### Abstract

This article reviews the properties of Tensorial Bernstein Basis (TBB) and its usage, with interval analysis, for solving systems of nonlinear, univariate or multivariate equations resulting from geometric constraints. TBB are routinely used in computerized geometry for geometric modelling in CAD-CAM, or in computer graphics. They provide sharp enclosures of polynomials and their derivatives. They are used to reduce domains while preserving roots of polynomial systems, to prove that domains do not contain roots, and to make existence and uniqueness tests. They are compatible with standard preconditioning methods and fit linear programming techniques. However, current Bernstein-based solvers are limited to small algebraic systems. We present Bernstein polytopes and show how combining them with linear programming allows us to solve larger systems as well. The article also gives a generalization of Bernstein polytopes to higher degrees and a comparison of polytopes-based versus TBB-based polynomial bounds.

Original language | English (US) |
---|---|

Pages (from-to) | 192-208 |

Number of pages | 17 |

Journal | Reliable Computing |

Volume | 17 |

State | Published - Dec 1 2012 |

### Fingerprint

### Keywords

- Algebraic systems
- Geometric constraint solving. bernstein polytope
- Tensorial Bernstein Basis
- Univariate and multivariate polynomials

### ASJC Scopus subject areas

- Software
- Computational Mathematics
- Applied Mathematics

### Cite this

*Reliable Computing*,

*17*, 192-208.

**The Bernstein Basis and its applications in solving geometric constraint systems.** / Foufou, Sebti; Michelucci, Dominique.

Research output: Contribution to journal › Article

*Reliable Computing*, vol. 17, pp. 192-208.

}

TY - JOUR

T1 - The Bernstein Basis and its applications in solving geometric constraint systems

AU - Foufou, Sebti

AU - Michelucci, Dominique

PY - 2012/12/1

Y1 - 2012/12/1

N2 - This article reviews the properties of Tensorial Bernstein Basis (TBB) and its usage, with interval analysis, for solving systems of nonlinear, univariate or multivariate equations resulting from geometric constraints. TBB are routinely used in computerized geometry for geometric modelling in CAD-CAM, or in computer graphics. They provide sharp enclosures of polynomials and their derivatives. They are used to reduce domains while preserving roots of polynomial systems, to prove that domains do not contain roots, and to make existence and uniqueness tests. They are compatible with standard preconditioning methods and fit linear programming techniques. However, current Bernstein-based solvers are limited to small algebraic systems. We present Bernstein polytopes and show how combining them with linear programming allows us to solve larger systems as well. The article also gives a generalization of Bernstein polytopes to higher degrees and a comparison of polytopes-based versus TBB-based polynomial bounds.

AB - This article reviews the properties of Tensorial Bernstein Basis (TBB) and its usage, with interval analysis, for solving systems of nonlinear, univariate or multivariate equations resulting from geometric constraints. TBB are routinely used in computerized geometry for geometric modelling in CAD-CAM, or in computer graphics. They provide sharp enclosures of polynomials and their derivatives. They are used to reduce domains while preserving roots of polynomial systems, to prove that domains do not contain roots, and to make existence and uniqueness tests. They are compatible with standard preconditioning methods and fit linear programming techniques. However, current Bernstein-based solvers are limited to small algebraic systems. We present Bernstein polytopes and show how combining them with linear programming allows us to solve larger systems as well. The article also gives a generalization of Bernstein polytopes to higher degrees and a comparison of polytopes-based versus TBB-based polynomial bounds.

KW - Algebraic systems

KW - Geometric constraint solving. bernstein polytope

KW - Tensorial Bernstein Basis

KW - Univariate and multivariate polynomials

UR - http://www.scopus.com/inward/record.url?scp=84872698533&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872698533&partnerID=8YFLogxK

M3 - Article

VL - 17

SP - 192

EP - 208

JO - Reliable Computing

JF - Reliable Computing

SN - 1385-3139

ER -