T2 in CGAL and T2Viewer

T2 in CGAL and T2Viewer

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.

References

You may read more about T² in:

This book: J. Stolfi. Oriented Projective Geometry: A Framework for Geometric Computations. Acad. Press, Inc., 1991.
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.
T²Viewer: 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:

	The source code for the Extension of CGAL to T² (including the geometric primitives, predicates and a few algorithms);
	The source code for the Dynamic Visualization System T²Viewer;
	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
Telephone: +55 19 3521-5860
FAX: +55 19 3521-5847