Parametric Surfaces for Elliptic and Hyperbolic Geometries

László Szirmay-Kalos (), András Fridvalszky, László Szécsi and Márton Vaitkus
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László Szirmay-Kalos: Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Műegyetem rkp. 3, 1111 Budapest, Hungary
András Fridvalszky: Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Műegyetem rkp. 3, 1111 Budapest, Hungary
László Szécsi: Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Műegyetem rkp. 3, 1111 Budapest, Hungary
Márton Vaitkus: Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Műegyetem rkp. 3, 1111 Budapest, Hungary

Mathematics, 2025, vol. 13, issue 21, 1-25

Abstract: Background/Objectives : In computer graphics, virtual worlds are constructed and visualized through algorithmic processes. These environments are typically populated with objects defined by mathematical models, traditionally based on Euclidean geometry. However, there is increasing interest in exploring non-Euclidean geometries, which require adaptations of the modeling techniques used in Euclidean spaces. Methods : This paper focuses on defining parametric curves and surfaces within elliptic and hyperbolic geometries. We explore free-form splines interpreted as hierarchical motions along geodesics. Translation, rotation, and ruling are managed through supplementary curves to generate surfaces. We also discuss how to compute normal vectors, which are essential for animation and lighting. The rendering approach we adopt aligns with physical principles, assuming that light follows geodesic paths. Results : We extend the Kochanek–Bartels spline to both elliptic and hyperbolic geometries using a sequence of geodesic-based interpolations. Simple recursive formulas are introduced for derivative calculations. With well-defined translation and rotation in these curved spaces, we demonstrate the creation of ruled, extruded, and rotational surfaces. These results are showcased through a virtual reality application designed to navigate and visualize non-Euclidean spaces.

Keywords: splines; parametric curves and surfaces; hyperbolic geometry; elliptic geometry; transformations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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