4) {
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distp3p4 = sqrt(distp3p4);
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vol1 = (p1 - p3) * normalize(p4 - p3);
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if (vol1 >= 0 && vol1 <= distp3p4) // Barycentric combinations valid
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{
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intersections.push_back(DoublePair(0.0, vol1 / distp3p4));
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++intersection;
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}
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vol1 = (p2 - p3) * normalize(p4 - p3);
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if (vol1 >= 0 && vol1 <= distp3p4) {
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intersections.push_back(DoublePair(1.0, vol1 / distp3p4));
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++intersection;
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}
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}
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if (distp2p1) {
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distp2p1 = sqrt(distp2p1);
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vol1 = (p3 - p1) * normalize(p2 - p1);
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if (dist2_p3p2 && dist2_p3p1)
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if (vol1 >= 0 && vol1 <= distp2p1) {
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intersections.push_back(DoublePair(vol1 / distp2p1, 0.0));
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++intersection;
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}
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vol1 = (p4 - p1) * normalize(p2 - p1);
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if (dist2_p4p2 && dist2_p4p1)
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if (vol1 >= 0 && vol1 <= distp2p1) {
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intersections.push_back(DoublePair(vol1 / distp2p1, 1.0));
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++intersection;
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}
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}
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return intersection;
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}
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return -1;
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}
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double par_s = d / f;
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double par_t = e / f;
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intersections.push_back(DoublePair(par_s, par_t));
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return 1;
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}
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//------------------------------------------------------------------------------------------------------------
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int intersectCloseControlPoints(const TQuadratic &c0, const TQuadratic &c1,
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std::vector<DoublePair> &intersections);
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int intersect(const TQuadratic &c0, const TQuadratic &c1,
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std::vector<DoublePair> &intersections, bool checksegments) {
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int ret;
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// Works baddly, sometimes patch intersections...
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if (checksegments) {
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ret = intersectCloseControlPoints(c0, c1, intersections);
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if (ret != -2) return ret;
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}
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double a = c0.getP0().x - 2 * c0.getP1().x + c0.getP2().x;
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double b = 2 * (c0.getP1().x - c0.getP0().x);
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double d = c0.getP0().y - 2 * c0.getP1().y + c0.getP2().y;
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double e = 2 * (c0.getP1().y - c0.getP0().y);
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double coeff = b * d - a * e;
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int i = 0;
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if (areAlmostEqual(coeff, 0.0)) // c0 is a Segment, or a single point!!!
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{
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TSegment s = TSegment(c0.getP0(), c0.getP2());
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ret = intersect(s, c1, intersections);
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if (a == 0 && d == 0) // values of t in s coincide with values of t in c0
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return ret;
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for (i = intersections.size() - ret; i < (int)intersections.size(); i++) {
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intersections[i].first = c0.getT(s.getPoint(intersections[i].first));
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}
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return ret;
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}
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double c = c0.getP0().x;
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double f = c0.getP0().y;
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double g = c1.getP0().x - 2 * c1.getP1().x + c1.getP2().x;
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double h = 2 * (c1.getP1().x - c1.getP0().x);
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double k = c1.getP0().x;
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double m = c1.getP0().y - 2 * c1.getP1().y + c1.getP2().y;
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double p = 2 * (c1.getP1().y - c1.getP0().y);
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double q = c1.getP0().y;
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if (areAlmostEqual(h * m - g * p,
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0.0)) // c1 is a Segment, or a single point!!!
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{
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TSegment s = TSegment(c1.getP0(), c1.getP2());
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ret = intersect(c0, s, intersections);
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if (g == 0 && m == 0) // values of t in s coincide with values of t in c0
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return ret;
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for (i = intersections.size() - ret; i < (int)intersections.size(); i++) {
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intersections[i].second = c1.getT(s.getPoint(intersections[i].second));
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}
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return ret;
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}
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double a2 = (g * d - a * m);
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double b2 = (h * d - a * p);
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double c2 = ((k - c) * d + (f - q) * a);
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coeff = 1.0 / coeff;
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double A = (a * a + d * d) * coeff * coeff;
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double aux = A * c2 + (a * b + d * e) * coeff;
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std::vector<double> t;
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std::vector<double> solutions;
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t.push_back(aux * c2 + a * c + d * f - k * a - d * q);
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aux += A * c2;
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t.push_back(aux * b2 - h * a - d * p);
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t.push_back(aux * a2 + A * b2 * b2 - g * a - d * m);
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t.push_back(2 * A * a2 * b2);
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t.push_back(A * a2 * a2);
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rootFinding(t, solutions);
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// solutions.push_back(0.0); //per convenzione; un valore vale l'altro....
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for (i = 0; i < (int)solutions.size(); i++) {
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if (solutions[i] < 0) {
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if (areAlmostEqual(solutions[i], 0, 1e-6))
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solutions[i] = 0;
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else
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continue;
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} else if (solutions[i] > 1) {
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if (areAlmostEqual(solutions[i], 1, 1e-6))
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solutions[i] = 1;
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else
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continue;
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}
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DoublePair tt;
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tt.second = solutions[i];
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tt.first = coeff * (tt.second * (a2 * tt.second + b2) + c2);
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if (tt.first < 0) {
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if (areAlmostEqual(tt.first, 0, 1e-6))
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tt.first = 0;
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else
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continue;
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} else if (tt.first > 1) {
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if (areAlmostEqual(tt.first, 1, 1e-6))
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tt.first = 1;
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else
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continue;
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}
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intersections.push_back(tt);
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assert(areAlmostEqual(c0.getPoint(tt.first), c1.getPoint(tt.second), 1e-1));
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}
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return intersections.size();
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}
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//=============================================================================
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// This function checks whether the control point
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// p1 is very close to p0 or p2.
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// In this case, we are approximated to the quadratic p0-p2 segment.
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// If p1 is near p0, the relationship between the original and the quadratic
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// segment:
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// tq = sqrt(ts),
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// If p1 is near p2, instead it's:
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// tq = 1-sqrt(1-ts).
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int intersectCloseControlPoints(const TQuadratic &c0, const TQuadratic &c1,
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std::vector<DoublePair> &intersections) {
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int ret = -2;
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double dist1 = tdistance2(c0.getP0(), c0.getP1());
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if (dist1 == 0) dist1 = 1e-20;
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double dist2 = tdistance2(c0.getP1(), c0.getP2());
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if (dist2 == 0) dist2 = 1e-20;
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double val0 = std::max(dist1, dist2) / std::min(dist1, dist2);
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double dist3 = tdistance2(c1.getP0(), c1.getP1());
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if (dist3 == 0) dist3 = 1e-20;
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double dist4 = tdistance2(c1.getP1(), c1.getP2());
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if (dist4 == 0) dist4 = 1e-20;
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double val1 = std::max(dist3, dist4) / std::min(dist3, dist4);
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if (val0 > 1000000 &&
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val1 > 1000000) // both c0 and c1 approximated by segments
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{
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TSegment s0 = TSegment(c0.getP0(), c0.getP2());
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TSegment s1 = TSegment(c1.getP0(), c1.getP2());
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ret = intersect(s0, s1, intersections);
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for (UINT i = intersections.size() - ret; i < (int)intersections.size();
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i++) {
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intersections[i].first = (dist1 < dist2)
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? sqrt(intersections[i].first)
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: 1 - sqrt(1 - intersections[i].first);
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intersections[i].second = (dist3 < dist4)
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? sqrt(intersections[i].second)
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: 1 - sqrt(1 - intersections[i].second);
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}
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// return ret;
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} else if (val0 > 1000000) // c0 only approximated segment
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{
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TSegment s0 = TSegment(c0.getP0(), c0.getP2());
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ret = intersect(s0, c1, intersections);
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for (UINT i = intersections.size() - ret; i < (int)intersections.size();
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i++)
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intersections[i].first = (dist1 < dist2)
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? sqrt(intersections[i].first)
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: 1 - sqrt(1 - intersections[i].first);
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// return ret;
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} else if (val1 > 1000000) // only c1 approximated segment
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{
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TSegment s1 = TSegment(c1.getP0(), c1.getP2());
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ret = intersect(c0, s1, intersections);
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for (UINT i = intersections.size() - ret; i < (int)intersections.size();
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i++)
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intersections[i].second = (dist3 < dist4)
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? sqrt(intersections[i].second)
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: 1 - sqrt(1 - intersections[i].second);
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// return ret;
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}
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/*
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if (ret!=-2)
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{
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std::vector<DoublePair> intersections1;
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int ret1 = intersect(c0, c1, intersections1, false);
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if (ret1>ret)
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{
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intersections = intersections1;
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return ret1;
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}
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}
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*/
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return ret;
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}
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//=============================================================================
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int intersect(const TQuadratic &q, const TSegment &s,
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std::vector<DoublePair> &intersections, bool firstIsQuad) {
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int solutionNumber = 0;
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// Note the line `a*x+b*y+c = 0` we search for solutions
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// di a*x(t)+b*y(t)+c=0 in [0,1]
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double a = s.getP0().y - s.getP1().y, b = s.getP1().x - s.getP0().x,
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c = -(a * s.getP0().x + b * s.getP0().y);
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// se il segmento e' un punto
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if (0.0 == a && 0.0 == b) {
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double outParForQuad = q.getT(s.getP0());
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if (areAlmostEqual(q.getPoint(outParForQuad), s.getP0())) {
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if (firstIsQuad)
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intersections.push_back(DoublePair(outParForQuad, 0));
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else
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intersections.push_back(DoublePair(0, outParForQuad));
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return 1;
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}
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return 0;
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}
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if (q.getP2() - q.getP1() ==
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q.getP1() - q.getP0()) { // the second is a segment....
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if (firstIsQuad)
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return intersect(TSegment(q.getP0(), q.getP2()), s, intersections);
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else
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return intersect(s, TSegment(q.getP0(), q.getP2()), intersections);
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}
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std::vector<TPointD> bez, pol;
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bez.push_back(q.getP0());
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bez.push_back(q.getP1());
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bez.push_back(q.getP2());
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bezier2poly(bez, pol);
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std::vector<double> poly_1(3, 0), sol;
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poly_1[0] = a * pol[0].x + b * pol[0].y + c;
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poly_1[1] = a * pol[1].x + b * pol[1].y;
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poly_1[2] = a * pol[2].x + b * pol[2].y;
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if (!(rootFinding(poly_1, sol))) return 0;
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double segmentPar, solution;
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TPointD v10(s.getP1() - s.getP0());
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for (UINT i = 0; i < sol.size(); ++i) {
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solution = sol[i];
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if ((0.0 <= solution && solution <= 1.0) ||
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areAlmostEqual(solution, 0.0, 1e-6) ||
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areAlmostEqual(solution, 1.0, 1e-6)) {
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segmentPar = (q.getPoint(solution) - s.getP0()) * v10 / (v10 * v10);
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if ((0.0 <= segmentPar && segmentPar <= 1.0) ||
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areAlmostEqual(segmentPar, 0.0, 1e-6) ||
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areAlmostEqual(segmentPar, 1.0, 1e-6)) {
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TPointD p1 = q.getPoint(solution);
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TPointD p2 = s.getPoint(segmentPar);
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assert(areAlmostEqual(p1, p2, 1e-1));
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if (firstIsQuad)
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intersections.push_back(DoublePair(solution, segmentPar));
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else
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intersections.push_back(DoublePair(segmentPar, solution));
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solutionNumber++;
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}
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}
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}
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return solutionNumber;
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}
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//=============================================================================
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bool isCloseToSegment(const TPointD &point, const TSegment &segment,
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double distance) {
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TPointD a = segment.getP0();
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TPointD b = segment.getP1();
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double length2 = tdistance2(a, b);
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if (length2 < tdistance2(a, point) || length2 < tdistance2(point, b))
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return false;
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if (a.x == b.x) return fabs(point.x - a.x) <= distance;
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if (a.y == b.y) return fabs(point.y - a.y) <= distance;
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// y=mx+q
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double m = (a.y - b.y) / (a.x - b.x);
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double q = a.y - (m * a.x);
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double d2 = pow(fabs(point.y - (m * point.x) - q), 2) / (1 + (m * m));
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return d2 <= distance * distance;
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}
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//=============================================================================
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double tdistance(const TSegment &segment, const TPointD &point) {
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TPointD v1 = segment.getP1() - segment.getP0();
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TPointD v2 = point - segment.getP0();
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TPointD v3 = point - segment.getP1();
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if (v2 * v1 <= 0)
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return tdistance(point, segment.getP0());
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else if (v3 * v1 >= 0)
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return tdistance(point, segment.getP1());
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return fabs(v2 * rotate90(normalize(v1)));
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}
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//-----------------------------------------------------------------------------
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/*
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This formule is derived from Graphic Gems pag. 600
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e = h^2 |a|/8
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e = pixel size
|
|
h = step
|
|
a = acceleration of curve (for a quadratic is a costant value)
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|
*/
|
|
|
|
double computeStep(const TQuadratic &quad, double pixelSize) {
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|
double step = 2;
|
|
|
|
TPointD A = quad.getP0() - 2.0 * quad.getP1() +
|
|
quad.getP2(); // 2*A is the acceleration of the curve
|
|
|
|
double A_len = norm(A);
|
|
|
|
/*
|
|
A_len is equal to 2*norm(a)
|
|
pixelSize will be 0.5*pixelSize
|
|
now h is equal to sqrt( 8 * 0.5 * pixelSize / (2*norm(a)) ) = sqrt(2) * sqrt(
|
|
pixelSize/A_len )
|
|
*/
|
|
|
|
if (A_len > 0) step = sqrt(2 * pixelSize / A_len);
|
|
|
|
return step;
|
|
}
|
|
|
|
//-----------------------------------------------------------------------------
|
|
|
|
double computeStep(const TThickQuadratic &quad, double pixelSize) {
|
|
TThickPoint cp0 = quad.getThickP0(), cp1 = quad.getThickP1(),
|
|
cp2 = quad.getThickP2();
|
|
|
|
TQuadratic q1(TPointD(cp0.x, cp0.y), TPointD(cp1.x, cp1.y),
|
|
TPointD(cp2.x, cp2.y)),
|
|
q2(TPointD(cp0.y, cp0.thick), TPointD(cp1.y, cp1.thick),
|
|
TPointD(cp2.y, cp2.thick)),
|
|
q3(TPointD(cp0.x, cp0.thick), TPointD(cp1.x, cp1.thick),
|
|
TPointD(cp2.x, cp2.thick));
|
|
|
|
return std::min({computeStep(q1, pixelSize), computeStep(q2, pixelSize),
|
|
computeStep(q3, pixelSize)});
|
|
}
|
|
|
|
//=============================================================================
|
|
|
|
/*
|
|
Explanation: The length of a Bezier quadratic can be calculated explicitly.
|
|
|
|
Let Q be the quadratic. The tricks to explicitly integrate | Q'(t) | are:
|
|
|
|
- The integrand can be reformulated as: | Q'(t) | = sqrt(at^2 + bt + c);
|
|
- Complete the square beneath the sqrt (add/subtract sq(b) / 4a)
|
|
and perform a linear variable change. We reduce the integrand to:
|
|
sqrt(kx^2 + k),
|
|
where k can be taken outside => sqrt(x^2 + 1)
|
|
- Use x = tan y. The integrand will yield sec^3 y.
|
|
- Integrate by parts. In short, the resulting primitive of sqrt(x^2 + 1) is:
|
|
|
|
F(x) = ( x * sqrt(x^2 + 1) + log(x + sqrt(x^2 + 1)) ) / 2;
|
|
*/
|
|
|
|
void TQuadraticLengthEvaluator::setQuad(const TQuadratic &quad) {
|
|
const TPointD &p0 = quad.getP0();
|
|
const TPointD &p1 = quad.getP1();
|
|
const TPointD &p2 = quad.getP2();
|
|
|
|
TPointD speed0(2.0 * (p1 - p0));
|
|
TPointD accel(2.0 * (p2 - p1) - speed0);
|
|
|
|
double a = accel * accel;
|
|
double b = 2.0 * accel * speed0;
|
|
m_c = speed0 * speed0;
|
|
|
|
m_constantSpeed = isAlmostZero(a); // => b isAlmostZero, too
|
|
if (m_constantSpeed) {
|
|
m_c = sqrt(m_c);
|
|
return;
|
|
}
|
|
|
|
m_sqrt_a_div_2 = 0.5 * sqrt(a);
|
|
|
|
m_noSpeed0 = isAlmostZero(m_c); // => b isAlmostZero, too
|
|
if (m_noSpeed0) return;
|
|
|
|
m_tRef = 0.5 * b / a;
|
|
double d = m_c - 0.5 * b * m_tRef;
|
|
|
|
m_squareIntegrand = (d < TConsts::epsilon);
|
|
if (m_squareIntegrand) {
|
|
m_f = (b > 0) ? -sq(m_tRef) : sq(m_tRef);
|
|
return;
|
|
}
|
|
|
|
m_e = d / a;
|
|
|
|
double sqrt_part = sqrt(sq(m_tRef) + m_e);
|
|
double log_arg = m_tRef + sqrt_part;
|
|
|
|
m_squareIntegrand = (log_arg < TConsts::epsilon);
|
|
if (m_squareIntegrand) {
|
|
m_f = (b > 0) ? -sq(m_tRef) : sq(m_tRef);
|
|
return;
|
|
}
|
|
|
|
m_primitive_0 = m_sqrt_a_div_2 * (m_tRef * sqrt_part + m_e * log(log_arg));
|
|
}
|
|
|
|
//-----------------------------------------------------------------------------
|
|
|
|
double TQuadraticLengthEvaluator::getLengthAt(double t) const {
|
|
if (m_constantSpeed) return m_c * t;
|
|
|
|
if (m_noSpeed0) return m_sqrt_a_div_2 * sq(t);
|
|
|
|
if (m_squareIntegrand) {
|
|
double t_plus_tRef = t + m_tRef;
|
|
return m_sqrt_a_div_2 *
|
|
(m_f + ((t_plus_tRef > 0) ? sq(t_plus_tRef) : -sq(t_plus_tRef)));
|
|
}
|
|
|
|
double y = t + m_tRef;
|
|
double sqrt_part = sqrt(sq(y) + m_e);
|
|
double log_arg =
|
|
y + sqrt_part; // NOTE: log_arg >= log_arg0 >= TConsts::epsilon
|
|
|
|
return m_sqrt_a_div_2 * (y * sqrt_part + m_e * log(log_arg)) - m_primitive_0;
|
|
}
|
|
|
|
//-----------------------------------------------------------------------------
|
|
// End Of File
|
|
//-----------------------------------------------------------------------------
|