Journal of Mathematical Imaging and Vision

, Volume 48, Issue 3, pp 432–450 | Cite as

Weighted Projective Spaces and a Generalization of Eves’ Theorem



For a certain class of configurations of points in space, Eves’ Theorem gives a ratio of products of distances that is invariant under projective transformations, generalizing the cross-ratio for four points on a line. We give a generalization of Eves’ theorem, which applies to a larger class of configurations and gives an invariant with values in a weighted projective space. We also show how the complex version of the invariant can be determined from classically known ratios of products of determinants, while the real version of the invariant can distinguish between configurations that the classical invariants cannot.


Invariant theory Weighted projective space Cross ratio 



The author was motivated to start writing this paper after attending a talk by M. Frantz [10], and he thanks Frantz for subsequent conversations and for pointing out the reference [13]. The Figures were generated using [1].

Supplementary material

10851_2013_417_MOESM1_ESM.pdf (54 kb)


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndiana University—Purdue University Fort WayneFort WayneUSA

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