Abstract
Optical space differs from physical space. The structure of optical space has generally been assumed to be metrical. In contradistinction, we do not assume any metric, but only incidence relations (i.e., we assume that optical points and lines exist and that two points define a unique line, and two lines a unique point). (The incidence relations have generally been assumed implicitly by earlier authors.) The condition that makes such anincidence structure into a projective space is the Pappus condition. The Pappus condition describes a projective relation between three collinear triples of points, whose validity can— in principle—be verified empirically. The Pappus condition is a necessary condition for optical space to be a homogeneous space (Lobatchevski hyperbolic or Riemann elliptic space) as assumed by, for example, the well-known Luneburg theory. We test the Pappus condition in a full-cue situation (open field, broad daylight, distances of up to 20 m, visual fields of up to 160° diameter). We found that although optical space is definitely not veridical, even under full-cue conditions, violations of the Pappus condition are the exception. Apparently optical space is not totally different from a homogeneous space, although it is in no way close to Euclidean.
Article PDF
Similar content being viewed by others
References
Ames, A., Ogle, K. N., &Glidden, G. H. (1932). Corresponding retinal points, the horopter and size and shape of ocular images.Journal of the Optical Society of America,22, 538–631.
Battro, A. M., di Piero Netto, S., &Rozestraten, R. J. A. (1976). Riemannian geometries of variable curvature in visual space: Visual alleys, horopters, and triangles in big open fields.Perception,5, 9–23.
Beltrami, E. (1865). Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappesentate da linee rette [Solution to the problem: Find the embedding of a surface such that the parameter curves are geodesics].Annali di Matematica Pura ed Applicata, ser. 1,7, 185–204.
Beltrami, E. (1868). Saggio di interpretazione della geometria noneuclidea [On the meaning of non-Euclidean geometry].Giornale di Matematiche, ad uso degli studenti delle università italiane,6, 284–312.
Bennett, M. K. (1995).Affine and projective geometry. New York: Wiley.
Blank, A. A. (1953). The Luneburg theory of binocular visual space.Journal of the Optical Society of America,43, 717–727.
Blank.A. A. (1958s). Analysis of experiments in binocular space perception.Journal of the Optical Society of America,48, 911–925.
Blank, A. A. (1958b). Axiomatics of binocular vision. The foundations of metric geometry in relation to space perception.Journal of the Optical Society of America,48, 328–334.
Blumenfeld, W. (1913). Untersuchungen über die scheinbare Grösse im Sehräume [Investigations on apparent size in visual spaces].Zeitschrift für Psychologie,65, 241–404.
Blumenthal, L. M. (1995).A modern view of geometry. New York: Dover.
Coxeter, H. S. M. (1961).Introduction to geometry. New York: Wiley.
Foley, J. M. (1964). Desarguesian property of visual space.Journal of the Optical Society of America,54, 684–692.
Gibson, J. J. (1950).The perception of the visual world. Boston: Houghton Mifflin.
Hauck, G. (1879).Die subjektive Perspektive und die horizontalen Curvaturen des Dorischen Styls [Subjective perspective and the curvature of horizontals in the Doric style]. Stuttgart, Germany: K. Wittwer.
Helmholtz, H. von (1866).Handbuch der physiologischen Optik [Handbook of physiological optics] (1st ed.). Hamburg and Leipzig: Leopold Voss.
Hilbert, D., &Cohn-Vossen, S. (1983).Geometry and the imagination. New York: Chelsea. (Original work published 1932)
Hillebrand, F. (1929).Lehre von den Gesichtsempfindungen auf Grund hinterlassener Aufzeichnungen (Herausgegeben von Dr. F. Hillebrand) [Studies in visual perception from the heritage of Hillebrand, communicated by F. Hillebrand]. Wien: Springer-Verlag.
Indow, T. (1990). On geometrical analysis of global structure of visual space. In H.-G. Geissler & M. H. Miller (Eds.),Psychophysical explorations of mental structures (pp. 172–180). Göttingen: Hogrefe & Huber.
Indow, T. (1991). A critical review of Luneburg’s model with regard to global structure of visual space.Psychological Review,98, 430–453.
Indow, T. (1997). Hyperbolic representation of global structure of visual space.Journal of Mathematical Psychology,14, 89–98.
Klein, E. (1932).Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (E. R. Hedrick & C. A. Noble, Trans.). New York: Macmillan.
Klein, E. (1939).Elementary mathematics from an advanced standpoint: Geometry (E. R. Hedrick & C. A. Noble, Trans.). New York: Macmillan.
Koenderink, J. J., van Doorn, A. J., &Lappin, J. S. (2000). Direct measurement of the curvature of visual space.Perception,29, 69–80.
Luneburg, R. K. (1947).Mathematical analysis of binocular vision. Princeton, NJ: Princeton University Press.
Ogle, K. N. (1950).Researches in binocular vision. Philadelphia and London: W. B. Saunders.
Pirenne, M. H. (1970).Optics, painting and photography. Cambridge: Cambridge University Press.
Suppes, P. (1977). Is visual space Euclidean?Synthèse,35, 397–421.
Wagner, M. (1985). The metric of visual space.Perception & Psychophysics,38, 483–495.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Koenderink, J.J., van Doorn, A.J., Kappers, A.M.L. et al. Pappus in optical space. Perception & Psychophysics 64, 380–391 (2002). https://doi.org/10.3758/BF03194711
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03194711