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Metric Multiview Geometry -- a Catalogue in Low Dimensions

Timothy Duff, Felix Rydell

Abstract

We systematically compile an exhaustive catalogue of multiview varieties and anchored multiview varieties arising from projections of points and lines in 1, 2, and 3-dimensional projective space. We say that two such varieties are ED-equivalent if there is a linear isomorphism between that that preserve ED-critical points. This gives rise to fourteen equivalence classes, and we determine various properties - dimension, set-theoretic equations, and multidegrees - for all varieties featured in our catalogue. In the case of points, we also present a complementary study of resectioning varieties and their singular loci. Finally, we propose conjectures for the Euclidean distance degrees of all varieties appearing in our comprehensive compilation.

Metric Multiview Geometry -- a Catalogue in Low Dimensions

Abstract

We systematically compile an exhaustive catalogue of multiview varieties and anchored multiview varieties arising from projections of points and lines in 1, 2, and 3-dimensional projective space. We say that two such varieties are ED-equivalent if there is a linear isomorphism between that that preserve ED-critical points. This gives rise to fourteen equivalence classes, and we determine various properties - dimension, set-theoretic equations, and multidegrees - for all varieties featured in our catalogue. In the case of points, we also present a complementary study of resectioning varieties and their singular loci. Finally, we propose conjectures for the Euclidean distance degrees of all varieties appearing in our comprehensive compilation.
Paper Structure (19 sections, 21 theorems, 60 equations)

This paper contains 19 sections, 21 theorems, 60 equations.

Table of Contents

  1. Acknoledgements.
  2. Preliminaries
  3. Wedge and cross products
  4. Smooth quadrics in $\mathbb{P}^3$
  5. The Cayley parametrization
  6. Euler characteristics
  7. Varieties from Vision
  8. Multiview Varieties
  9. Anchored Multiview Varieties
  10. Schubert varieties that are linearly isomorphic to projective subspaces
  11. ED-Equivalence Catalogue
  12. Proof of main theorem
  13. Set-Theoretic Equations
  14. Multidegrees
  15. Resectioning Varieties
  16. ...and 4 more sections

Key Result

Theorem 1.2

A smooth quadric in $\mathbb{P}^3$ contains two 1-dimensional families of lines. Any two lines in the same family are disjoint, and any two lines from different families meet in exactly one point.

Theorems & Definitions (52)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • proof
  • Lemma 2.1
  • ...and 42 more