dts2 - Spherical (constrained) delaunay triangulations on the unit sphere

Introduction

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 snapped on the sphere using libratss. It essentially 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 as well.

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 simplifies many tasks in the construction of algorithm. Since we want to map the sphere on the 2D triangulation we have to deal with these peculiarities. 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 north-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.

Scope

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.

Terminology

• Snapping is the process of converting an input point to a rational point that is exactly on the unit sphere
• Distances are measured in maximum norm.

Snap options

libratss can use multiple floating point computation back ends in conjunction with different snapping modes.

• PLANE or SPHERE uses mpfr as back end, option -c gives the number of mantissa bits of the used floating point representation.
• PAPER uses CGAL's bundled Core 1.x to compute points that obey the requested quality measure
• PAPER2 the same as PAPER but uses Core 2. Can also compute snapped points from spherical/geo coordinates with quality guarantees

These can be combined with different snap modes:

• FX uses fix-point snapping and guarantees denominators of roughly twice the requested precision

• FL uses floating-point snapping which may produce large results for points close to zero

• CF uses the continuos-fraction algorithm to compute a snapped point

• JP uses the jacobi-perron algorithm (only useful for 3D data)

• FPLLL uses fplll to compute a simultaneous approximation

• BRUTE_FORCE uses brute force to compute a simultaneous approximation

• GUARANTEE_SIZE is used in conjunction with JP, CF, FPLLL and ensures that the resulting point has at most 2n+1 many bits

• GUARANTEE_DISTANCE is used in conjunction with JP, CF, FPLLL and ensures the the resulting point is at most eps=2^-n away from the input point

Dependecies

• CGAL
• gmpxx
• mpfr

Quickstart guide

Command line tools

git clone --recursive <https://github.com/fmi-alg/libdts2.git> libdts2
cd libdts2 && mkdir build && cd build
cmake -DCMAKE_BUILD_TYPE=lto ../
cd dts2tools && make
./triang -h

Graphs from OpenStreetMap data

There is a tool to create triangulations from OpenStreetMap street data:

1. Download and build https://github.com/fmi-alg/OsmGraphCreator.git

2. Get appropriate data in osm.pbf format

3. Create graph of the type topotext and add the --no-reverse-edge switch

   ./creator -g topotext -c configs/car.cfg --no-reverse-edge data.osm.pbf -o graph.topog

4. For triang select geo as an input type: -if geo

triang -t cxspk64 -go none -gi ne -io ne --verbose --progress -c 53 -p 31 -if geo -of geo -s FX\|PLANE -i graph.topog

Visualizing graphs

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

1. Download and build the dev-branch of https://github.com/invor/simplestGraphRendering
2. create an output of type -go 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 you can simply link to dts2
3. 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);

License

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.