Torus

A torus (plural tori, from Latin torus, bead ') is a bead- shaped geometric shape that can be compared to the shape of a rescue ring floating tire or donuts. Specifically, three related terms are distinguished:

Embedded Tori

An embedded torus can be described as a set of points, having r of a circle having a radius of the fixed distance, which is.

Toruskoordinaten

One can in the torus surface, a surface of genus 1 is topologically ( ie, it has a hole ), a toroidal coordinate and introduce a perpendicular poloidal coordinate. The surface can be thought of as arising from a circle that is rotated about an axis which lies in the county level. The radius of the original circle we call this circle is also simultaneously a coordinate line. The distance of the circle center from the axis is called here the coordinates of lines are circles around the axis of rotation. Both coordinates are angular and run from 0 to.

A possible conversion to Cartesian (three-dimensional ) coordinates is (here is the position vector )

This representation is recovered for example from the parameterization of the position vector in the xy-plane and the XZ - plane.

Volume and surface area

Since the torus is a body of revolution, one can calculate volume and surface area by means of the guldinschen rule. Seen very simply, is a torus, a fully curved circular cylinder. Its lateral surface and volume change during the " kink-free bending " not because the inside is thereby compressed linearly by the same factor by which the outside is stretched - corresponding to the distances of the rectangular cylinder cross sections for large cylindrical cross- section passing through the focal points of the cylinder circuits and parallel to the torus axis (see principle of Cavalieri ).

The outwardly facing surface normal is in Cartesian coordinates

The surface element

By integration we obtain the surface of the torus:

A partial surface of the torus is obtained by integration into the confines to ( horizontally ) and up (vertical):

Results

To calculate the volume of the Volltorus one replaces the variable r r ' and leaves it from 0 ( degenerate to circular torus, no volume ) vary to r:

The Torusvolumen is the integral of the surface (above ).

External volume of the Spindeltorus

A circle with a radius and center point, the equation depending on the size of displays and in the first quadrant of a Cartesian coordinate system different arcs. Lets you rotate these arcs around the vertical coordinate axis, the result half Spindeltori which can be supplemented perpendicular to the axis of rotation to complete Spindeltori by reflection in the plane. In a Spindeltorus shows two peaks, at the degeneracy of the sphere and at the indentations ( apple shape ) which open from the Torusloch. The volume element is the distance from the rotational axis, h is the height, and designate the angle of rotation. Due to the existing cylindrical symmetry one finds the outer volume in the range as

From the volume is then ( the lower limit in the integral is now instead of 0 ). The surface also arises here from the derivative of volume with respect to the radius.

Moment of inertia of a Volltorus

The moment of inertia of a Volltorus with the density with respect to the axis ( axis of symmetry ) can be

Be calculated. Now you can perform the transformation to Toruskoordinaten. Here, in addition, the Jacobian comes into the integral.

With partial Integrate and Torusmasse one obtains:

Algebraic equation

The Rotationstorus can also be described by the following equation in the coordinates:

Or

You can, for example, from the equation

Derive the results from the theorem of Pythagoras.

Types of Tori

A flat torus can be described by a parallelogram, the opposite sides are glued together. Equivalently, can be described as a topological factor groups for two linearly independent vectors flat tori. In the special case and one obtains the quotient.

This Tori hot flat because their metric locally corresponds to the metric of the plane and its sectional curvature therefore disappears.

Elliptic curves over the complex numbers (with a translation-invariant metric ) Examples of flat tori.

Torus topology

In contrast to the surface of a sphere of the torus can be mapped without singularities on a flat, rectangular surface.

The right edge of the rectangle or square is stapled with its left edge and its lower edge is stapled with its upper edge. This topology also have a lot of computer games, such as Pacman, or the Game of Life.

Volltori

A Volltorus is a handle body of genus.

Embedded Volltori can be described as embedded tori in the parametric representation given above should be replaced only by a parameter with a range of values. Topologically is a Volltorus homeomorphic to the product of circular disk with the circle line.

The 3- sphere, ie the three-dimensional space along with a point at infinity, can be represented as a union of two Volltori, the only overlap in their surface. They are obtained for example from the Hopf fibration by conceives the base space as a union of northern and southern hemispheres; over both halves of the fibration is trivial. The decomposition of the 3- sphere in two Volltori is utilized for example in the construction of Reeb foliation.

Higher-Dimensional Tori

In the three-dimensional torus or 3 - torus is a square or cube whose six opposing faces are pairwise stapled together.

In the four-dimensional torus or 4- torus is a tesseract, whose eight opposing cube pairs stapled together.

In general, the dimensional torus is an n- dimensional cube, its opposite ( n -1) hypercubes are identified with each other in pairs. One can also pose as him.

The (n 1) -dimensional " volume " of an n- torus

The n-dimensional "surface"