### Abstract

It is known that the region V(s) of a simple polygon P, directly visible (illuminable) from an internal point s, is simply connected. Aronov et al. [2] established that the region V_{1}(s) of a simple polygon visible from an internal point s due to at most one diffuse reflection on the boundary of the polygon P, is also simply connected. In this paper we establish that the region V_{2}(s), visible from s due to at most two diffuse reflections may be multiply connected; we demonstrate the construction of an n-sided simple polygon with a point s inside it so that the region of P visible from s after at most two diffuse reflections is multiply connected. We also show that V-_{3}(s), the region of P visible from s after at most three diffuse reflections, can have ω(n) holes.

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

Pages (from-to) | 5-14 |

Number of pages | 10 |

Journal | Journal of Geometry |

Volume | 81 |

Issue number | 1-2 |

DOIs | |

State | Published - Dec 1 2004 |

### Fingerprint

### Keywords

- Diffuse reflection
- Multiple connectedness
- Simple polygon
- Visibility

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Geometry*,

*81*(1-2), 5-14. https://doi.org/10.1007/S00022-004-1726-2

**A linear worst-case lower bound on the number of holes inside regions visible due to multiple diffuse reflections.** / Brahma, Siddhartha; Pal, Sudebkumar Prasant; Sarkar, Dilip.

Research output: Contribution to journal › Article

*Journal of Geometry*, vol. 81, no. 1-2, pp. 5-14. https://doi.org/10.1007/S00022-004-1726-2

}

TY - JOUR

T1 - A linear worst-case lower bound on the number of holes inside regions visible due to multiple diffuse reflections

AU - Brahma, Siddhartha

AU - Pal, Sudebkumar Prasant

AU - Sarkar, Dilip

PY - 2004/12/1

Y1 - 2004/12/1

N2 - It is known that the region V(s) of a simple polygon P, directly visible (illuminable) from an internal point s, is simply connected. Aronov et al. [2] established that the region V1(s) of a simple polygon visible from an internal point s due to at most one diffuse reflection on the boundary of the polygon P, is also simply connected. In this paper we establish that the region V2(s), visible from s due to at most two diffuse reflections may be multiply connected; we demonstrate the construction of an n-sided simple polygon with a point s inside it so that the region of P visible from s after at most two diffuse reflections is multiply connected. We also show that V-3(s), the region of P visible from s after at most three diffuse reflections, can have ω(n) holes.

AB - It is known that the region V(s) of a simple polygon P, directly visible (illuminable) from an internal point s, is simply connected. Aronov et al. [2] established that the region V1(s) of a simple polygon visible from an internal point s due to at most one diffuse reflection on the boundary of the polygon P, is also simply connected. In this paper we establish that the region V2(s), visible from s due to at most two diffuse reflections may be multiply connected; we demonstrate the construction of an n-sided simple polygon with a point s inside it so that the region of P visible from s after at most two diffuse reflections is multiply connected. We also show that V-3(s), the region of P visible from s after at most three diffuse reflections, can have ω(n) holes.

KW - Diffuse reflection

KW - Multiple connectedness

KW - Simple polygon

KW - Visibility

UR - http://www.scopus.com/inward/record.url?scp=55449088668&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55449088668&partnerID=8YFLogxK

U2 - 10.1007/S00022-004-1726-2

DO - 10.1007/S00022-004-1726-2

M3 - Article

AN - SCOPUS:55449088668

VL - 81

SP - 5

EP - 14

JO - Journal of Geometry

JF - Journal of Geometry

SN - 0047-2468

IS - 1-2

ER -