Interactive 4-D Visualization of Stereographic Images From the Double Orthogonal Projection

TL;DR

This paper introduces an interactive method for visualizing four-dimensional objects using double orthogonal and stereographic projections, with applications to hyperspherical shapes and Hopf tori.

Contribution

It presents a novel interactive animation technique for visualizing 4D objects through stereographic projections and double orthogonal projections.

Findings

Developed synthetic constructions for stereographic images of hyperspherical objects.

Enabled inverse creation of double-orthogonal projections from stereographic images.

Applied visualization techniques to Hopf tori and spherical inversions.

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.