## Abstract

Given a fixed origin o in the d-dimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment op̄ between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log n) bound in the n × n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital star-shaped regions centered at o, which we use to design efficient algorithms for image processing problems.

Original language | English |
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Pages (from-to) | 359-378 |

Number of pages | 20 |

Journal | Discrete and Computational Geometry |

Volume | 42 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 |

## Keywords

- Digital geometry
- Discrete geometry
- Star-shaped regions
- Tree embedding

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics