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Issue No.02 - March/April (2003 vol.23)

pp: 68-78

Daniel Fontijne , University of Amsterdam

Leo Dorst , University of Amsterdam

ABSTRACT

<p>Computations of 3D Euclidean geometry can be performed using various computational models of different effectiveness. In this article, the authors compare five alternatives: 3D linear algebra, 3D geometric algebra, a mix of 4D homogeneous coordinates and Pl?cker coordinates, a 4D homogeneous model using geometric algebra, and the 5D conformal model using geometric algebra. Higher dimensional models and models using geometric algebra can express geometric primitives, computations, and constructions more elegantly, but this elegance might come at a performance penalty. The authors explore these issues using the implementation of a simple ray tracer as a practical goal and guide and show how to implement the most important geometric computations of the ray-tracing algorithm using each of the five models as well as benchmark each implementation.</p>

CITATION

Daniel Fontijne, Leo Dorst, "Modeling 3D Euclidean Geometry",

*IEEE Computer Graphics and Applications*, vol.23, no. 2, pp. 68-78, March/April 2003, doi:10.1109/MCG.2003.1185582REFERENCES

- 3. S. Mann, N. Litke,, and T. DeRose,
A Coordinate Free Geometry ADT, research report CS-97-15, Computer Science Dept., Univ. of Waterloo, 1997.- 4. T. DeRose,
Coordinate-Free Geometric Programming, tech. report 89-09-16, Dept. of Computer Science, Univ. of Washington, Sept. 1989.- 7. D. Hestenes, H. Li,, and A. Rockwood,"A Unified Algebraic Framework for Classical Geometry,"
Geometric Computing with Clifford Algebra, G. Sommer, ed., Springer, 1999, http://modelingnts.la.asu.edu/htmlUAFCG.html . - 9. D. Fontijne, T. Bouma,, and L. Dorst,
Gaigen: a Geometric Algebra Implementation Generator, http://www.science.uva.nl/~fontijneraytracer . |