Torus and Toroids: visualization of solids 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
Torus and Toroids
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.
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.
Augmented Reality | 3D Models | Home
Immersive room
Augmented Reality
To view torus and toroids in AR, simply visit the pages indicated in the 3D solid models using any browser with a webcam device (smartphone, tablet or notebook).
Access to the VR pages is done by clicking on the blue circle that appears on top of the marker.
3D models
1. Torus
The torus is a surface of revolution obtained by rotating a circumference of radius r around an axis coplanar with the circumference. The distance from the circumference center to the rotation center measures the tube radius R. The parametric equations for a torus azimuthally symmetric about the z-axis are: x = (R + r·cos(v))·cos(u), y = (R + r·cos(v))·sin(u) and z = r·sin(v), for u, v ∈ [0, 2π).
2. Polyhedral torus
Consider n equal cylinders frustums, equidistant from a point and with coplanar axes. The generated solid by the union of these cylinders frustums is a polyhedral torus with n sides, and the intersections of the cylinders frustums are circles with equal radii.
3. Torus knot
A (p, q) torus knot is obtained by winding a rope through the hole of a torus q times, with p revolutions before joining its ends, where p and q are relative prime numbers. The parametric equations for a torus azimuthally symmetric about the z-axis are: x = (R + r·cos(q·u))·cos(p·u), y = (R + r·cos(q·u))·sin(p·u) and z = r·sin(q·u), for u ∈ [0, 2π).
4. Polygonal toroid
The polygonal toroid is a surface of revolution obtained by rotating a polygon around an axis coplanar with the polygon.
5. Polyhedral toroid
Consider n equal regular prisms frustums, equidistant from a point P and with lateral edges orthogonal to the axis passing through P. The solid generated by the union of these prisms frustums is a polyhedral toroid with n sides, and the intersections of the prisms frustums are congruent regular polygons.
6. Polyhedral toroidal knot
A polyhedral toroidal knot (p, q) is obtained by winding a chain through the hole of a torus q times, with p revolutions before joining its ends, where p and q are relative prime numbers. The links of the chain are formed by prisms and prisms frustums.
7. Borromean rings: torus knot
The Borromean rings, also called Borromean links, are three interlocking rings named after the Italian Renaissance family that used them in their coat of arms. Removing any one ring leaves the other two unconnected. In this example, we have the Borromean rings made with torus knots with p = 1 and q = 2.
8. Borromean rings: polyhedral toroid
The Borromean rings, also called Borromean links, are three interlocking rings named after the Italian Renaissance family that used them in their coat of arms. Removing any one ring leaves the other two unconnected. In this example, we have the Borromean rings made with polyhedral toroids with n = 4.
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/basic/>, 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/