### Abstract

We have developed a qualitative calculus for three-dimensional directions and rotations. A direction is characterized in terms of the signs of its components relative to an absolute coordinate system. A rotation is characterized in terms of the signs of the components of the associated 3 × 3 rotation matrix. A system has been implemented that can solve the following problems: 1. Given the signs of direction and rotation matrix P, find the possible signs of the image of under P. Moreover, for each possible sign vector of · P, generate numerical instantiations of and P that yields that result. 2. Given the signs of rotation matrices P and Q, find the possible signs of the composition P · Q. Moreover, for each possible sign matrix for the composition, generate numerical instantiations of P and Q that yield that result. We have also proved some related complexity and expressivity results. The satisfiability problem for a qualitative rotation constraint network is NP-complete in two dimensions and NP-hard in three dimensions. In three dimensions, any two directions are distinguishable by a qualitative rotation constraint network.

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

Pages (from-to) | 18-57 |

Number of pages | 40 |

Journal | Spatial Cognition and Computation |

Volume | 14 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

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### Keywords

- qualitative calculus
- qualitative spatial reasoning
- three-dimensional rotation

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Computer Vision and Pattern Recognition
- Modeling and Simulation
- Earth-Surface Processes
- Experimental and Cognitive Psychology

### Cite this

**A Qualitative Calculus for Three-Dimensional Rotations.** / Asl, Azam; Davis, Ernest.

Research output: Contribution to journal › Article

*Spatial Cognition and Computation*, vol. 14, no. 1, pp. 18-57. https://doi.org/10.1080/13875868.2013.807811

}

TY - JOUR

T1 - A Qualitative Calculus for Three-Dimensional Rotations

AU - Asl, Azam

AU - Davis, Ernest

PY - 2014

Y1 - 2014

N2 - We have developed a qualitative calculus for three-dimensional directions and rotations. A direction is characterized in terms of the signs of its components relative to an absolute coordinate system. A rotation is characterized in terms of the signs of the components of the associated 3 × 3 rotation matrix. A system has been implemented that can solve the following problems: 1. Given the signs of direction and rotation matrix P, find the possible signs of the image of under P. Moreover, for each possible sign vector of · P, generate numerical instantiations of and P that yields that result. 2. Given the signs of rotation matrices P and Q, find the possible signs of the composition P · Q. Moreover, for each possible sign matrix for the composition, generate numerical instantiations of P and Q that yield that result. We have also proved some related complexity and expressivity results. The satisfiability problem for a qualitative rotation constraint network is NP-complete in two dimensions and NP-hard in three dimensions. In three dimensions, any two directions are distinguishable by a qualitative rotation constraint network.

AB - We have developed a qualitative calculus for three-dimensional directions and rotations. A direction is characterized in terms of the signs of its components relative to an absolute coordinate system. A rotation is characterized in terms of the signs of the components of the associated 3 × 3 rotation matrix. A system has been implemented that can solve the following problems: 1. Given the signs of direction and rotation matrix P, find the possible signs of the image of under P. Moreover, for each possible sign vector of · P, generate numerical instantiations of and P that yields that result. 2. Given the signs of rotation matrices P and Q, find the possible signs of the composition P · Q. Moreover, for each possible sign matrix for the composition, generate numerical instantiations of P and Q that yield that result. We have also proved some related complexity and expressivity results. The satisfiability problem for a qualitative rotation constraint network is NP-complete in two dimensions and NP-hard in three dimensions. In three dimensions, any two directions are distinguishable by a qualitative rotation constraint network.

KW - qualitative calculus

KW - qualitative spatial reasoning

KW - three-dimensional rotation

UR - http://www.scopus.com/inward/record.url?scp=84890923739&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890923739&partnerID=8YFLogxK

U2 - 10.1080/13875868.2013.807811

DO - 10.1080/13875868.2013.807811

M3 - Article

AN - SCOPUS:84890923739

VL - 14

SP - 18

EP - 57

JO - Spatial Cognition and Computation

JF - Spatial Cognition and Computation

SN - 1387-5868

IS - 1

ER -