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Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

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Computer Vision

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Synonyms

Surfaces

Related Concepts

Curvature; Differential Invariants; Euclidean Geometry; Osculating Paraboloids; Parametric Curve

Definition

Differential geometry studies spatial entities in local (infinitesimal) neighborhoods. This approach enables one to exploit the power of (multi-)linear algebra. Geometrical entities are differential invariants, the generic example being the curvature of planar curves. The number of relevant differential invariants increases in more complicated settings like – in this entry – that of surfaces in three-dimensional Euclidean space.

Background

The differential geometry of surfaces in three-dimensional Euclidean space is often called “classical differential geometry” as its history goes back to the founding fathers of infinitesimal calculus, Newton and Leibnitz. There exists a huge literature, although novel additions are still forthcoming. This entry reviews the basics. There is ample literature for those needing to delve deeper.

Theory

Surfaces are...

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References

  1. do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs

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  2. Eisenhart LP (2004) A treatise on the differential geometry of curves and surfaces. Dover, New York

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  5. Hilbert D, Cohn-Vossen S (1952) Geometry and the imagination, 2nd edn. Chelsea, New York

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  6. Koenderink JJ (1990) Solid shape. MIT Press, Cambridge, MA

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  7. Kreyszig E (1991) Differential geometry, Chapter II. Dover, New York

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  9. Spivak M (1979) Comprehensive introduction to differential geometry, 5 vols. Publish or Perish, Houston

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Author information

Authors and Affiliations

  1. Faculty of EEMSC, Delft University of Technology, P.O. Box 5031, 2600, Delft, The Netherlands

    Jan J. Koenderink

Authors
  1. Jan J. Koenderink
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Editor information

Editors and Affiliations

  1. Institute of Industrial Science, The University of Tokyo, Tokyo, Japan

    Katsushi Ikeuchi

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Koenderink, J.J. (2014). Differential Geometry of Surfaces in Three-Dimensional Euclidean Space. In: Ikeuchi, K. (eds) Computer Vision. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-31439-6_643

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