Hyperboloid



Hyperboloid of one sheet
conical surface in between
Hyperboloid of two sheets

In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
A hyperboloid is a quadric surface, that is, a surface defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry.
Given a hyperboloid, if one chooses a Cartesian coordinate system whose axes are the axes of symmetry of the hyperboloid and the origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations:
or
Both surfaces are asymptotic to the cone of the equation
The surface is a hyperboloid of revolution if and only if Otherwise, the axes are uniquely defined.
There are two kinds of hyperboloids. In the first case : a one-sheet hyperboloid, also called a hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies near every point the intersection of the hyperboloid and its tangent plane at the point consists of two branches of curve that have distinct tangents at the point. In the case of the one-sheet hyperboloid, these branches of curves are lines and thus the one-sheet hyperboloid is a doubly ruled surface.
In the second case : a two-sheet hyperboloid, also called an elliptic hyperboloid. The surface has two connected components and a positive Gaussian curvature at every point. Thus the surface is convex in the sense that the tangent plane at every point intersects the surface only in this point.

Parametric representations

Cartesian coordinates for the hyperboloids can be defined, similar to spherical coordinates, keeping the azimuth angle, but changing inclination into hyperbolic trigonometric functions:
One-surface hyperboloid:
Two-surface hyperboloid:

Properties of a hyperboloid of one sheet

Lines on the surface

If the hyperboloid has the equation
then the lines
are contained in the surface.
In case the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines or, which are skew to the rotation axis. This property is called Wren's theorem. The more common generation of a one-sheet hyperboloid of revolution is rotating a hyperbola around its semi-minor axis.
A hyperboloid of one sheet is projectively equivalent to a hyperbolic paraboloid.

Plane sections

For simplicity the plane sections of the unit hyperboloid with equation are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too.
Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case.

Properties of a hyperboloid of two sheets

The hyperboloid of two sheets does not contain lines. The discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation
which can be generated by a rotating hyperbola around one of its axes
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case.
Remark: A hyperboloid of two sheets is projectively equivalent to a sphere.

Common parametric representation

The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the -axis as the axis of symmetry:
One can obtain a parametric representation of a hyperboloid with a different coordinate axis as the axis of symmetry by shuffling the position of the term to the appropriate component in the equation above.

Symmetries of a hyperboloid

The hyperboloids with equations
are
Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.

Generalised equations

More generally, an arbitrarily oriented hyperboloid, centered at, is defined by the equation
where is a matrix and, are vectors.
The eigenvectors of define the principal directions of the hyperboloid and the eigenvalues of A are the reciprocals of the squares of the semi-axes:, and. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.

In more than three dimensions

Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a pseudo-Euclidean space one has the use of a quadratic form:
When is any constant, then the part of the space given by
is called a hyperboloid. The degenerate case corresponds to.
As an example, consider the following passage:
However, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality.

Hyperboloid structures

One-sheeted hyperboloids are used in construction, with the structures called hyperboloid structures. A hyperboloid is a doubly ruled surface; thus, it can be built with straight steel beams, producing a strong structure at a lower cost than other methods. Examples include cooling towers, especially of power stations, and many other structures.

Relation to the sphere

In 1853 William Rowan Hamilton published his Lectures on Quaternions which included presentation of biquaternions. The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from quaternions to produce hyperboloids from the equation of a sphere:
In this passage is the operator giving the scalar part of a quaternion, and is the "tensor", now called norm, of a quaternion.
A modern view of the unification of the sphere and hyperboloid uses the idea of a conic section as a slice of a quadratic form. Instead of a conical surface, one requires conical hypersurfaces in four-dimensional space with points determined by quadratic forms. First consider the conical hypersurface
Then is the sphere with radius. On the other hand, the conical hypersurface
In the theory of quadratic forms, a unit quasi-sphere is the subset of a quadratic space consisting of the such that the quadratic norm of is one.