\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Spherical Geometry Kernel

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Circular_kernel_3/functor_compare_theta_3.cpp
#include <CGAL/Exact_spherical_kernel_3.h>
typedef CGAL::Exact_spherical_kernel_3 SK;
int main(){
//construction of 3 spheres from their centers and squared radii
SK::Sphere_3 s1(SK::Point_3(0,0,0),2);
SK::Sphere_3 s2(SK::Point_3(0,1,0),1);
SK::Sphere_3 s3(SK::Point_3(1,0,0),3);
//construct two circles lying on sphere s1
SK::Circle_3 C1(s1,s2);
SK::Circle_3 C2(s1,s3);
SK::Intersect_3 inter;
//create a functor to compare theta-coordinates on sphere s1
SK::Compare_theta_z_3 cmp(s1);
std::vector< CGAL::Object > intersections;
inter(C1,C2,std::back_inserter(intersections));
//unsigned integer indicates multiplicity of intersection point
std::pair<SK::Circular_arc_point_3,unsigned> p1=
CGAL::object_cast< std::pair<SK::Circular_arc_point_3,unsigned> >(intersections[0]);
std::pair<SK::Circular_arc_point_3,unsigned> p2=
CGAL::object_cast< std::pair<SK::Circular_arc_point_3,unsigned> >(intersections[1]);
SK::Circular_arc_point_3 t_extreme[2];
//Compute theta extremal points of circle C1 on sphere s1
CGAL::theta_extremal_points(C1,s1,t_extreme);
//The theta coordinates of theta extremal points of C1 enclose that of each intersection point.
assert(cmp(t_extreme[0],p1.first)==CGAL::SMALLER);
assert(cmp(t_extreme[0],p2.first)==CGAL::SMALLER);
assert(cmp(t_extreme[1],p1.first)==CGAL::LARGER);
assert(cmp(t_extreme[1],p2.first)==CGAL::LARGER);
return 0;
}