Torus

A torus is a surface having Genus 1, and therefore possessing a single ``Hole.'' The usual torus in 3-D space is shaped like a donut, but the concept of the torus is extremely useful in higher dimensional space as well. One of the more common uses of $n$-D tori is in Dynamical Systems. A fundamental result states that the Phase Space trajectories of a Hamiltonian System with $n$ Degrees of Freedom and possessing $n$ Integrals of Motion lie on an $n$-D Manifold which is topologically equivalent to an $n$-torus (Tabor 1989).

The usual 3-D ``ring'' torus is known in older literature as an ``Anchor Ring.'' Let the radius from the center of the hole to the center of the torus tube be $c$, and the radius of the tube be $a$. Then the equation in Cartesian Coordinates is

$$(c-\sqrt{x^2+y^2}\,)^2+z^2=a^2.$$ (1)

The parametric equations of a torus are

$$x = (c+a\cos v)\cos u$$ (2)

$$y = (c+a\cos v)\sin u$$ (3)

$$z = a\sin v$$ (4)

for $u,v\in [0,2\pi)$. Three types of torus, known as the Standard Tori, are possible, depending on the relative sizes of $a$ and $c$. $c>a$ corresponds to the Ring Torus (shown above), $c=a$ corresponds to a Horn Torus which is tangent to itself at the point (0, 0, 0), and $c<a$ corresponds to a self-intersecting Spindle Torus (Pinkall 1986).

If no specification is made, ``torus'' is taken to mean Ring Torus. The three Standard Tori are illustrated below, where the first image shows the full torus, the second a cut-away of the bottom half, and the third a Cross-Section of a plane passing through the z-Axis.

The Standard Tori and their inversions are Cyclides. If the coefficient of $\sin v$ in the formula for $z$ is changed to $b\not=a$, an Elliptic Torus results.

To compute the metric properties of the ring torus, define the inner and outer radii by

$$r \equiv c-a$$ (5)

$$R \equiv c+a.$$ (6)

Solving for $a$ and $c$ gives

$$a = {\textstyle{1\over 2}}(R-r)$$ (7)

$$c = {\textstyle{1\over 2}}(R+r).$$ (8)

Then the Surface Area of this torus is

$$S = (2\pi a)(2\pi c) = 4\pi^2 ac$$ (9)

$$= \pi^2(R+r)(R-r),$$ (10)

and the Volume can be computed from Pappus's Centroid Theorem

$$V = (\pi a^2)^2\pi c = 2\pi^2a^2c$$ (11)

$$= {\textstyle{1\over 4}}\pi^2(R+r)(R-r)^2.$$ (12)

The coefficients of the first and second Fundamental Forms of the torus are given by

$$e = -(c+a\cos v)\cos v$$ (13)

$$f = 0$$ (14)

$$g = -a$$ (15)

$$E = (c+a\cos v)^2$$ (16)

$$F = 0$$ (17)

$$G = a^2,$$ (18)

giving Riemannian Metric

$$ds^2=(c+a\cos v)^2\,du^2+a^2\,dv^2,$$ (19)

Area Element

$$dA=a(c+a\cos v)\,du\wedge dv$$ (20)

(where $du\wedge dv$ is a Wedge Product), and Gaussian and Mean Curvatures as

$$K = {\cos v\over a(c+a\cos v)}$$ (21)

$$H = -{c+2a\cos v\over 2a(c+a\cos v)}$$ (22)

(Gray 1993, pp. 289-291).

A torus with a Hole in its surface can be turned inside out to yield an identical torus. A torus can be knotted externally or internally, but not both. These two cases are Ambient Isotopies, but not Regular Isotopies. There are therefore three possible ways of embedding a torus with zero or one Knot.

An arbitrary point $P$ on a torus (not lying in the $xy$-plane) can have four Circles drawn through it. The first circle is in the plane of the torus and the second is Perpendicular to it. The third and fourth Circles are called Villarceau Circles (Villarceau 1848, Schmidt 1950, Coxeter 1969, Melnick 1983).

To see that two additional Circles exist, consider a coordinate system with origin at the center of torus, with $\hat{\bf z}$ pointing up. Specify the position of $P$ by its Angle $\phi$ measured around the tube of the torus. Define $\phi=0$ for the circle of points farthest away from the center of the torus (i.e., the points with $x^2+y^2=R^2$), and draw the x-Axis as the intersection of a plane through the $z$-axis and passing through $P$ with the $xy$-plane. Rotate about the y-Axis by an Angle $\theta$, where

$$\theta=\sin^{-1}\left({a\over c}\right).$$ (23)

In terms of the old coordinates, the new coordinates are

$$x = x_1\cos\theta-z_1\sin\theta$$ (24)

$$z = x_1\sin\theta+z_1\cos\theta.$$ (25)

So in $(x_1, y_1, z_1)$ coordinates, equation (1) of the torus becomes

$$[\sqrt{(x_1\cos\theta-z_1\sin\theta)^2+{y_1}^2}-c]^2+(x_1\sin\theta+z_1\cos\theta)^2=a^2.$$ (26)

Squaring both sides gives

$(x_1\cos\theta-z_1\sin\theta)^2+{y_1}^2+c^2$
$-2c\sqrt{(x_1\cos\theta-z_1\sin\theta)^2+{y_1}^2}$
$+(x_1\sin\theta+z_1\cos\theta)^2=a^2.\quad$	(27)

But

$$(x_1\cos\theta-z_1\sin\theta)^2+(x_1\sin\theta+z_1\cos\theta)^2={x_1}^2+{z_1}^2,$$ (28)

so

$${x_1}^2+{y_1}^2+{z_1}^2+c^2-2c\sqrt{(x_1\cos\theta-z_1\sin\theta)^2+{y_1}^2}=a^2.$$ (29)

In the $z_1=0$ plane, plugging in (23) and factoring gives

$$[{x_1}^2+(y_1-a)^2-c^2][{x_1}^2+(y_1+a)-c^2]=0.$$ (30)

This gives the Circles

$${x_1}^2+(y_1-a)^2=c^2$$ (31)

and

$${x_1}^2+(y_1+a)^2=c^2$$ (32)

in the $z_1$ plane. Written in Matrix form with parameter $t\in [0,2\pi)$, these are

$$C_1 = \left[\begin{array}{c}c\cos t\\ c\sin t+a\\ 0\end{array}\right]$$ (33)

$$C_2 = \left[\begin{array}{c}c\cos t\\ c\sin t-a\\ 0\end{array}\right].$$ (34)

In the original $(x,y,z)$ coordinates,

$$C_1 = \left[\begin{array}{ccc}\cos\theta & 0 & -\sin\theta\\ 0 & 1 & 0... ...array}\right] \left[\begin{array}{c}c\cos t\\ c\sin t+a\\ 0\end{array}\right]$$

$$= \left[\begin{array}{c}c\cos\theta\cos t\\ c\sin t+a\\ -c\sin\theta\cos t\end{array}\right]$$ (35)

$$C_2 = \left[\begin{array}{ccc}\cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\... ...array}\right] \left[\begin{array}{c}c\cos t\\ c\sin t-a\\ 0\end{array}\right]$$

$$= \left[\begin{array}{c}c\cos\theta\cos t\\ c\sin t-a\\ -c\sin\theta\cos t\end{array}\right].$$ (36)

The point $P$ must satisfy

$$z=a\sin\phi=c\sin\theta\cos t,$$ (37)

so

$$\cos t={a\sin\phi\over c\sin\theta}.$$ (38)

Plugging this in for $x_1$ and $y_1$ gives the Angle $\psi$ by which the Circle must be rotated about the z-Axis in order to make it pass through $P$,

$$\psi=\tan^{-1}\left({y\over x}\right)={c\sin t+a\over c\cos\theta\cos t}={c\sqrt{1-\cos^2t}+a\over c\cos\theta\cos t}.$$ (39)

The four Circles passing through $P$ are therefore

$$C_1 = \left[\begin{array}{ccc}\cos\psi & \sin\psi & 0\\ -\sin\psi & \c... ...array}{c}c\cos\theta\cos t\\ c\sin t+a\\ -c\sin\theta\cos t\end{array}\right]$$ (40)

$$C_2 = \left[\begin{array}{ccc}\cos\psi & \sin\psi & 0\\ -\sin\psi & \c... ...array}{c}c\cos\theta\cos t\\ c\sin t-a\\ -c\sin\theta\cos t\end{array}\right]$$ (41)

$$C_3 = \left[\begin{array}{c}(c+a\cos\phi)\cos t\\ (c+a\cos\phi)\sin t\\ a\sin\phi\end{array}\right]$$ (42)

$$C_4 = \left[\begin{array}{c}c+a\cos t\\ 0\\ a\sin t\end{array}\right].$$ (43)

See also Apple, Cyclide, Elliptic Torus, Genus (Surface), Horn Torus, Klein Quartic, Lemon, Ring Torus, Spindle Torus, Spiric Section, Standard Tori, Toroid, Torus Coloring, Torus Cutting

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131-132, 1987.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 132-133, 1969.

Geometry Center. ``The Torus.'' http://www.geom.umn.edu/zoo/toptype/torus/.

Gray, A. ``Tori.'' §11.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 218-220 and 289-290, 1993.

Melzak, Z. A. Invitation to Geometry. New York: Wiley, pp. 63-72, 1983.

Pinkall, U. ``Cyclides of Dupin.'' §3.3 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 28-30, 1986.

Schmidt, H. Die Inversion und ihre Anwendungen. Munich: Oldenbourg, p. 82, 1950.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 71-74, 1989.

Villarceau, M. ``Théorème sur le tore.'' Nouv. Ann. Math. 7, 345-347, 1848.