~~==Definition==~~ ~~The '''torus''' in <math>\R^3</math> is constructed as follows:~~ ~~* Take a ''base circle'' centered at a point which we shall call the ''origin''. We shall call the radius of the base circle the ''average radius''.~~~~* Fix a length smaller than the average radius, called the ''tube radius''.~~~~* For each point on the torus, consider the circle centered at that point of radius equal to the tube radius, in the plane perpendicular to the plane of the base circle, and containing that center and the origin~~~~* The union of all such circles is termed the torus~~ ~~The torus can thus be thought of as the trace of a circle whose center is itself moving on a base circle, such that the plane of the circle always contains the origin of the base circle.~~ ~~Some further terminology:~~ ~~* The ''inner radius'' is the average radius minus the tube radius. It is the shortest possible distance between the origin and points on the torus. The set of points at this minimum distance forms a circle, called the ''inner rim'' or ''inner circle''.~~~~* The ''outer radius'' is the average radius plus the tube radius. It is the maximum distance between the origin and points on the torus. The set of points at this maximum distance form a circle, called the ''outer rim'' or ''outer circle''.~~~~* Consider two planes parallel to the plane containing the base circle, with distance from it equal to the tube radius. Both these planes are tangent to the torus, meeting it at circles. These circles are termed the ''top circle'' and ''bottom circle'' (interchangeably). The radius of these circles is equal to the average radius.~~ ~~==Equational descriptions==~~ ~~===Cartesian parametric equation===~~ ~~Let <math>r_1</math> denote the average radius and <math>r_2</math> denote the tube radius. Suppose the base circle is in the <math>xy</math>-plane and the origin of the torus is the origin. Then the parametric equations aer in terms of two angles, <math>\alpha</math> and <math>\beta</math>, where:~~ ~~* <math>x = r_1 \cos \alpha + r_2 \cos \alpha \cos \beta</math>~~~~* <math>y = r_1 \sin \alpha + r_2 \sin \alpha \cos \beta</math>~~~~* <math>z = r_2 \sin \beta</math>~~ ~~Both <math>\alpha</math> and <math>\beta</math> are modulo <math>2\pi</math>.~~~~==Abstract structure==~~ ~~Topologically, and even differentially, the torus is isomorphic to the direct product of ~~#redirect [[~~circle]]s, viz <math>S^1 \times S^1</math>. One coordinate describes the position of the center on the base circle, and the other coordinate describes the position of the point on that particular small circle. The Cartesian parametric equations given above make this explicit.~~ ~~However, the metric structure on the torus is very different from that on <math>S^1 \times S^1</math> with the direct product metric. The latter is actually the [[flat ~~ring torus]]~~, which cannot be embedded in <math>\R^3</math>, but needs to be embedded in <math>\R^4</math>.~~ ~~==Structure and symmetry==~~ ~~The isometry group of the torus is the semidirect product of the following two groups:~~ ~~* The group <math>S^1</math>, which acts on the first coordinate by rotating the base circle.~~~~* The group of order two which reflects the whole configuration about the <math>xy</math>-plane~~ ~~All these can be extended uniquely to isometries of <math>\R^3</math>.~~ ~~In fact, this is precisely the group of isometries of <math>\R^3</math> which fix the origin and the preserve the <math>z</math>-axis (though they may not preserve direction).~~ ~~==Constructions==~~ ~~{{surface of revolution of|circle}}~~ ~~The torus is obtained as the surface of revolution of a circle. The average radius of the torus equals the distance of the center of the circle from the axis of ervolution, and the tube radius equals the radius of the circle being revolved.~~ ~~{{tube of curve of|circle}}~~ ~~The torus is the tube of its base circle.~~