Journal for Geometry and Graphics 29 (2025), No. 2, 199--221
Copyright by the authors licensed under CC BY SA 4.0
3-D Shadows of 4-D Algebraic Hypersurfaces in a 4-D Perspective
Michal Zamboj
Faculty of Education, Charles University, Prague, Czech Republic
michal.zamboj@pedf.cuni.cz
Jakub Rada
Faculty of Architecture, Technical University, Prague, Czech Republic
jakub.rada@cvut.cz
Visualizing a scene of objects in 4-D space faces several challenges. Mere projections into 3- or less-dimensional spaces usually contain overlapping parts, making them difficult to comprehend or study. Illuminating the scene can enhance intuition about its "dimensionality". Our contribution describes a geometric approach to creating visualizations of 4-D hypersurfaces represented by implicit algebraic equations without their parametrization. By geometric, we mean methods using constructions of geometric objects without their approximation, for example, by polyhedral meshes. Therefore, instead of sets of many points and operating with meshes, we work with implicitly represented hypersurfaces, their projections, contours, intersections, etc. We provide a general algorithm to find shadow boundaries in an arbitrary dimension and apply it in a 4-D space. Furthermore, we design a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective. The results of our algorithm are components of the 3-D model of a scene image represented by polynomial equations and inequalities prepared for plotting by standard computer algebra systems with visualization tools. The method is presented on three 4-D scenes with gradual complexity created in Wolfram Mathematica. Since algebraic methods preserve many properties of the visualized shapes, they are suitable for precise mathematical or scientific visualization. On the other hand, processing higher-degree polynomials using elimination methods places greater demands on computational time.
Keywords: Four-dimensional visualization, algebraic hypersurface, shadow, implicit equation, elimination methods.
MSC: 51N15; 68U05, 14J70.
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