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

CGAL 4.14.2 - 2D Circular Geometry Kernel
Circular_kernel_2/intersecting_arcs.cpp
#include <CGAL/Exact_circular_kernel_2.h>
#include <CGAL/point_generators_2.h>
#include <iostream>
typedef CGAL::Exact_circular_kernel_2 Circular_k;
typedef CGAL::Point_2<Circular_k> Point_2;
typedef CGAL::Circle_2<Circular_k> Circle_2;
typedef CGAL::Circular_arc_2<Circular_k> Circular_arc_2;
template <typename T>
double prob_2() {
CGAL::Random_points_in_square_2<Point_2> g(1.0);
double prob = 0.0;
for (int i = 0; i < 10000; i++) {
Point_2 p1, p2, p3, p4, p5, p6;
p1 = *g++; p2 = *g++; p3 = *g++;
p4 = *g++; p5 = *g++; p6 = *g++;
// the pi's are points inherited from the Cartesian kernel Point_2, so,
// the orientation predicate can be called on them
if(CGAL::orientation(p1, p2, p3) != CGAL::COUNTERCLOCKWISE) std::swap(p1, p3);
T o1 = T(p1, p2, p3);
if(CGAL::orientation(p4, p5, p6) != CGAL::COUNTERCLOCKWISE) std::swap(p4, p6);
T o2 = T(p4, p5, p6);
typedef typename CGAL::CK2_Intersection_traits<Circular_k, T, T>::type
Intersection_result;
std::vector<Intersection_result> res;
CGAL::intersection(o1, o2, std::back_inserter(res));
prob += (res.size() != 0) ? 1.0 : 0.0;
}
return prob/10000.0;
}
int main()
{
std::cout << "What is the probability that two arcs formed by" << std::endl;
std::cout << "three random counterclockwise-oriented points on" << std::endl;
std::cout << "an unit square intersect? (wait a second please)" << std::endl;
std::cout << "The probability is: " << prob_2<Circular_arc_2>() <<
std::endl << std::endl;
std::cout << "And what about the probability that two circles formed by"
<< std::endl;
std::cout << "three random counterclockwise-oriented points on" << std::endl;
std::cout << "an unit square intersect? (wait a second please)" << std::endl;
std::cout << "The probability is: " << prob_2<Circle_2>() << std::endl;
return 0;
}