### Abstract

Let [a,b] be a line segment with end points a, b and ν a point at which a viewer is located, all in R ^{3}. The aperture angle of [a,b] from point ν, denoted by θ(ν), is the interior angle at ν of the triangle Δ(a,b,ν). Given a convex polyhedron P not intersecting a given segment [a,b] we consider the problem of computing θ_{max}(ν) and θ_{min}(ν), the maximum and minimum values of θ(ν) as ν varies over all points in P. We obtain two characterizations of θ_{max}(ν). Along the way we solve several interesting special cases of the above problems and establish linear upper and lower bounds on their complexity under several models of computation.

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

Pages (from-to) | 301-329 |

Number of pages | 29 |

Journal | Journal of Mathematical Modelling and Algorithms |

Volume | 1 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2002 |

### Fingerprint

### Keywords

- algorithms
- aperture-angle
- computational-complexity
- convexity
- discrete-optimization
- geometry

### ASJC Scopus subject areas

- Modeling and Simulation
- Applied Mathematics

### Cite this

*Journal of Mathematical Modelling and Algorithms*,

*1*(4), 301-329. https://doi.org/10.1023/A:1021666512528

**Aperture-Angle Optimization Problems in Three Dimensions.** / Omaña-Pulido, Elsa; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Journal of Mathematical Modelling and Algorithms*, vol. 1, no. 4, pp. 301-329. https://doi.org/10.1023/A:1021666512528

}

TY - JOUR

T1 - Aperture-Angle Optimization Problems in Three Dimensions

AU - Omaña-Pulido, Elsa

AU - Toussaint, Godfried

PY - 2002/12/1

Y1 - 2002/12/1

N2 - Let [a,b] be a line segment with end points a, b and ν a point at which a viewer is located, all in R 3. The aperture angle of [a,b] from point ν, denoted by θ(ν), is the interior angle at ν of the triangle Δ(a,b,ν). Given a convex polyhedron P not intersecting a given segment [a,b] we consider the problem of computing θmax(ν) and θmin(ν), the maximum and minimum values of θ(ν) as ν varies over all points in P. We obtain two characterizations of θmax(ν). Along the way we solve several interesting special cases of the above problems and establish linear upper and lower bounds on their complexity under several models of computation.

AB - Let [a,b] be a line segment with end points a, b and ν a point at which a viewer is located, all in R 3. The aperture angle of [a,b] from point ν, denoted by θ(ν), is the interior angle at ν of the triangle Δ(a,b,ν). Given a convex polyhedron P not intersecting a given segment [a,b] we consider the problem of computing θmax(ν) and θmin(ν), the maximum and minimum values of θ(ν) as ν varies over all points in P. We obtain two characterizations of θmax(ν). Along the way we solve several interesting special cases of the above problems and establish linear upper and lower bounds on their complexity under several models of computation.

KW - algorithms

KW - aperture-angle

KW - computational-complexity

KW - convexity

KW - discrete-optimization

KW - geometry

UR - http://www.scopus.com/inward/record.url?scp=73349108020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73349108020&partnerID=8YFLogxK

U2 - 10.1023/A:1021666512528

DO - 10.1023/A:1021666512528

M3 - Article

AN - SCOPUS:73349108020

VL - 1

SP - 301

EP - 329

JO - Journal of Mathematical Modelling and Algorithms

JF - Journal of Mathematical Modelling and Algorithms

SN - 1570-1166

IS - 4

ER -