next surface previous surface 2D curves 3D curves surfaces fractals polyhedra

TORUS (GEOMETRIC NOTION)
Tore ouvertTore à trou nulTore croisé

From the Latin torus "cushion, bulge".
Illustrative names: inner tube, life preserver, etc...

 
Cylindrical equation:  (with  a = major radius, b = minor radius)
Toroidal parametrization: .
Cartesian parametrization the coordinate lines of which are the meridian and parallel circles: .
Cartesian parametrization the coordinate lines of which are the Villarceau and parallel circles, in the case a > b  (e  = 1 for one family of circles, -1 for the other one): 
Cartesian equation:  i.e. .
Rational quartic surface.
With the first parametrization:
First fundamental quadratic form: .
Surface element: .
Second fundamental quadratic form: .
Total curvature: .
Mean curvature: .
Volume and area for .

The torus is the surface generated by the revolution of a circle (C) around a line (D) of its plane; it is therefore a tube with constant diameter and circular bore.
Here (D) is the axis Oz, b (minor radius of the torus) the radius of (C) and a (major radius of the torus) the distance from its center to (D).
If (D) is secant to the circle (a £ b), we get the spindle torus, shaped like a pumpkin or a cherry with the limit cases of the sphere, if (D) is a diameter (a = 0), and the horn torus if (D) is a tangent of the circle (a = b).
Otherwise (usual case a > b) we get a ring torus, shaped like an inner tube.

The torus is a fourfold circled surface: except the meridians (sections by the planes passing by the axis of revolution) and the parallels (sections by the planes orthogonal to the axis), there exist two families of circles obtained by the sections by the bitangent planes of the torus, called Villarceau circles:
 
The strip located between two neighboring Villarceau circles looks like a Möbius strip but is not one since it has two edges. It has one twist.

The curves traced on the torus are the spirics (or toroidal curves).
See, in particular, the geodesics, the asymptotic lines and the rhumb lines of the torus.

For the contour of the projection of a torus, see toroid.
The inverse surfaces of the torus are the Dupin cyclides.
For a special torus, see Wilmore torus.

For the torus as a topological notion, see the next surface.

See also the Bohemian dome, Clifford's torus, and the sine tori.

A torus with its Villarceau circles, museum of the charity of Notre-Dame, Strasbourg, 16th century.
See also this beautiful virtual sculpture


      
Americans call the torus donut like the pastry, but in gastronomy, there are other tori like the onion rings... Lots of fruits are shaped (very approximately) like the internal part of a spindle torus (above, the fruit of the cacao tree). For the external part, think of a tomato. These fan vaults are shaped like the internal parts of the horn torus.

A pulley is a half-torus


 
net surface previous surface 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL, Jacques MANDONNET 2017