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On taxicab distance mean functions and their geometric applications: methods, implementations and examples

Csaba Vincze, Ábris Nagy

TL;DR

This work develops and surveys the theory of distance mean functions under taxicab geometry and their geometric-tomography applications. It establishes that distance-mean level sets, or generalized conics, are convex and possess global minimizers, enabling unique reconstructions under certain X-ray data; the coordinate X-rays are encoded in the distance-mean framework, linking forbidden and allowed region growth to reconstruction. The authors present both analytic results (unarity, gradient-based bisecting methods) and computational schemes (Maple implementations, 0–1 programming, and network-flow approaches) for planar and lattice settings, including least-average-value principles and switching-chain mechanisms. Collectively, the paper provides a cohesive toolkit for reconstructing planar bodies from coordinate X-rays and for approximating focal-set level sets via distance-mean functions, with practical algorithms and illustrative examples that extend classical geometric tomography results to taxicab geometry.

Abstract

A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points the average distance is typically given by integration over the focal set. The paper contains a survey on the applications of taxicab distance mean functions and generalized conics' theory in geometric tomography: bisection of the focal set and reconstruction problems by coordinate X-rays. The theoretical results are illustrated by implementations in Maple, methods and examples as well.

On taxicab distance mean functions and their geometric applications: methods, implementations and examples

TL;DR

This work develops and surveys the theory of distance mean functions under taxicab geometry and their geometric-tomography applications. It establishes that distance-mean level sets, or generalized conics, are convex and possess global minimizers, enabling unique reconstructions under certain X-ray data; the coordinate X-rays are encoded in the distance-mean framework, linking forbidden and allowed region growth to reconstruction. The authors present both analytic results (unarity, gradient-based bisecting methods) and computational schemes (Maple implementations, 0–1 programming, and network-flow approaches) for planar and lattice settings, including least-average-value principles and switching-chain mechanisms. Collectively, the paper provides a cohesive toolkit for reconstructing planar bodies from coordinate X-rays and for approximating focal-set level sets via distance-mean functions, with practical algorithms and illustrative examples that extend classical geometric tomography results to taxicab geometry.

Abstract

A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points the average distance is typically given by integration over the focal set. The paper contains a survey on the applications of taxicab distance mean functions and generalized conics' theory in geometric tomography: bisection of the focal set and reconstruction problems by coordinate X-rays. The theoretical results are illustrated by implementations in Maple, methods and examples as well.
Paper Structure (13 sections, 14 theorems, 72 equations, 6 figures)

This paper contains 13 sections, 14 theorems, 72 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Some general observations
  3. The problem of unicity
  4. Bisection of bodies by coordinate hyperplanes
  5. Implementation and examples
  6. Applications in geometric tomography
  7. Reconstruction of planar sets by their coordinate X-rays
  8. An algorithm for the reconstruction
  9. Reconstruction by the least average values
  10. The least average value principle for the reconstruction of planar lattice sets by coordinate X-rays
  11. Example.
  12. Switching chains
  13. The existence of the switching chain

Key Result

Theorem 1

VinNagy12VinNagy17 The sublevel sets of a distance mean function are convex and compact.

Figures (6)

  • Figure 1: The sequence of points $X_k$ generated by the above procedure for $k=1,2,\ldots 50$. Darker points present elements $X_k$ with higher indices $k$. Notice how these sequences of points converge to the minimizer of the taxicab distance mean function \ref{['unweighted']}.
  • Figure 2: X-rays of compact planar bodies.
  • Figure 3: The set we are looking for.
  • Figure 4: The coordinate X-rays.
  • Figure 5: The optimal solution under low resolution (left) and a greedy version under high resolution (right).
  • ...and 1 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Definition 1
  • Theorem 5
  • Theorem 6
  • Corollary 1
  • Example 1
  • ...and 8 more