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Pappus in optical space.

Jan J Koenderink1, Andrea J van Doorn, Astrid M L Kappers

  • 1Helmholtz Instituut, University of Utrecht, The Netherlands. j.j.koenderink@phys.uu.nl

Perception & Psychophysics
|June 7, 2002
PubMed
Summary

Optical space, while not perfectly accurate, largely adheres to projective geometry principles. Empirical tests show violations of the Pappus condition are rare, suggesting optical space is not entirely dissimilar to homogeneous spaces.

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Area of Science:

  • Visual perception
  • Geometric psychology
  • Spatial cognition

Background:

  • Optical space structure is typically assumed to be metrical.
  • Previous research implicitly assumed incidence relations for optical space.
  • Homogeneous spaces (e.g., Luneburg theory) are often used to model optical space.

Purpose of the Study:

  • To investigate the geometric structure of optical space without assuming a metric.
  • To empirically test the Pappus condition in optical space under full-cue conditions.
  • To determine if optical space conforms to projective geometry principles.

Main Methods:

  • Assumed only incidence relations (points and lines) for optical space structure.
  • Empirically tested the Pappus condition, a key property of projective spaces.
  • Conducted tests in a full-cue environment (open field, daylight, up to 20m distances, 160-degree visual fields).

Main Results:

  • Violations of the Pappus condition in optical space were found to be exceptional.
  • Optical space demonstrated adherence to projective geometry, despite not being veridical.
  • Results indicate optical space is not entirely dissimilar to homogeneous spaces.

Conclusions:

  • Optical space, under empirical testing, largely satisfies the Pappus condition.
  • This suggests optical space possesses a projective structure, deviating from purely metrical assumptions.
  • While not Euclidean, optical space exhibits properties closer to homogeneous spaces than previously assumed.

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