Torus and Toroids

Visualization of Torus and Toroids with Augmented Reality (AR) and Virtual Reality (VR) in A-frame

author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português

An ordinary torus is considered a genus 1 surface. This solid has a single "hole", and can be constructed from a rectangle by gluing together both pairs of opposite edges without twists. The usual torus embedded in three-dimensional space is shaped like a donut, but the concept of a torus is also extremely useful in higher-dimensional space.
In general, torus can also have multiple holes, with the term n-torus being used for a torus with n holes. The torus can be defined as the geometric locus formed by the rotation of a flat circular surface of radius r, around a circle of radius R.
The toroid is a surface of revolution obtained by rotating a closed plane curve, or a polygon, around an axis parallel to the plane that does not intersect the curve. The simplest toroid is the torus and the term toroid is used to refer to a toroidal polyhedron.
This work shows torus and toroids modeled in 3D, with views that can be accessed with Augmented Reality resources and also in immersive Virtual Reality rooms.

Immersive rooms

Augmented Reality and 3D models

The Augmented Reality environments were created with the Jerome Etienne scripts: AR.js - Augmented Reality for the Web.
The orbit scripts developed by Kevin Ngo were used in the Virtual Reality pages of the 3D models: Orbit controls for A-Frame.
The immersive rooms use the physical properties of 3D objects developed by Micah Blumberg: Physics for A-Frame VR
The interaction controls used in the immersive rooms were developed by Will Murphy: Super Hands

Torus and Toroids: visualization of solids with Augmented Reality and Virtual Reality by Paulo Henrique Siqueira is licensed with a license Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International.

How to cite this work:

Siqueira, P.H., "Torus and Toroids: visualization of solids with Augmented Reality and Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/>, February 2025.

References:
Weisstein, Eric W. “Torus” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/Torus.html
Weisstein, Eric W. “Toroid” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/Toroid.html
McCooey, D. I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/

Torus and Toroids