2D Circular Geometry Kernel ( Geometry Kernels） CGAL 4.13 -User Manual

1 Introduction

The goal of the circular kernel is to offer to the user a large set of functionalities on circles and circular arcs in the plane. All the choices (interface, robustness, representation, and so on) made here are consistent with the choices made in the CGAL kernel, for which we refer the user to the 2D kernel manual.

In this first release, all functionalities necessary for computing an arrangement of circular arcs and these line segments are defined. Three traits classes are provided for the CGAL arrangement package.

circular kernel 的目标是提供一个用于平面圆和圆弧的大量的函数集。本版中，所有计算圆弧和这些线段排列的函数全部提供。三个traits类提供给了这个CGAL的arrangement 包。

2 Software Design

The design is done in such a way that the algebraic concepts and the geometric concepts are clearly separated. Circular_kernel_2 has therefore two template parameters:

• the first parameter must model the CGAL three dimensional Kernel concept. The circular kernel derives from it, and it provides all elementary geometric objects like points, lines, circles, and elementary functionality on them.
• the second parameter is the algebraic kernel, which is responsible for computations on polynomials and algebraic numbers. It has to be a model of concept AlgebraicKernelForCircles. The robustness of the package relies on the fact that the algebraic kernel provides exact computations on algebraic objects.

The circular kernel uses the extensibility scheme presented in the 2D kernel manual (see Section Extensible Kernel). The types of Kernel are inherited by the circular kernel and some types are taken from the AlgebraicKernelForCircles parameter. Three new main geometric objects are introduced by Circular_kernel_2: circular arcs, points of circular arcs (used in particular for endpoints of arcs and intersection points between arcs) and line segments whose endpoints are points of this new type.

In fact, the circular kernel is documented as a concept, CircularKernel, and two models are provided:

• Circular_kernel_2<Kernel,AlgebraicKernelForCircles>, the basic kernel,
• and a predefined filtered kernel Exact_circular_kernel_2, that is based on similar techniques as Exact_predicates_exact_constructions_kernel.

设计将代数概念和几何概念截然分开，所以 Circular_kernel_2 类有两个模板参数：

　　第一个参数必须是CGAL 3 维 kernel概念的模型。本circular kernel 由此概念继承而来， 它提供了所有基本的几何体，包括点、线、圆和它们的基本方程。

　　第二个参数是代数内核（algebraic kernel），它负责计算多项式和代数数。它必须是AlgebraicKernelForCircles概念的模型。本包的健壮性依赖于代数内核提供的对代数对象的精确计算。

circular kernel 使用2D内核手册中提到的可扩展的scheme（see Section Extensible Kernel）。Kernel的类型由 circular kernel继承，有些类型来自于AlgebraicKernelForCircles参数。三个新的几何对象由 Circular_kernel_2引入：圆弧（circular arcs），圆弧的点（points of circular arcs，主要用于弧的端点和弧与弧的交点），和端点是圆弧上点的线段（ line segments whose endpoints are points of this new type）。

实际上，circular kernel 是一个概念CircularKernel，它提供了2个模型：


　　Circular_kernel_2<Kernel,AlgebraicKernelForCircles>是基本内核

　　一个预先定义的过滤内核（filtered kernel）Exact_circular_kernel_2，它是基于与Exact_predicates_exact_constructions_kernel相似的技术。

3 Examples

The first example shows how to construct circles or circular arcs from points, and how to compute intersections between them using the global function.

这个例子展示如何创建一个弧，如何通过全局函数求弧的交点。

File Circular_kernel_2/intersecting_arcs.cpp

#include <CGAL/Exact_circular_kernel_2.h>

#include <CGAL/point_generators_2.h>

 

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;

}

 

 

The following example shows how to use a functor of the kernel.

下面 例子演示如何使用内核中的一个函子。

File Circular_kernel_2/functor_has_on_2.cpp

#include <CGAL/Exact_circular_kernel_2.h>

#include <CGAL/point_generators_2.h>

 

typedef CGAL::Exact_circular_kernel_2 Circular_k;

 

typedef CGAL::Point_2<Circular_k> Point_2;

typedef CGAL::Circular_arc_2<Circular_k> Circular_arc_2;

 

int main()

{

int n = 0;

Circular_arc_2 c = Circular_arc_2(Point_2(10,0), Point_2(5,5), Point_2(0, 0));

 

for(int i = 0; i <= 10; i++) {

for(int j = 0; j <= 10; j++) {

Point_2 p = Point_2(i, j);

if(Circular_k().has_on_2_object()(c,p)) {

n++;

std::cout << "(" << i << "," << j << ")" << std::endl;

}

}

}

std::cout << "There are " << n << " points in the [0,..,10]x[0,..,10] "

<< "grid on the circular" << std::endl

<< " arc defined counterclockwisely by the points (0,0), (5,5), (10,0)"

<< std::endl << "See the points above." << std::endl;

return 0;

}

4 Design and Implementation History

The first pieces of prototype code were comparisons of algebraic numbers of degree 2, written by Olivier Devillers [1],cgal:dfmt-amafe-02.

Some work was then done in the direction of a "kernel" for CGAL.^[1] and the first design emerged in [2].

The code of this package was initially written by Sylvain Pion and Monique Teillaud who also wrote the manual. Athanasios Kakargias had worked on a prototype version of this kernel in 2003. Julien Hazebrouck participated in the implementation in July and August

1. The contribution of Pedro Machado Manhães de Castro in summer 2006 improved significantly the efficiency of this kernel. He also added more functionality in 2008.

This work was partially supported by the IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-2000-26473 (ECG - Effective Computational Geometry for Curves and Surfaces) and by the IST Programme of the 6th Framework Programme of the EU as a STREP (FET Open Scheme) Project under Contract No IST-006413 (ACS - Algorithms for Complex Shapes).

1. ^Monique Teillaud, First Prototype of a CGAL Geometric Kernel with Circular Arcs, Technical Report ECG-TR-182203-01, 2002 Sylvain Pion and Monique Teillaud, Towards a CGAL-like kernel for curves, Technical Report ECG-TR-302206-01, 2003