Abstrakt
A classical sphere (2-sphere) is naturally embedded in a 3-dimensional Euclidean space. Analogically, in a higher dimension, a 3-sphere might be embedded in a 4-dimensional Euclidean space.
Computer visualizations and constructive methods of examining properties of a 3-sphere and quadric sections of hypercones have already been studied in a double orthogonal projection of a 4-space onto two mutually perpendicular 3-spaces. Based on its double orthogonal projection, a synthetic construction of a 3-sphere in a 4-dimensional perspective is proposed in this contribution.
The method of visualization is a 4-dimensional generalization of linear perspective. Instead of the picture plane, the image is a 3-dimensional model in the modeling 3-space, and the use of a double orthogonal projection of a 4-space onto two mutually perpendicular 3-spaces is an analogy to Monge's projection.
We discuss a perspective construction of a section of a 3-sphere with a 3-space in various positions. Consequently, we provide a construction of shades of a 3-sphere in central and directional lighting using polar properties of quadrics.
The boundary of a shade of a 3-sphere is a 2-spherical intersection of the 3-sphere and its polar 3-space, with the pole being the proper or improper source of light. The perspective image of a shade is a quadric embedded in the 3-dimensional modeling space.
Our virtual 3-D models and synthetic constructions are created in the interactive environment of the dynamic geometry software GeoGebra. The results are supplemented with visualizations in Wolfram Mathematica based on analytic representation.