Abstract
A method for determining loci without using a deep algebraic background is presented. It uses pseudodivision techniques (Wu's algorithm). The key idea is to make the hypothesis conditions depend on an indeterminate point, X. When forcing the thesis condition to be a consequence of hypothesis conditions, a new condition involving X appears. That condition leads to the locus. The method is applied to prove a new theorem: the generalization of Simson-Steiner Theorem to 3D.
Partially supported by project DGES PB96-0098-C04-03 (Spain).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Cox, J. Little, D. O’shea: Ideals, Varieties, and Algorithms. Springer, 1991.
S. C. Chou: Mechanical Geometry Theorem Proving. Reidel, 1988.
M. de Guzmán: An Extension of the Wallace-Simson Theorem: Projecting in Arbitrary Directions. Mathematical Monthly, 106,6, June 1999 (pages 574–580).
D. Kapur: Automated Geometric Reasoning: Dixon Resultants, Groebner Bases and Characteristic Sets. In: D. Wang (ed.): Proceedings International Workshop on Automated Deduction in Geometry. Springer-Verlag, LNAI 1360.
D. Kapur, J.L. Mundy: Wu’s method and its application to perspective viewing. In: D. Kapur and J.L. Mundy (eds.): Geometric Reasoning. MIT Press, 1989.
W. Koepf: Gröbner bases and triangles. Int. J. of Comput. Algebra in Math. Education, 4,4, 1998 (pages 371–386).
T. Recio, M. P. Vélez: Automatic Discovery of Theorems in Elementary Geometry. Journal of Automated Reasoning, 23, 1999(pages 63–82).
E. Roanes M., E. Roanes L.: Cálculos autom′aticos por ordenador con MapleV.5. Rubiños, 1999.
E. Roanes M., E. Roanes L.: BÚsqueda Automática de Lugares Geométricos. Bol. Soc. Puig Adam de Profs. de Matemáticas, 53, 1999(pages 67–77).
W. T. Wu: Mechanical Theorem Proving in Geometries. Text and Monographs in Symbolic Computation. Springer-Verlag, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Roanes-Macías, E., Roanes-Lozano, E. (2001). Automatic Determination of Geometric Loci. 3D-Extension of Simson-Steiner Theorem. In: Campbell, J.A., Roanes-Lozano, E. (eds) Artificial Intelligence and Symbolic Computation. AISC 2000. Lecture Notes in Computer Science(), vol 1930. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44990-6_12
Download citation
DOI: https://doi.org/10.1007/3-540-44990-6_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42071-2
Online ISBN: 978-3-540-44990-4
eBook Packages: Springer Book Archive