### Abstract

The key to a robust and efficient implementation of a computational geometry algorithm is an efficient algorithm for detecting degenerate predicates. We study degeneracy detection in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to another polyhedron. The structure of the free space is determined by the signs of univariate polynomials, called angle polynomials, whose coefficients are polynomials in the coordinates of the vertices of the polyhedra. Every predicate is expressible as the sign of an angle polynomial f evaluated at a zero t of an angle polynomial g. A predicate is degenerate (the sign is zero) when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of every possible angle polynomial. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

Original language | English (US) |
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Title of host publication | 34th International Symposium on Computational Geometry, SoCG 2018 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 611-6114 |

Number of pages | 5504 |

Volume | 99 |

ISBN (Electronic) | 9783959770668 |

DOIs | |

State | Published - Jun 1 2018 |

Event | 34th International Symposium on Computational Geometry, SoCG 2018 - Budapest, Hungary Duration: Jun 11 2018 → Jun 14 2018 |

### Other

Other | 34th International Symposium on Computational Geometry, SoCG 2018 |
---|---|

Country | Hungary |

City | Budapest |

Period | 6/11/18 → 6/14/18 |

### Fingerprint

### Keywords

- Degenerate predicates
- Free space construction
- Robustness

### ASJC Scopus subject areas

- Software

### Cite this

*34th International Symposium on Computational Geometry, SoCG 2018*(Vol. 99, pp. 611-6114). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2018.61

**Table based detection of degenerate predicates in free space construction.** / Milenkovic, Victor; Sacks, Elisha; Butt, Nabeel.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*34th International Symposium on Computational Geometry, SoCG 2018.*vol. 99, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 611-6114, 34th International Symposium on Computational Geometry, SoCG 2018, Budapest, Hungary, 6/11/18. https://doi.org/10.4230/LIPIcs.SoCG.2018.61

}

TY - GEN

T1 - Table based detection of degenerate predicates in free space construction

AU - Milenkovic, Victor

AU - Sacks, Elisha

AU - Butt, Nabeel

PY - 2018/6/1

Y1 - 2018/6/1

N2 - The key to a robust and efficient implementation of a computational geometry algorithm is an efficient algorithm for detecting degenerate predicates. We study degeneracy detection in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to another polyhedron. The structure of the free space is determined by the signs of univariate polynomials, called angle polynomials, whose coefficients are polynomials in the coordinates of the vertices of the polyhedra. Every predicate is expressible as the sign of an angle polynomial f evaluated at a zero t of an angle polynomial g. A predicate is degenerate (the sign is zero) when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of every possible angle polynomial. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

AB - The key to a robust and efficient implementation of a computational geometry algorithm is an efficient algorithm for detecting degenerate predicates. We study degeneracy detection in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to another polyhedron. The structure of the free space is determined by the signs of univariate polynomials, called angle polynomials, whose coefficients are polynomials in the coordinates of the vertices of the polyhedra. Every predicate is expressible as the sign of an angle polynomial f evaluated at a zero t of an angle polynomial g. A predicate is degenerate (the sign is zero) when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of every possible angle polynomial. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

KW - Degenerate predicates

KW - Free space construction

KW - Robustness

UR - http://www.scopus.com/inward/record.url?scp=85048948026&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048948026&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2018.61

DO - 10.4230/LIPIcs.SoCG.2018.61

M3 - Conference contribution

VL - 99

SP - 611

EP - 6114

BT - 34th International Symposium on Computational Geometry, SoCG 2018

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -