Abstract
The double orthogonal projection of the 4-space onto two mutually perpendicular 3-spaces is a method of visualization of four-dimensional objects in a three-dimensional space. We present an interactive animation of the stereographic projection of a hyperspherical hexahedron on a 3-sphere embedded in the 4-space. Described are synthetic constructions of stereographic images of a point, hyperspherical tetrahedron, and 2-sphere on a 3-sphere from their double orthogonal projections. Consequently, the double-orthogonal projection of a freehand curve on a 3-sphere is created inversely from its stereographic image. Furthermore, we show an application to a synthetic construction of a spherical inversion and visualizations of double orthogonal projections and stereographic images of Hopf tori on a 3-sphere generated from Clelia curves on a 2-sphere.
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Notes
- 1.
The faces are not depicted due to insufficient possibilities of the surface parametrization in GeoGebra, but the reader can turn on the visibility of the corresponding spheres in the stereographic projection in the online model.
- 2.
This choice reflects the possibility of a definition of the Hopf fibration in the complex number plane, and so the common plane \(\pi (x,z)\) corresponds to the real parts of coordinates \((x+\i y, z+\i w)\) of points.
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Zamboj, M. (2021). Interactive 4-D Visualization of Stereographic Images from the Double Orthogonal Projection. In: Cheng, LY. (eds) ICGG 2020 - Proceedings of the 19th International Conference on Geometry and Graphics. ICGG 2021. Advances in Intelligent Systems and Computing, vol 1296. Springer, Cham. https://doi.org/10.1007/978-3-030-63403-2_11
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