Hyperboloid

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.

Hyperboloid of one sheet

conical surface in between

Hyperboloid of two sheets

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, one can choose a Cartesian coordinate system such that the hyperboloid is defined by one of the following equations: $${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1,}$$ or $${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1.}$$ The coordinate axes are axes of symmetry of the hyperboloid and the origin is the center of symmetry of the hyperboloid. In any case, the hyperboloid is asymptotic to the cone of the equations: $${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0.}$$

One has a hyperboloid of revolution if and only if ${\displaystyle a^{2}=b^{2}.}$ 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): 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 (−1 in the right-hand side of the equation): a two-sheet hyperboloid, also called an elliptic hyperboloid. The surface has two connected components and a positive Gaussian curvature at every point. 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 ∈ (−∞, ∞) $${\displaystyle {\begin{aligned}x&=a\cosh v\cos \theta \\y&=b\cosh v\sin \theta \\z&=c\sinh v\end{aligned}}}$$

Two-surface hyperboloid: v ∈ [0, ∞) $${\displaystyle {\begin{aligned}x&=a\sinh v\cos \theta \\y&=b\sinh v\sin \theta \\z&=\pm c\cosh v\end{aligned}}}$$

The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the ${\displaystyle z}$-axis as the axis of symmetry: $${\displaystyle \mathbf {x} (s,t)=\left({\begin{array}{lll}a{\sqrt {s^{2}+d}}\cos t\\b{\sqrt {s^{2}+d}}\sin t\\cs\end{array}}\right)}$$

• For ${\displaystyle d>0}$0}"> one obtains a hyperboloid of one sheet,
• For ${\displaystyle d<0}$ a hyperboloid of two sheets, and
• For ${\displaystyle d=0}$ a double cone.

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 ${\displaystyle cs}$ term to the appropriate component in the equation above.

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: $${\displaystyle q(x)=\left(x_{1}^{2}+\cdots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\cdots +x_{n}^{2}\right),\quad k<n.}$$ When c is any constant, then the part of the space given by $${\displaystyle \lbrace x\$$ :\ q(x)=c\rbrace } is called a hyperboloid. The degenerate case corresponds to c = 0.

As an example, consider the following passage:^[4]

... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates (y₁, ..., y₄), its equation is y²
₁ + y²
₂ + y²
₃ − y²
₄ = −1, analogous to the hyperboloid y²
₁ + y²
₂ − y²
₃ = −1 of three-dimensional space.^[6]

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.

• Gallery of one sheet hyperboloid structures
• 
  The Adziogol Lighthouse, Ukraine, 1911.
• 
  The first 1916 patented Van Iterson cooling tower of DSM Emma in Heerlen, The Netherlands, 1918
• 
  Kobe Port Tower, Japan, 1963.
• 
  Saint Louis Science Center's James S. McDonnell Planetarium, St. Louis, Missouri, 1963.
• 
  Newcastle International Airport control tower, Newcastle upon Tyne, England, 1967.
• 
  Ještěd Transmission Tower, Czech Republic, 1968.
• 
  Cathedral of Brasília, Brazil, 1970.
• 
  Hyperboloid water tower with toroidal tank, Ciechanów, Poland, 1972.
• 
  Roy Thomson Hall, Toronto, Canada, 1982.
• 
  The THTR-300 cooling tower for the now decommissioned thorium nuclear reactor in Hamm-Uentrop, Germany, 1983.
• 
  The Corporation Street Bridge, Manchester, England, 1999.
• 
  The Killesberg observation tower, Stuttgart, Germany, 2001.
• 
  BMW Welt, (BMW World), museum and event venue, Munich, Germany, 2007.
• 
  The Canton Tower, China, 2010.
• 
  The Essarts-le-Roi water tower, France.

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:

... the equation of the unit sphere ρ² + 1 = 0, and change the vector ρ to a bivector form, such as σ + τ √−1. The equation of the sphere then breaks up into the system of the two following,

σ² − τ² + 1 = 0, S.στ = 0;

and suggests our considering σ and τ as two real and rectangular vectors, such that

Tτ = (Tσ² − 1 )^1/2.

Hence it is easy to infer that if we assume σ || λ, where λ is a vector in a given position, the new real vector σ + τ will terminate on the surface of a double-sheeted and equilateral hyperboloid; and that if, on the other hand, we assume τ || λ, then the locus of the extremity of the real vector σ + τ will be an equilateral but single-sheeted hyperboloid. The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the 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) ∈ R⁴ determined by quadratic forms. First consider the conical hypersurface

• ${\displaystyle P=\left\{p\;:\;w^{2}=x^{2}+y^{2}+z^{2}\right\}}$ and
• ${\displaystyle H_{r}=\lbrace p\$ :\ w=r\rbrace ,} which is a hyperplane.

Then ${\displaystyle P\cap H_{r}}$ is the sphere with radius r. On the other hand, the conical hypersurface

${\displaystyle Q=\lbrace p\$ :\ w^{2}+z^{2}=x^{2}+y^{2}\rbrace } provides that ${\displaystyle Q\cap H_{r}}$ is a hyperboloid.

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.^[7]

• De Sitter space
• Ellipsoid
• List of surfaces
• Paraboloid / Hyperbolic paraboloid
• Regulus
• Rotation of axes
• Split-quaternion § Profile
• Translation of axes