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Complex brackets, balanced complex differences, and applications in symbolic geometric computing

Authors:

Hongbo Li,

Lei HuangAuthors Info & Claims

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

Pages 181 - 188

https://doi.org/10.1145/1390768.1390795

Published: 20 July 2008 Publication History

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Abstract

In advanced invariant algebras such as null bracket algebra (NBA), symmetries of algebraic operators are the most important devices of encoding and employing syzygies of advanced geometric invariants. The larger the symmetry group, the more powerful the computing devices. In this paper, the largest symmetry group of the two kinds of bracket operators in the NBA of plane geometry is found. An algorithm of complexity O(N log N) is proposed to reduce a bracket of length N to its normal form, and then decide the congruence of two brackets of length N. By writing the two bracket operators as the real and pure imaginary parts of a complex bracket operator, their normal forms can be translated into a class of complex polynomials whose variables are first-order differences, called balanced complex difference (BCD) polynomials. BCD polynomials provide a complex-numbers-based invariant language for advanced algebraic manipulations of geometric problems. A simplification algorithm is proposed for making symbolic geometric computing with NBA and BCD polynomials, with the unique feature of controlling the expression size by avoiding multilinear expansions of the first-order difference variables of complex polynomials.

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Cited By

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Li HCao Y(2011)On Geometric Theorem Proving with Null Geometric AlgebraGuide to Geometric Algebra in Practice10.1007/978-0-85729-811-9_10(195-215)Online publication date: 2011
https://doi.org/10.1007/978-0-85729-811-9_10
Rui Ling Yuanjun He Kairen Deng (2010)Automatic generation of geometric base sequences2010 IEEE International Conference on Progress in Informatics and Computing10.1109/PIC.2010.5687899(693-696)Online publication date: Dec-2010
https://doi.org/10.1109/PIC.2010.5687899

Index Terms

Complex brackets, balanced complex differences, and applications in symbolic geometric computing
1. Computing methodologies
  1. Artificial intelligence
    1. Knowledge representation and reasoning
  2. Symbolic and algebraic manipulation
    1. Representation of mathematical objects
    2. Symbolic and algebraic algorithms
      1. Theorem proving algorithms

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Published In

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

July 2008

348 pages

ISBN:9781595939043

DOI:10.1145/1390768

General Chair:
J. Rafael Sendra
Universidad de Alcalá, Spain
,
Program Chair:
Laureano Gonzalez-Vega
Universidad de Cantabria, Spain

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 20 July 2008

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ISSAC '08

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ISSAC '08: International Symposium on Symbolic and Algebraic Computation

July 20 - 23, 2008

Linz/Hagenberg, Austria

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

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Li HCao Y(2011)On Geometric Theorem Proving with Null Geometric AlgebraGuide to Geometric Algebra in Practice10.1007/978-0-85729-811-9_10(195-215)Online publication date: 2011
https://doi.org/10.1007/978-0-85729-811-9_10
Rui Ling Yuanjun He Kairen Deng (2010)Automatic generation of geometric base sequences2010 IEEE International Conference on Progress in Informatics and Computing10.1109/PIC.2010.5687899(693-696)Online publication date: Dec-2010
https://doi.org/10.1109/PIC.2010.5687899

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Computing rational Gauss-Chebyshev quadrature formulas with complex poles: The algorithm

Computing the Roots of Complex Orthogonal and Kernel Polynomials

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