### Abstract

It is known that a general polyhedral scene of complexity n has at most O(n ^{6}) combinatorially different orthographic views and at most O(n ^{9}) combinatorially different perspective views, and that these bounds are tight in the worst case. In this paper we show that, for the special case of scenes consisting of a collection of n translates of a cube, these bounds improve to O(n ^{4+ε}) and O(n ^{6+ε}), for any ε>0, respectively. In addition, we present constructions inducing Ω(n ^{4}) combinatorially different orthographic views and Ω(n ^{6}) combinatorially different perspective views, thus showing that these bounds are nearly tight in the worst case. Finally, we show how to extend the upper and lower bounds to several classes of related scenes.

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

Pages (from-to) | 179-182 |

Number of pages | 4 |

Journal | Computational Geometry: Theory and Applications |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2004 |

### Fingerprint

### Keywords

- Arrangements
- Aspect graphs
- Combinatorial geometry
- Envelopes
- Fat objects
- Orthographic views
- Perspective views
- Polyhedral terrains
- Visibility

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Science Applications
- Computational Mathematics
- Control and Optimization
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*27*(2), 179-182. https://doi.org/10.1016/j.comgeo.2003.10.001

**On the number of views of translates of a cube and related problems.** / Aronov, Boris; Schiffenbauer, Robert; Sharir, Micha.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 27, no. 2, pp. 179-182. https://doi.org/10.1016/j.comgeo.2003.10.001

}

TY - JOUR

T1 - On the number of views of translates of a cube and related problems

AU - Aronov, Boris

AU - Schiffenbauer, Robert

AU - Sharir, Micha

PY - 2004/2

Y1 - 2004/2

N2 - It is known that a general polyhedral scene of complexity n has at most O(n 6) combinatorially different orthographic views and at most O(n 9) combinatorially different perspective views, and that these bounds are tight in the worst case. In this paper we show that, for the special case of scenes consisting of a collection of n translates of a cube, these bounds improve to O(n 4+ε) and O(n 6+ε), for any ε>0, respectively. In addition, we present constructions inducing Ω(n 4) combinatorially different orthographic views and Ω(n 6) combinatorially different perspective views, thus showing that these bounds are nearly tight in the worst case. Finally, we show how to extend the upper and lower bounds to several classes of related scenes.

AB - It is known that a general polyhedral scene of complexity n has at most O(n 6) combinatorially different orthographic views and at most O(n 9) combinatorially different perspective views, and that these bounds are tight in the worst case. In this paper we show that, for the special case of scenes consisting of a collection of n translates of a cube, these bounds improve to O(n 4+ε) and O(n 6+ε), for any ε>0, respectively. In addition, we present constructions inducing Ω(n 4) combinatorially different orthographic views and Ω(n 6) combinatorially different perspective views, thus showing that these bounds are nearly tight in the worst case. Finally, we show how to extend the upper and lower bounds to several classes of related scenes.

KW - Arrangements

KW - Aspect graphs

KW - Combinatorial geometry

KW - Envelopes

KW - Fat objects

KW - Orthographic views

KW - Perspective views

KW - Polyhedral terrains

KW - Visibility

UR - http://www.scopus.com/inward/record.url?scp=84867994588&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867994588&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2003.10.001

DO - 10.1016/j.comgeo.2003.10.001

M3 - Article

VL - 27

SP - 179

EP - 182

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 2

ER -