### Abstract

This paper discusses how a line segment and segment pair in a 3‐dimensional space are constrained by an image obtained by their perspective projection. Also, it introduces the concept of polarity in the interpretation of a line‐segment image obtained under perspective projection and gives a definition of a spherical PAT curve. In addition, the relationship is explained between a PAT constraint [5], which is of an angle formed by two line segments on a plane, and the spherical PAT curve [7], which is a constraint of two line segments at an arbitrary position in a 3‐dimensional space (an extended idea of the former). Although the spherical PAT curve an be obtained formally by projecting a plane PAT curve onto a sphere, and this is justified for a semisphere, the meaning of this on the opposite semisphere has not been clarified with a justification. This paper justifies the sphere PAT curve by discussing the polarity of the segment interpretation, and explains its accurate meaning. This paper also describes the characteristics of a spherical PAT curve. The idea will have many applications to image understanding. Copyright © 1994 Wiley Periodicals, Inc., A Wiley Company

Original language | English |
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Pages (from-to) | 22-32 |

Number of pages | 11 |

Journal | Systems and Computers in Japan |

Volume | 25 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1994 |

### Keywords

- line segment pair
- PAT constraint
- Perspective projection
- single image

### ASJC Scopus subject areas

- Theoretical Computer Science
- Information Systems
- Hardware and Architecture
- Computational Theory and Mathematics

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## Cite this

*Systems and Computers in Japan*,

*25*(4), 22-32. https://doi.org/10.1002/scj.4690250403