The Torus and Super-torus

Written by Paul Bourke
May 1990

The torus is perhaps the least used object in real modelling applications but it still appears as a standard form in modelling and rendering packages ahead of far more useful geometric primitives. In any case, given the conventions below, that is, defining the radius (r0) from the center to the middle of the ring of the torus and the radius (r1) of the cross section of ring of the torus.

The points on the torus are defined as

x = cos(theta) * [ r0 + r1 * cos(phi) ]
y = sin(theta) * [ r0 + r1 * cos(phi) ]
z = r1 * sin(phi)

Or in an implicit form

z2 + (sqrt(x2 + y2) - a)2 - b2 = 0

where theta and phi range from 0 to 2pi. Using the above formulation a torus with center radius r0 = 1 and outer radius r1 = 0.25 will look something like this


The code for generating a facet approximation to a torus as described later involves creating facets with vertices defined by

theta,phi
theta + dtheta,phi
theta + dtheta,phi + dphi
theta,phi + dphi

If a facet approximation to a torus is required then a large number of facets are generally necessary to produce a smooth rendered surface. In the wire frame example above dphi and dtheta were 10 degrees, this results in 36x36=1296 facets.

Another approach is to create a torus from a number of spheres which rendering packages often support more efficiently. This is achieved by simply packing spheres around a circular path, radius r0. The radius of the sphere being the radius r1. For example, the following shows the same torus dimension as in the earlier example created with 50 and 100 spheres.

C Source

This C code generated the facet representation shown above, the planar facets are exported in DXF for this example.

Super-toroid - A Geometric Primitive for Computer Aided Design

The super-toroid is a family of geometric primitives based on the torus. Let
r0 the radius of the inner ring
r1 the radius of the outer ring
as illustrated below where theta and phi range from 0 to 2pi.

The equation for the super-toroid is the same as that for the torus except that the sin and cosine terms are raised to powers.

It is different values of these powers which give rise to a family of 3D shapes all basically toroidal in shape. The value of n1 determines the shape of the torus ring, n2 determines the shape of the cross section of the ring. Note that with the normal definition of power the above equations are only defined for a single quadrant, namely that where the cosine and sine terms are positive. When creating the super-torus in practice one has two choices, either replicate the result in the one quadrant to the other three with the correct mirror operations, or define xn = sign(x) * abs(x)n.

Examples of the super-toroid generated for different values of n1 and n2 are shown below, of course, the legal values of n1 and n2 form a continuum of values from 0 to infinity (although there are representation issues near 0 and above 4).

An idea of the continuum of cross sectional shapes offered by squaring the sin() and cosine() terms consider the following

C source

Code which generates a facet based super-toroid in DXF format is given here. (Written for clarity not efficiency). Something similar for stl is here.





Elliptic Torus

Written by Paul Bourke
December 2000
x = (c + cos(v)) cos(u)
y = (c + cos(v)) sin(u)
z = sin(v) + cos(v)
-pi <= u,v <= pi

The surface doesn't intersect itself for c > 1.

c = 0.5

c = 1.0

c = 1.5

c = 2.5

Solid
Physical version created using the ZCorp Z406 colour rapid prototyping machine.




Limpet Torus

Graphics by Paul Bourke
February 2005

Attributed to Roger Bagula

x = cos(u) / [sqrt(2) + sin(v)]
y = sin(u) / [sqrt(2) + sin(v)]
z = 1 / [sqrt(2) + cos(v)]




Gumdrop torus

Graphics by Paul Bourke
March 2003

4 [x4 + (y2 + z2)2] + 17 x2 (y2 + z2) - 20 (x2 + y2 + z2) + 17 = 0

PovRay scene: gumdrop.pov




Figure 8 Torus

Written by Paul Bourke
December 2000
x = cos(u) (c + sin(v) cos(u) - sin(2 v) sin(u) / 2)
y = sin(u) (c + sin(v) cos(u) - sin(2 v) sin(u) / 2)
z = sin(u) sin(v) + cos(u) sin(2 v) / 2
-pi <= u,v <= pi

c = 1

c = 1.5

c = 0.5




2-Torus

By Roger Bagula
Graphics by Paul Bourke
July 2007
x = cos(u) (21/4 + cos(v))
y = sin(u) (21/4 + sin(v))
z = sqrt((u/pi)2 + (v/pi)2)




Saddle torus

By Roger Bagula
Graphics by Paul Bourke
August 2002
x = [2 + cos(u)] cos(v)
y = [2 + cos(u + 2 pi / 3)] cos(v + 2 pi / 3)
z = [2 + sign(F(u)) sqrt(|F(u)|)] sign(F(v)) sqrt(|F(v)|)
where F(s) = 1 - cos(s)2 - cos(s + 2 pi / 3)2
0 <= u <= 2 pi, 0 <= v <= 2 pi



Kinky Torus

Graphics by Paul Bourke
June 2003
Equations by Roger Bagula
x = sech(u) - cos(v)
y = sin(v)
z = u / PI - tanh(u)
-2 pi <= u <= 2 pi, -pi < v < pi



Klein Bottle

Written by Paul Bourke
October 1996

A German topologist named Klein
Thought the Möbius Loop was divine
Said he, "If you glue
The edges of two
You get a weird bottle like mine."

Most containers have an inside and an outside, a Klein bottle is a closed surface with no interior and only one surface. It is unrealisable in 3 dimensions without intersecting surfaces. It can be realised in 4 dimensions. The classical representation is shown below.

Featured in "mama 27", November 2000, pp 91, Figure 18.

The parametric equations to produce the above are:

A Klein bottle can be formed by taking two Möbius strips and join them along their boundaries, this so called "Figure-8 Klein Bottle" can be parameterized as follows:

   x = cos(u) (a + sin(v) cos(u/2) - sin(2v) sin(u/2)/2)
   y = sin(u) (a + sin(v) cos(u/2) - sin(2v) sin(u/2)/2)
   z = sin(u/2) sin(v) + cos(u/2) sin(2v)/2
where
   -pi <= u <= pi
and
   -pi <= v <= pi

Source code

This source code creates geometry files in the "geom" format. For a description of this file format see the specifications in the 3D file formats section. You may choose to translate the resulting geom file into your favourite file format or modify the code to create the format of your choice, all you need to do is modify the bit where the polygon is written to a file. This source code creates the classical Klein bottle shape as described at the top of this document.

If you are used to using GeomView then this code creates a CMESH that can be opened directly in that package. Alternatively if you have a package that allows plotting of regular meshes then this is the code to grab and modify.

If you would like to create a Klein bottle for a CAD package, this source code creates a DXF file of the classical Klein bottle. Alternatively, here is a sample DXF file created using the above program and tessellating the bottle by a 50x50 mesh.

Tessel version




Gray's Klein bottle

Inspired by Roger Bagula, Graphics by Paul Bourke

September 2012

p.x = (a + cos(n*u/2.0) * sin(v) - sin(n*u/2.0) * sin(2*v)) * cos(m*u/2.0)
p.y = (a + cos(n*u/2.0) * sin(v) - sin(n*u/2.0) * sin(2*v)) * sin(m*u/2.0)
p.z = sin(n*u/2.0) * sin(v) + cos(n*u/2.0) * sin(2*v)
0 <= u <= 4 pi
0 <= u <= 2 pi

a = 2, n = 2, m = 1


a = 1.5, n = 2, m = 1


a = 3.0, n = 2, m = 1


a = 3.0, n = 3, m = 1


a = 2.5, n = 2, m = 2


a = 2.5, n = 3, m = 3


a = 2, n = 2, m = 1


a = 2, n = 2, m = 1




Möbius (Möbius) strip

Written by Paul Bourke
May 1996

The Möbius strip is the simplest geometric shape which has only one surface and only one edge. It can be created by taking a strip of paper, giving it a half twist along its long axis, and then joining the two narrow ends together.

The Möbius strip in 3 dimensions can be represented parametrically f(s,t) as follows


where s ranges from 0 to 2*pi and t ranges typically from -0.4 to 0.4

An example of such a strip is shown below.



Featured in "mama 27", November 2000, pp 91, Figure 17.

The band for different values of t are illustrated below.

t = -1 -> 1

t = -0.1 -> 0.1

t = -0.5 -> 0.5

Increasing the range of t even further yields interesting folded and increasingly convoluted forms.




Bow Tie

Graphics by Paul Bourke
February 2005

Attributed to Roger Bagula

x = sin(u) / [sqrt(2) + cos(v)]
y = sin(u) / [sqrt(2) + sin(v)]
z = cos(u) / [1 + sqrt(2)]




Triaxial Tritorus

Supplied by Roger Bagula
Further discussion by Roger Bagula Mathematica code by Roger Bagula.

Graphics by Paul Bourke
September 1998
C Code by Paul Bourke.


The Triaxial Tritorus is defined parametrically as

x = sin(u) (1 + cos(v))
y = sin(u + 2 / 3) (1 + cos(v + 2 / 3))
z = sin(u + 4 / 3) (1 + cos(v + 4 / 3))

Where

- <= u <= and - <= v <=


Appeared on stickers for the MAA
(Mathematical Association of America).


Bohemian Dome

Graphics by Paul Bourke
February 2003
x = a cos(u)
y = a sin(u) + b cos(v)
z = c sin(v)
0 <= u <= 2 pi, 0 <= v <= 2 pi


Surface obtained by moving a circle that remains parallel to a plane along a curve that is perpendicular to the same plane.



Triaxial Hexatorus

Graphics by Paul Bourke
February 2005

Attributed to Roger Bagula

x = sin(u) / [sqrt(2) + cos(v)]
y = sin(u+2pi/3) / [sqrt(2) + cos(v+2pi/3)]
z = cos(u-2pi/3) / [sqrt(2) + cos(v-2pi/3)]




Twisted Pipe Surface

By Roger Bagula
Graphics by Paul Bourke
August 2002
x = cos(v) * (2 + cos(u)) / sqrt(1 + sin(v)2)
y = sin(v + 2pi/3) * (2 + cos(u + 2pi/3)) / sqrt(1 + sin(v)2)
z = sin(v - 2pi/3) * (2 + cos(u - 2pi/3)) / sqrt(1 + sin(v)2)
0 <= u <= 2 pi, 0 <= v <= 2 pi




Double torus

Graphics by Paul Bourke
April 2004
z2 = 0.04 - x4 + 2 x6 - x8 + 2 x2 y2 - 2 x4 y2 - y4
Credited to H. Karcher

PovRay scene: doubletorus.pov




Bow Curve

By Paul Bourke
August 2002
x = (2 + T sin(2 pi u)) sin(4 pi v)
y = (2 + T sin(2 pi u)) cos(4 pi v)
z = T cos(2 pi u) + 3 cos(2 pi v)
0 <= u <= 1, 0 <= v <= 1, 0 < T (Thickness)




Pretzel

Graphics by Paul Bourke
November 2004

[ ((x-1)2 + y2 - c2) ((x+1)2 + y2 - c2) ]2 + z2 = d



Bretzel

Graphics by Paul Bourke
November 2004

(x2 (1-x2) - y2)2 + z2 = 0.01



Pisot Triaxial

By Roger Bagula
Graphics by Paul Bourke

August 2002

x = 0.655866 cos(1.03002 + u) (2 + cos(v))
y = 0.754878 cos(1.40772 - u) (2 + 0.868837 cos(2.43773 + v))
z = 0.868837 cos(2.43773 + u) (2 + 0.495098 cos(0.377696 - v))
0 <= u <= 2 pi, 0 <= v <= 2 pi