Library to compute constrained delaunay triangulations with intersections on the sphere using CGAL
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This is a small library to compute a delaunay triangulation of points that are on the sphere. If input points are not exactly on the sphere then they are snaped on the sphere using libratss. It essentialy provides an adaptor class for the (Delaunay) triangulation algorithms that are provided by CGAL. Any other triangulation algorithms/data structures following the triangulation concepts of CGAL should work aswell.

Basic idea

The 2D triangulation data structure of CGAL distinguishes between inner faces and outer faces. An inner face is a triangle that is within the convex hull of the point set. An outer face on the other hand consists of two vertices of the convex hull and the infinite vertex. This design simplifys many tasks in the construction of algorithm. Since we want to map the sphere on the 2D triangulation we have to deal with these pecularities. To this end we introduce 4 new vertices into our initial triangulation. One is at the south-pole and the other 3 form a small triangle around the north-pole. Furthermore we place the infinite vertex at the nort-pole. In essence we now have 3 inner faces each having the south-pole as a vertex and two vertices of the northern points. Furthermore there are 3 outer faces connected to the infinite vertex at the north-pole and two vertices of the northern points. We now implement all necessary predicates based on this setting. The downside of this is of course that it is not possible to add points that would fall inside of the outer faces. It is also not possible to create a triangulation that does not have these initial triangles and points.


This library currently supports Delaunay and Constrained Delaunay Triangulations without intersecting segments. There is also support for Constrained Delaunay Triangulations where segments are allowed to intersect. There is the possibility to use exact arithmetic to compute the intersecting point using CORE::Expr. Approximate intersections are of course faster.

Things to consider

Computing points that are exactly on the sphere is hard. libdts2 relies on libratss to compute points on the sphere given a point that is near the sphere. Usually one has to deal with geographic coordinates or spherical coordinates. These have to be mapped to cartesian coordinates in 3D. Since floating-point computations are inexact this will not yield a point that is on the sphere. The distance to the sphere depends on the precision of the computation. The higher the precision, the closer the point to the sphere, the closer the snapped point to the original point.

Usually a computation precision of 53 is more than enough to represent points on a sphere representing our earth. This translates to a Point occupying 3 * 2 * 8 Bytes.


  • CGAL
  • gmpxx
  • mpfr

Quickstart guide

Building a triangulation

  1. git clone --recursive libdts2
  2. cd libdts2 && mkdir build && cd build
  3. cmake -DCMAKE_BUILD_TYPE=lto ../
  4. cd dts2tools && make
  5. ./triang -h

Getting sample graphs

There is a tool to create sample graphs out of OpenStreetMap data:

  1. Download and build
  2. Create graph of the type topotext and add the --no-reverse-edge switch
  3. For triang select geo as an input type: -if geo

Visualizing graphs

There is a tool to visualize graphs create by the triang tool:

  1. Download and build the dev-branch of
  2. create an output of type -g simplest_andre
  3. Start simplestGraphRendering with the options -f sg

Use the library

  1. add the repository to your project, make sure to checkout the subprojects of libdts2
  2. make sure to add the appropriate compile definitions to your project. If you're using CMake and you're adding libdts2 as a subdirectory then these are exported. See the CMakeLists.txt for a list of exported variables.
  3. add <libdts2/Constrained_delaunay_triangulation_s2.h> as include
  4. select an appropriate triangulation class, see below for a small example

Input formats

Currently input points can be any of:

  • ratss::GeoCoord provides a small wrapper to handle wgs84 coordinates
  • ratss::SphericalCoord provides a small wrapper to handle spherical coordinates
  • Triangulation::Point provides a 3d Point

A small example

//This file has predefined typedefs for various triangulations.
#include <libdts2/Constrained_delaunay_triangulation_s2.h>

using namespace dts2;

//The following triangulation supports in-exact intersections and can store information in its vertices and faces:

template<typename VertexInfo = void, typename FaceInfo = void>
using MyTriangulation = Constrained_Delaunay_triangulation_with_inexact_intersections_with_info_s2<VertexInfo, FaceInfo>;

int significands = 53;
MyTriangulation<void, void> tr(significands);
auto vh1 = tr.insert(ratss::GeoCoord(48.0, 8.0));
auto vh2 = tr.insert(ratss::GeoCoord(49.0, 9.0));
auto vh3 = tr.insert(ratss::GeoCoord(48.0, 9.0));
auto vh4 = tr.insert(ratss::GeoCoord(49.0, 8.0));
tr.insert(vh1, vh2);
tr.insert(vh3, vh4);


libdts2 is licensed under LGPL v2.1 Since libdts2 links with mpfr, mpfr C++, gmp, gmpxx and cgal the resulting binaries are usually governed by the GPL v3