Hyperboloid of one sheet |
conical surface in between |
Hyperboloid of two sheets |
In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. An hyperboloid is a surface that may be obtained from a paraboloid 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 that may be 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 also 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 axes of symmetry of the hyperboloid, and origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations:
or
Both of these surfaces are asymptotic to the cone of equation
One has an hyperboloid of revolution if and only if $a^{2}=b^{2}.$ Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis.
There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation), one has a one-sheet hyperboloid, also called hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies that the tangent plane at any point intersect the hyperboloid into two lines, and thus that the one-sheet hyperboloid is a doubly ruled surface.
In the second case (−1 in the right-hand side of the equation), one has a two-sheet hyperboloid, also called 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.
Cartesian coordinates for the hyperboloids can be defined, similar to spherical coordinates, keeping the azimuth angle θ ∈ [0, 2π), but changing inclination v into hyperbolic trigonometric functions:
One-surface hyperboloid: v ∈ (−∞, ∞)
Two-surface hyperboloid: v ∈ [0, ∞)
If the hyperboloid has the equation ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1$ then the lines
are contained in the surface.
In case of $a=b$ the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines $g_{0}^{+}$ or $g_{0}^{-}$, which are skew to the rotation axis (see picture). The more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis (see picture).
Remark: A hyperboloid of two sheets is projectively equivalent to a hyperbolic paraboloid.
For simplicity the plane sections of the unit hyperboloid with equation $\ H_{1}:x^{2}+y^{2}-z^{2}=1$ 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 (see circular section).
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 (the one that cuts the hyperbola)
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see circular section).
Remark: A hyperboloid of two sheets is projectively equivalent to a sphere.
The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the $z$-axis as the axis of symmetry:
${\vec {x}}(s,t)=\left({\begin{array}{lll}a{\sqrt {s^{2}+d}}\cos t\\b{\sqrt {s^{2}+d}}\sin t\\cs\end{array}}\right)$
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 $cs$ term to the appropriate component in the equation above.
The hyperboloids with equations ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1,\quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\$ 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.
More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation
where A is a matrix and x, v are vectors.
The eigenvectors of A define the principal directions of the hyperboloid and the eigenvalues of A are the reciprocals of the squares of the semi-axes: ${1/a^{2}}$, ${1/b^{2}}$ and ${1/c^{2}}$. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.
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 c is any constant, then the part of the space given by
is called a hyperboloid. The degenerate case corresponds to c = 0.
As an example, consider the following passage:^{[3]}
However, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality (See § Relation to the sphere below).
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.
Canton Tower, a one-sheet hyperboloid
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 S is the operator giving the scalar part of a quaternion, and T 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 p = (w, x, y, z) ∈ R^{4} determined by quadratic forms. First consider the conical hypersurface
Then $P\cap H_{r}$ is the sphere with radius r. On the other hand, the conical hypersurface
In the theory of quadratic forms, a unit quasi-sphere is the subset of a quadratic space X consisting of the x ∈ X such that the quadratic norm of x is one.^{[4]}
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