Logical Interpretation of the Algorithm for Solving Apollonius' Problem

Denis Voloshinov; S. Shchur

doi:10.12737/2308-4898-2026-14-1-3-14

UDC 004.925.8

LOGICAL INTERPRETATION OF THE ALGORITHM FOR SOLVING APOLLONIUS' PROBLEM Logical Interpretation of the Algorithm for Solving Apollonius' Problem

АА Аа

Published in Geometry & Graphics · Volume 14, Issue 1 · Pages 3–14 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/2308-4898-2026-14-1-3-14

Received: 25.03.2026 Accepted: 25.03.2026 Published: 25.03.2026

Authors

Denis Voloshinov

S. Shchur

Denis Voloshinov

Academic degree: doctor of technical sciences

The Bonch-Bruevich St.-Petersburg State University of Telecommunications Professor · Russian Federation · St.-Petersburg

S. Shchur

Peter the Great Saint-Petersburg Polytechnic University assistant professor · Higher School of Design and Architecture · St.Petersburg

ABSTRACT

Algorithmization of solving geometric modeling problems is closely related to a complex of applied research. The article shows the advantages of the automated approach of logic programming in matters of geometric modeling in comparison with standard approaches. The research is based on the systematization of projection models of a four-dimensional space of a class of homogeneous and equally connected discrete-continuous structures. The rationale for the relevance of developing algorithms for creating computer-aided design systems for geometric shapes using software tools that provide graphical and grapho-analytical solutions is presented. The options for applying the principles of logic programming in solving geometric problems are outlined. Examples of solving problems on models of four-dimensional space in three flat fields are given. Attention is focused on solving similar problems for higher-dimensional spaces with testing using the example of solving multidimensional analogues of the Apollonius problem. Other projection models of multidimensional projective space are considered. It is established that this algorithm for solving the projection problem makes it possible to speed up and automate the construction of projection models of n-dimensional space and increase the visibility of the construction. The results of the study can form the basis for the development of new interfaces for interaction with geometric structures for solving surface design problems. The properties and methods of declarative programming in the construction of geometric models are defined. A software solution has been defined for working with geometric models in terms of automating the construction process, analyzing the algorithm content, searching, visualizing the result, and storing geometric information. The results of generating a solution to the Apollonius problem are presented, as well as a sample declarative program for running in the Simplex software environment (https://voloshinov. ru/simplex/), the Logic module.

KEYWORDS

geometric modeling, multidimensional space, Simplex, Apollonius problem, logic programming language, Prolog, predicate

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Voloshinov D. V., Shchur S. Y. Logical Interpretation of the Algorithm for Solving Apollonius' Problem // Geometry & Graphics. 2026. no. 1. pp. 3-14. DOI: https://doi.org/10.12737/2308-4898-2026-14-1-3-14. (Date of access 06.10.2026).

MLA

Denis Voloshinov, Shchur S. Y.. "Logical Interpretation of the Algorithm for Solving Apollonius' Problem." Geometry & Graphics 14.1 (2026): 3-14. DOI: https://doi.org/10.12737/2308-4898-2026-14-1-3-14.

APA

Voloshinov D. V., Shchur S. Y. (2026). Logical Interpretation of the Algorithm for Solving Apollonius' Problem. Geometry & Graphics, 14(1), 3-14. https://doi.org/10.12737/2308-4898-2026-14-1-3-14

Chicago

Denis Voloshinov, Shchur S. Y.. "Logical Interpretation of the Algorithm for Solving Apollonius' Problem." Geometry & Graphics 14, no. 1 (2026): 3-14. https://doi.org/10.12737/2308-4898-2026-14-1-3-14.

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{{cite journal | author = Voloshinov D. V., Shchur S. Y. | title = Logical Interpretation of the Algorithm for Solving Apollonius' Problem | journal = Geometry & Graphics | year = 2026 | volume = 14 | issue = 1 | pages = 3-14 | doi = 10.12737/2308-4898-2026-14-1-3-14 }}

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LOGICAL INTERPRETATION OF THE ALGORITHM FOR SOLVING APOLLONIUS' PROBLEM Logical Interpretation of the Algorithm for Solving Apollonius' Problem

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