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320 lines
13 KiB
C++
320 lines
13 KiB
C++
// This file is part of libigl, a simple c++ geometry processing library.
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//
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// Copyright (C) 2016 Michael Rabinovich
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//
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// This Source Code Form is subject to the terms of the Mozilla Public License
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// v. 2.0. If a copy of the MPL was not distributed with this file, You can
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// obtain one at http://mozilla.org/MPL/2.0/.
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#include "flip_avoiding_line_search.h"
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#include "line_search.h"
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#include "PI.h"
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#include <Eigen/Dense>
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#include <vector>
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namespace igl
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{
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namespace flip_avoiding
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{
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//---------------------------------------------------------------------------
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// x - array of size 3
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// In case 3 real roots: => x[0], x[1], x[2], return 3
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// 2 real roots: x[0], x[1], return 2
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// 1 real root : x[0], x[1] ± i*x[2], return 1
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// http://math.ivanovo.ac.ru/dalgebra/Khashin/poly/index.html
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IGL_INLINE int SolveP3(std::vector<double>& x,double a,double b,double c)
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{ // solve cubic equation x^3 + a*x^2 + b*x + c
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using namespace std;
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double a2 = a*a;
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double q = (a2 - 3*b)/9;
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double r = (a*(2*a2-9*b) + 27*c)/54;
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double r2 = r*r;
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double q3 = q*q*q;
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double A,B;
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if(r2<q3)
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{
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double t=r/sqrt(q3);
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if( t<-1) t=-1;
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if( t> 1) t= 1;
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t=acos(t);
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a/=3; q=-2*sqrt(q);
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x[0]=q*cos(t/3)-a;
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x[1]=q*cos((t+(2*igl::PI))/3)-a;
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x[2]=q*cos((t-(2*igl::PI))/3)-a;
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return(3);
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}
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else
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{
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A =-pow(fabs(r)+sqrt(r2-q3),1./3);
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if( r<0 ) A=-A;
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B = A==0? 0 : B=q/A;
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a/=3;
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x[0] =(A+B)-a;
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x[1] =-0.5*(A+B)-a;
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x[2] = 0.5*sqrt(3.)*(A-B);
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if(fabs(x[2])<1e-14)
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{
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x[2]=x[1]; return(2);
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}
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return(1);
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}
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}
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IGL_INLINE double get_smallest_pos_quad_zero(double a,double b, double c)
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{
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using namespace std;
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double t1, t2;
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if(std::abs(a) > 1.0e-10)
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{
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double delta_in = pow(b, 2) - 4 * a * c;
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if(delta_in <= 0)
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{
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return INFINITY;
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}
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double delta = sqrt(delta_in); // delta >= 0
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if(b >= 0) // avoid subtracting two similar numbers
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{
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double bd = - b - delta;
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t1 = 2 * c / bd;
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t2 = bd / (2 * a);
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}
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else
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{
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double bd = - b + delta;
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t1 = bd / (2 * a);
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t2 = (2 * c) / bd;
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}
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assert (std::isfinite(t1));
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assert (std::isfinite(t2));
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if(a < 0) std::swap(t1, t2); // make t1 > t2
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// return the smaller positive root if it exists, otherwise return infinity
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if(t1 > 0)
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{
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return t2 > 0 ? t2 : t1;
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}
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else
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{
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return INFINITY;
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}
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}
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else
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{
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if(b == 0) return INFINITY; // just to avoid divide-by-zero
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t1 = -c / b;
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return t1 > 0 ? t1 : INFINITY;
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}
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}
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IGL_INLINE double get_min_pos_root_2D(const Eigen::MatrixXd& uv,
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const Eigen::MatrixXi& F,
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Eigen::MatrixXd& d,
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int f)
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{
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using namespace std;
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/*
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Finding the smallest timestep t s.t a triangle get degenerated (<=> det = 0)
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The following code can be derived by a symbolic expression in matlab:
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Symbolic matlab:
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U11 = sym('U11');
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U12 = sym('U12');
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U21 = sym('U21');
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U22 = sym('U22');
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U31 = sym('U31');
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U32 = sym('U32');
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V11 = sym('V11');
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V12 = sym('V12');
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V21 = sym('V21');
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V22 = sym('V22');
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V31 = sym('V31');
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V32 = sym('V32');
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t = sym('t');
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U1 = [U11,U12];
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U2 = [U21,U22];
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U3 = [U31,U32];
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V1 = [V11,V12];
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V2 = [V21,V22];
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V3 = [V31,V32];
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A = [(U2+V2*t) - (U1+ V1*t)];
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B = [(U3+V3*t) - (U1+ V1*t)];
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C = [A;B];
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solve(det(C), t);
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cf = coeffs(det(C),t); % Now cf(1),cf(2),cf(3) holds the coefficients for the polynom. at order c,b,a
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*/
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int v1 = F(f,0); int v2 = F(f,1); int v3 = F(f,2);
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// get quadratic coefficients (ax^2 + b^x + c)
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const double& U11 = uv(v1,0);
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const double& U12 = uv(v1,1);
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const double& U21 = uv(v2,0);
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const double& U22 = uv(v2,1);
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const double& U31 = uv(v3,0);
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const double& U32 = uv(v3,1);
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const double& V11 = d(v1,0);
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const double& V12 = d(v1,1);
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const double& V21 = d(v2,0);
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const double& V22 = d(v2,1);
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const double& V31 = d(v3,0);
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const double& V32 = d(v3,1);
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double a = V11*V22 - V12*V21 - V11*V32 + V12*V31 + V21*V32 - V22*V31;
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double b = U11*V22 - U12*V21 - U21*V12 + U22*V11 - U11*V32 + U12*V31 + U31*V12 - U32*V11 + U21*V32 - U22*V31 - U31*V22 + U32*V21;
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double c = U11*U22 - U12*U21 - U11*U32 + U12*U31 + U21*U32 - U22*U31;
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return get_smallest_pos_quad_zero(a,b,c);
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}
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IGL_INLINE double get_min_pos_root_3D(const Eigen::MatrixXd& uv,
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const Eigen::MatrixXi& F,
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Eigen::MatrixXd& direc,
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int f)
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{
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using namespace std;
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/*
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Searching for the roots of:
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+-1/6 * |ax ay az 1|
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|bx by bz 1|
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|cx cy cz 1|
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|dx dy dz 1|
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Every point ax,ay,az has a search direction a_dx,a_dy,a_dz, and so we add those to the matrix, and solve the cubic to find the step size t for a 0 volume
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Symbolic matlab:
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syms a_x a_y a_z a_dx a_dy a_dz % tetrahedera point and search direction
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syms b_x b_y b_z b_dx b_dy b_dz
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syms c_x c_y c_z c_dx c_dy c_dz
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syms d_x d_y d_z d_dx d_dy d_dz
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syms t % Timestep var, this is what we're looking for
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a_plus_t = [a_x,a_y,a_z] + t*[a_dx,a_dy,a_dz];
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b_plus_t = [b_x,b_y,b_z] + t*[b_dx,b_dy,b_dz];
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c_plus_t = [c_x,c_y,c_z] + t*[c_dx,c_dy,c_dz];
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d_plus_t = [d_x,d_y,d_z] + t*[d_dx,d_dy,d_dz];
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vol_mat = [a_plus_t,1;b_plus_t,1;c_plus_t,1;d_plus_t,1]
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//cf = coeffs(det(vol_det),t); % Now cf(1),cf(2),cf(3),cf(4) holds the coefficients for the polynom
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[coefficients,terms] = coeffs(det(vol_det),t); % terms = [ t^3, t^2, t, 1], Coefficients hold the coeff we seek
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*/
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int v1 = F(f,0); int v2 = F(f,1); int v3 = F(f,2); int v4 = F(f,3);
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const double& a_x = uv(v1,0);
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const double& a_y = uv(v1,1);
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const double& a_z = uv(v1,2);
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const double& b_x = uv(v2,0);
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const double& b_y = uv(v2,1);
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const double& b_z = uv(v2,2);
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const double& c_x = uv(v3,0);
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const double& c_y = uv(v3,1);
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const double& c_z = uv(v3,2);
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const double& d_x = uv(v4,0);
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const double& d_y = uv(v4,1);
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const double& d_z = uv(v4,2);
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const double& a_dx = direc(v1,0);
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const double& a_dy = direc(v1,1);
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const double& a_dz = direc(v1,2);
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const double& b_dx = direc(v2,0);
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const double& b_dy = direc(v2,1);
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const double& b_dz = direc(v2,2);
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const double& c_dx = direc(v3,0);
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const double& c_dy = direc(v3,1);
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const double& c_dz = direc(v3,2);
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const double& d_dx = direc(v4,0);
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const double& d_dy = direc(v4,1);
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const double& d_dz = direc(v4,2);
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// Find solution for: a*t^3 + b*t^2 + c*d +d = 0
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double a = a_dx*b_dy*c_dz - a_dx*b_dz*c_dy - a_dy*b_dx*c_dz + a_dy*b_dz*c_dx + a_dz*b_dx*c_dy - a_dz*b_dy*c_dx - a_dx*b_dy*d_dz + a_dx*b_dz*d_dy + a_dy*b_dx*d_dz - a_dy*b_dz*d_dx - a_dz*b_dx*d_dy + a_dz*b_dy*d_dx + a_dx*c_dy*d_dz - a_dx*c_dz*d_dy - a_dy*c_dx*d_dz + a_dy*c_dz*d_dx + a_dz*c_dx*d_dy - a_dz*c_dy*d_dx - b_dx*c_dy*d_dz + b_dx*c_dz*d_dy + b_dy*c_dx*d_dz - b_dy*c_dz*d_dx - b_dz*c_dx*d_dy + b_dz*c_dy*d_dx;
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double b = a_dy*b_dz*c_x - a_dy*b_x*c_dz - a_dz*b_dy*c_x + a_dz*b_x*c_dy + a_x*b_dy*c_dz - a_x*b_dz*c_dy - a_dx*b_dz*c_y + a_dx*b_y*c_dz + a_dz*b_dx*c_y - a_dz*b_y*c_dx - a_y*b_dx*c_dz + a_y*b_dz*c_dx + a_dx*b_dy*c_z - a_dx*b_z*c_dy - a_dy*b_dx*c_z + a_dy*b_z*c_dx + a_z*b_dx*c_dy - a_z*b_dy*c_dx - a_dy*b_dz*d_x + a_dy*b_x*d_dz + a_dz*b_dy*d_x - a_dz*b_x*d_dy - a_x*b_dy*d_dz + a_x*b_dz*d_dy + a_dx*b_dz*d_y - a_dx*b_y*d_dz - a_dz*b_dx*d_y + a_dz*b_y*d_dx + a_y*b_dx*d_dz - a_y*b_dz*d_dx - a_dx*b_dy*d_z + a_dx*b_z*d_dy + a_dy*b_dx*d_z - a_dy*b_z*d_dx - a_z*b_dx*d_dy + a_z*b_dy*d_dx + a_dy*c_dz*d_x - a_dy*c_x*d_dz - a_dz*c_dy*d_x + a_dz*c_x*d_dy + a_x*c_dy*d_dz - a_x*c_dz*d_dy - a_dx*c_dz*d_y + a_dx*c_y*d_dz + a_dz*c_dx*d_y - a_dz*c_y*d_dx - a_y*c_dx*d_dz + a_y*c_dz*d_dx + a_dx*c_dy*d_z - a_dx*c_z*d_dy - a_dy*c_dx*d_z + a_dy*c_z*d_dx + a_z*c_dx*d_dy - a_z*c_dy*d_dx - b_dy*c_dz*d_x + b_dy*c_x*d_dz + b_dz*c_dy*d_x - b_dz*c_x*d_dy - b_x*c_dy*d_dz + b_x*c_dz*d_dy + b_dx*c_dz*d_y - b_dx*c_y*d_dz - b_dz*c_dx*d_y + b_dz*c_y*d_dx + b_y*c_dx*d_dz - b_y*c_dz*d_dx - b_dx*c_dy*d_z + b_dx*c_z*d_dy + b_dy*c_dx*d_z - b_dy*c_z*d_dx - b_z*c_dx*d_dy + b_z*c_dy*d_dx;
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double c = a_dz*b_x*c_y - a_dz*b_y*c_x - a_x*b_dz*c_y + a_x*b_y*c_dz + a_y*b_dz*c_x - a_y*b_x*c_dz - a_dy*b_x*c_z + a_dy*b_z*c_x + a_x*b_dy*c_z - a_x*b_z*c_dy - a_z*b_dy*c_x + a_z*b_x*c_dy + a_dx*b_y*c_z - a_dx*b_z*c_y - a_y*b_dx*c_z + a_y*b_z*c_dx + a_z*b_dx*c_y - a_z*b_y*c_dx - a_dz*b_x*d_y + a_dz*b_y*d_x + a_x*b_dz*d_y - a_x*b_y*d_dz - a_y*b_dz*d_x + a_y*b_x*d_dz + a_dy*b_x*d_z - a_dy*b_z*d_x - a_x*b_dy*d_z + a_x*b_z*d_dy + a_z*b_dy*d_x - a_z*b_x*d_dy - a_dx*b_y*d_z + a_dx*b_z*d_y + a_y*b_dx*d_z - a_y*b_z*d_dx - a_z*b_dx*d_y + a_z*b_y*d_dx + a_dz*c_x*d_y - a_dz*c_y*d_x - a_x*c_dz*d_y + a_x*c_y*d_dz + a_y*c_dz*d_x - a_y*c_x*d_dz - a_dy*c_x*d_z + a_dy*c_z*d_x + a_x*c_dy*d_z - a_x*c_z*d_dy - a_z*c_dy*d_x + a_z*c_x*d_dy + a_dx*c_y*d_z - a_dx*c_z*d_y - a_y*c_dx*d_z + a_y*c_z*d_dx + a_z*c_dx*d_y - a_z*c_y*d_dx - b_dz*c_x*d_y + b_dz*c_y*d_x + b_x*c_dz*d_y - b_x*c_y*d_dz - b_y*c_dz*d_x + b_y*c_x*d_dz + b_dy*c_x*d_z - b_dy*c_z*d_x - b_x*c_dy*d_z + b_x*c_z*d_dy + b_z*c_dy*d_x - b_z*c_x*d_dy - b_dx*c_y*d_z + b_dx*c_z*d_y + b_y*c_dx*d_z - b_y*c_z*d_dx - b_z*c_dx*d_y + b_z*c_y*d_dx;
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double d = a_x*b_y*c_z - a_x*b_z*c_y - a_y*b_x*c_z + a_y*b_z*c_x + a_z*b_x*c_y - a_z*b_y*c_x - a_x*b_y*d_z + a_x*b_z*d_y + a_y*b_x*d_z - a_y*b_z*d_x - a_z*b_x*d_y + a_z*b_y*d_x + a_x*c_y*d_z - a_x*c_z*d_y - a_y*c_x*d_z + a_y*c_z*d_x + a_z*c_x*d_y - a_z*c_y*d_x - b_x*c_y*d_z + b_x*c_z*d_y + b_y*c_x*d_z - b_y*c_z*d_x - b_z*c_x*d_y + b_z*c_y*d_x;
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if (std::abs(a)<=1.e-10)
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{
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return get_smallest_pos_quad_zero(b,c,d);
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}
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b/=a; c/=a; d/=a; // normalize it all
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std::vector<double> res(3);
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int real_roots_num = SolveP3(res,b,c,d);
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switch (real_roots_num)
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{
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case 1:
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return (res[0] >= 0) ? res[0]:INFINITY;
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case 2:
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{
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double max_root = std::max(res[0],res[1]); double min_root = std::min(res[0],res[1]);
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if (min_root > 0) return min_root;
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if (max_root > 0) return max_root;
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return INFINITY;
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}
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case 3:
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default:
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{
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std::sort(res.begin(),res.end());
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if (res[0] > 0) return res[0];
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if (res[1] > 0) return res[1];
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if (res[2] > 0) return res[2];
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return INFINITY;
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}
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}
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}
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IGL_INLINE double compute_max_step_from_singularities(const Eigen::MatrixXd& uv,
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const Eigen::MatrixXi& F,
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Eigen::MatrixXd& d)
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{
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using namespace std;
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double max_step = INFINITY;
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// The if statement is outside the for loops to avoid branching/ease parallelizing
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if (uv.cols() == 2)
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{
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for (int f = 0; f < F.rows(); f++)
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{
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double min_positive_root = get_min_pos_root_2D(uv,F,d,f);
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max_step = std::min(max_step, min_positive_root);
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}
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}
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else
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{ // volumetric deformation
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for (int f = 0; f < F.rows(); f++)
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{
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double min_positive_root = get_min_pos_root_3D(uv,F,d,f);
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max_step = std::min(max_step, min_positive_root);
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}
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}
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return max_step;
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}
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}
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}
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IGL_INLINE double igl::flip_avoiding_line_search(
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const Eigen::MatrixXi F,
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Eigen::MatrixXd& cur_v,
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Eigen::MatrixXd& dst_v,
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std::function<double(Eigen::MatrixXd&)> energy,
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double cur_energy)
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{
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using namespace std;
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Eigen::MatrixXd d = dst_v - cur_v;
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double min_step_to_singularity = igl::flip_avoiding::compute_max_step_from_singularities(cur_v,F,d);
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double max_step_size = std::min(1., min_step_to_singularity*0.8);
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return igl::line_search(cur_v,d,max_step_size, energy, cur_energy);
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}
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#ifdef IGL_STATIC_LIBRARY
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#endif
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