T2 in CGAL and T2Viewer

Extension of CGAL to T2

What is it about

Keywords: Computational Geometry, Dynamic Visualization, Oriented Projective Plane

In the Institute of Computing at UNICAMP (State University of Campinas), we have extended the Computational Geometry Algorithms Library (CGAL) to allow for the implementation of geometric algorithms on the Oriented Projective Plane T2.

The oriented projective plane T2 is an extension of the Euclidean plane E2 and comprises a number of advantages for algorithm design and implementation. It consists of an alternative geometric model that combines the elegance and efficiency of projective geometry with the consistent handling of oriented lines and planes, signed angles, segments, convex sets, and many other concepts that the classical theory doesn't support. The value of this model for practical computing is well known.

T2 can be viewed by means of two very insightful models. By representing each point with signed homogeneous coordinates [w,x,y] of R3*, one naturally arrives at the planar model of T2 which consists of two copies of E2, representing the two ranges of T2 (w>0 and w<0), and a circle S1, representing the points at infinity (w=0). An equally natural visualization of T2 is the spherical model which consists of the surface of the sphere S2 whose points (w,x,y) belong to the front range when w>0, to the back range when w<0 and to the line at infinity when w=0.


bullet You may read more about T2 in:
  1. This book: J. Stolfi. Oriented Projective Geometry: A Framework for Geometric Computations. Acad. Press, Inc., 1991.
  2. This book: P. J. de Rezende and J. Stolfi. Fundamentos de Geometria Computacional. UFPE-DI, IV Escola de Computação, 1994.
Extension of CGAL to the Oriented Projective Plane
CGAL is a library of computational geometry algorithms developed by a consortium of universities. At UNICAMP, CGAL has been extended in order to allow for the implementation of geometric algorithms in the Oriented Projective Plane, T2.

In order to benefit from the extension of CGAL to the oriented projective plane in the context of the classroom, the need for visualization arose. So, we developed a dynamic visualization system, T2Viewer.

What is available

You may download:

bullet The source code for the Extension of CGAL to T2 (including the geometric primitives, predicates and a few algorithms);
bullet The source code for the Dynamic Visualization System T2Viewer;
bullet A couple of Videos that illustrate the use of T2Viewer.

Contact Information

If you have questions not answered in these pages, feel free to contact us. However, we should point out that all the documentation we have available is posted here as well as all the sources we have.

People involved
  • Alessandra G. de Oliveira
  • Fábio P. Selmi-Dei
  • Pedro J. de Rezende
  • Electronic mail
    General Information:
    Postal address
    Institute of Computing, UNICAMP, Campinas SP, Brazil
    +55 19 3521-5860
    +55 19 3521-5847


    (c) 1998-2008 Pedro J. de Rezende. Last modified: 2008.08.06.