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On flat shadow boundaries from point light sources and the characterization of ellipsoids

Bartłomiej Zawalski

Abstract

In his classical work, W. Blaschke proved that a convex body whose shadow boundaries are flat for every direction of parallel illumination must be an ellipsoid. An extension recently proposed by I. Gonzalez-García, J. Jerónimo-Castro, E. Morales-Amaya, and D.J. Verdusco-Hernández predicts that the same conclusion holds for illumination by point light sources located on a hypersurface enclosing the body. We confirm this conjecture for convex bodies with sufficiently smooth boundaries. We further develop a duality framework relating illumination by point light sources to classical symmetry properties of hyperplane sections, linking several known and conjectured characterizations of quadrics from these complementary viewpoints.

On flat shadow boundaries from point light sources and the characterization of ellipsoids

Abstract

In his classical work, W. Blaschke proved that a convex body whose shadow boundaries are flat for every direction of parallel illumination must be an ellipsoid. An extension recently proposed by I. Gonzalez-García, J. Jerónimo-Castro, E. Morales-Amaya, and D.J. Verdusco-Hernández predicts that the same conclusion holds for illumination by point light sources located on a hypersurface enclosing the body. We confirm this conjecture for convex bodies with sufficiently smooth boundaries. We further develop a duality framework relating illumination by point light sources to classical symmetry properties of hyperplane sections, linking several known and conjectured characterizations of quadrics from these complementary viewpoints.

Paper Structure

This paper contains 19 sections, 17 theorems, 83 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Duality between supporting cones and affine sections
  3. Flat shadow boundaries
  4. Centrally symmetric affine sections and axially symmetric supporting cones
  5. Linearly equivalent supporting cones
  6. Flat intersections of supporting cones
  7. Definitions and basic concepts
  8. Infinitesimal properties of flat shadow boundaries
  9. Local parametrization of a shadow boundary
  10. Local parametrization of a section
  11. Constraints resulting from flatness of the shadow boundary
  12. Proofs of the main theorems
  13. Bounds on the number of point light sources
  14. Upper bound
  15. Lower bound
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $K\subset\mathbb R^n$, $n\geq 4$, be a convex body with boundary of class $C^3$. Suppose that for every point $p\in\partial K$ on the boundary, there are at least $L(n)$ point light sources on the tangent hyperplane $T_p\partial K$ in a general linear position with respect to $p$, that create fl

Figures (6)

  • Figure 1.4: The planet Venus illuminated by a point light source (the Sun), with the visible terminator representing the boundary between illuminated and shadowed regions. (Steven Molina/shutterstock.com)
  • Figure 4.0: Notations used in the proof
  • Figure 6.5: List of specific graphs used in the proof of \ref{['lem:04']}
  • Figure 6.6: Operation \ref{['it:02']} of splitting a vertex incident with $4$ or more edges
  • Figure 6.6: Minimal $4$-connected graphs on $7$ vertices
  • ...and 1 more figures

Theorems & Definitions (48)

  • Conjecture 1.1: cf. Gonzalez_Garcia2022, https://doi.org/10.1112/mtk.12176, flatgrazesconvexbodies
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 2.1: cf. Bianchi1987
  • Theorem 2.2: False Center Theorem Larman
  • Conjecture 2.3: cf. BARKER200179
  • Theorem 2.4: Shaken False Center Theorem S002557930000019X
  • Theorem 2.5: moralesamaya2023
  • Theorem 2.6: cf. Morales_Jeronimo_Verdusco_2022
  • ...and 38 more