double Bernsteins::secant(Bezier bz) { double s = 0, t = 1; double e = 1e-14; int side = 0; double r, fr, fs = bz.at0(), ft = bz.at1(); for (size_t n = 0; n < 100; ++n) { r = (fs*t - ft*s) / (fs - ft); if (fabs(t-s) < e * fabs(t+s)) { debug(std::cout << "error small " << fabs(t-s) << ", accepting solution " << r << "after " << n << "iterations\n"); return r; } fr = horner(bz, r); if (fr * ft > 0) { t = r; ft = fr; if (side == -1) fs /= 2; side = -1; } else if (fs * fr > 0) { s = r; fs = fr; if (side == +1) ft /= 2; side = +1; } else break; } return r; }
// suggested by Sederberg. double Bernsteins::horner(Bezier bz, double t) { double u, tn, tmp; u = 1.0 - t; tn = 1.0; tmp = bz.at0() * u; for(size_t i = 1; i < bz.degree(); ++i) { tn *= t; tmp = (tmp + tn*choose<double>(bz.order(), (unsigned)i)*bz[i]) * u; } return (tmp + tn*t*bz.at1()); }
vector<double> find_all_roots(Bezier b) { vector<double> rts = b.roots(); if(b.at0() == 0) rts.push_back(0); if(b.at1() == 0) rts.push_back(1); return rts; }
if(a.size() != b.size()) return; for(int i = 0; i < a.size(); i++) { EXPECT_FLOAT_EQ(a[i], b[i]); } } vector<double> find_all_roots(Bezier b) { vector<double> rts = b.roots(); if(b.at0() == 0) rts.push_back(0); if(b.at1() == 0) rts.push_back(1); return rts; } TEST_F(ChainTest, Deflate) { Bezier b = array_roots(vector_from_array((const double[]){0,0.25,0.5})); EXPECT_FLOAT_EQ(0, b.at0()); b = b.deflate(); EXPECT_FLOAT_EQ(0, b.valueAt(0.25)); b = b.subdivide(0.25).second; EXPECT_FLOAT_EQ(0, b.at0()); b = b.deflate(); const double rootposition = (0.5-0.25) / (1-0.25); EXPECT_FLOAT_EQ(0, b.valueAt(rootposition)); b = b.subdivide(rootposition).second; EXPECT_FLOAT_EQ(0, b.at0()); } TEST_F(ChainTest, Roots) { expect_array((const double[]){0.5}, wiggle.roots()); Bezier bigun(Bezier::Order(30));
void Bernsteins::find_bernstein_roots(Bezier bz, unsigned depth, double left_t, double right_t) { debug(std::cout << left_t << ", " << right_t << std::endl); size_t n_crossings = 0; int old_sign = SGN(bz[0]); //std::cout << "w[0] = " << bz[0] << std::endl; int sign; for (size_t i = 1; i < bz.size(); i++) { //std::cout << "w[" << i << "] = " << w[i] << std::endl; sign = SGN(bz[i]); if (sign != 0) { if (sign != old_sign && old_sign != 0) { ++n_crossings; } old_sign = sign; } } //std::cout << "n_crossings = " << n_crossings << std::endl; if (n_crossings == 0) return; // no solutions here if (n_crossings == 1) /* Unique solution */ { //std::cout << "depth = " << depth << std::endl; /* Stop recursion when the tree is deep enough */ /* if deep enough, return 1 solution at midpoint */ if (depth > MAX_DEPTH) { //printf("bottom out %d\n", depth); const double Ax = right_t - left_t; const double Ay = bz.at1() - bz.at0(); solutions.push_back(left_t - Ax*bz.at0() / Ay); return; } double r = secant(bz); solutions.push_back(r*right_t + (1-r)*left_t); return; } /* Otherwise, solve recursively after subdividing control polygon */ Bezier::Order o(bz); Bezier Left(o), Right = bz; double split_t = (left_t + right_t) * 0.5; // If subdivision is working poorly, split around the leftmost root of the derivative if (depth > 2) { debug(std::cout << "derivative mode\n"); Bezier dbz = derivative(bz); debug(std::cout << "initial = " << dbz << std::endl); std::vector<double> dsolutions = dbz.roots(Interval(left_t, right_t)); debug(std::cout << "dsolutions = " << dsolutions << std::endl); double dsplit_t = 0.5; if(!dsolutions.empty()) { dsplit_t = dsolutions[0]; split_t = left_t + (right_t - left_t)*dsplit_t; debug(std::cout << "split_value = " << bz(split_t) << std::endl); debug(std::cout << "spliting around " << dsplit_t << " = " << split_t << "\n"); } std::pair<Bezier, Bezier> LR = bz.subdivide(dsplit_t); Left = LR.first; Right = LR.second; } else { // split at midpoint, because it is cheap Left[0] = Right[0]; for (size_t i = 1; i < bz.size(); ++i) { for (size_t j = 0; j < bz.size()-i; ++j) { Right[j] = (Right[j] + Right[j+1]) * 0.5; } Left[i] = Right[0]; } } debug(std::cout << "Solution is exactly on the subdivision point.\n"); debug(std::cout << Left << " , " << Right << std::endl); Left = reverse(Left); while(Right.order() > 0 and fabs(Right[0]) <= 1e-10) { debug(std::cout << "deflate\n"); Right = Right.deflate(); Left = Left.deflate(); solutions.push_back(split_t); } Left = reverse(Left); if (Right.order() > 0) { debug(std::cout << Left << " , " << Right << std::endl); find_bernstein_roots(Left, depth+1, left_t, split_t); find_bernstein_roots(Right, depth+1, split_t, right_t); } }