Hyperboloid
A hyperboloid is a quadric surface in three-dimensional Euclidean space defined by a second-degree equation, existing in two distinct forms: the one-sheeted hyperboloid, which is a connected surface resembling a tube or hourglass, and the two-sheeted hyperboloid, consisting of two separate bowl-shaped components.[1] The standard equation for a one-sheeted hyperboloid aligned along the z-axis is \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1, where a, b, and c are positive constants determining the scale along each axis, producing elliptical cross-sections parallel to the xy-plane and hyperbolic cross-sections in planes parallel to the xz- or yz-planes.[2] Similarly, the two-sheeted hyperboloid has the equation \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1, yielding no real intersection with planes parallel to the xy-plane for small |z| but hyperbolas elsewhere, with the sheets separated along the z-axis.[2] Both types of hyperboloids are ruled surfaces, meaning they can be entirely generated by the motion of straight lines, a property that distinguishes them among quadric surfaces and enables their construction with minimal material while maintaining structural integrity.[3] The one-sheeted hyperboloid, in particular, is doubly ruled, allowing two distinct families of lines to cover the surface, which facilitates applications in architecture and engineering.[4] For instance, Russian engineer Vladimir Shukhov pioneered the use of hyperboloid lattice structures in the late 19th century, constructing the first such tower—a 37-meter water tower—at the 1896 All-Russian Industrial Exhibition in Nizhny Novgorod, leveraging the form's aerodynamic stability and ease of assembly without extensive scaffolding.[5] This innovation influenced modern designs, including nuclear power plant cooling towers, which adopt the one-sheeted hyperboloid shape for optimal heat dissipation and wind resistance.[2] In mathematics, hyperboloids arise as surfaces of revolution by rotating a hyperbola around one of its axes—the transverse axis for the one-sheeted form and the conjugate axis for the two-sheeted form—and play key roles in analytic geometry, differential geometry, and relativity, where the two-sheeted hyperboloid models hyperbolic space in Minkowski spacetime.[6] Parametric representations, such as x = a \cosh u \cos v, y = a \cosh u \sin v, z = c \sinh u for the one-sheeted case (with u \in \mathbb{R}, v \in [0, 2\pi)), allow for detailed study of their curvature and geodesics.[7] These surfaces exemplify the diversity of quadric forms, bridging pure mathematics with practical engineering feats.Definition and Classification
Canonical Forms
The one-sheeted hyperboloid is a quadric surface that is doubly ruled by two families of straight lines, distinguishing it from ellipsoids, which are compact and non-ruled, and paraboloids, which exhibit parabolic rather than hyperbolic profiles. The two-sheeted hyperboloid is not ruled.[1] These surfaces arise as special cases of the general quadric equation and represent unbounded structures with hyperbolic cross-sections in principal planes. The canonical equation for the hyperboloid of one sheet, aligned with the coordinate axes, is \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1, where a, b, c > 0 are positive parameters that define the semi-axes lengths: a and b along the x- and y-directions, and c scaling the hyperbolic opening along the z-axis.[8] This form describes a connected surface that intersects the xy-plane in an ellipse with semi-axes lengths a and b.[7] For the hyperboloid of two sheets, the canonical equation is -\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, with the same parameters a, b, c > 0 interpreting the semi-axes similarly, though here the sheets separate along the z-axis, each extending infinitely in opposite directions without connecting.[9] The minimum distance between the sheets is $2c, occurring at the vertices (0, 0, \pm c).[6] Leonhard Euler introduced the hyperboloid in the 18th century, recognizing it as the hyperbolic counterpart to the ellipsoid within his classification of quadric surfaces.Distinction from Other Quadrics
Quadric surfaces are classified according to the signature of the quadratic form defining them, which corresponds to the number of positive, negative, and zero eigenvalues of the associated symmetric matrix.[10] This signature determines the geometric type: definite forms (all eigenvalues positive or all negative) yield ellipsoids, while indefinite forms (mixed signs) produce hyperboloids, and forms with a zero eigenvalue lead to paraboloids or cylinders.[11] Hyperboloids specifically exhibit indefinite signatures without zero eigenvalues in their non-degenerate cases, setting them apart from the positive definite signature of bounded ellipsoids and the signatures involving zeros in unbounded paraboloids.[10] The one-sheet hyperboloid has a signature of (2,1), meaning two positive and one negative eigenvalue, resulting in a connected, ruled surface.[11] In contrast, the two-sheet hyperboloid has a signature of (1,2), with one positive and two negative eigenvalues, producing two disconnected sheets.[10] Eigenvalue analysis of the quadratic form matrix distinguishes these types precisely; for example, in the canonical form \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1, the eigenvalues are \lambda_1 = 1/a^2 > 0, \lambda_2 = 1/b^2 > 0, and \lambda_3 = -1/c^2 < 0, confirming the one-sheet hyperboloid.[10] Ellipsoids require all \lambda_i > 0, such as in \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, while paraboloids involve one zero eigenvalue, as in the elliptic paraboloid z = \frac{x^2}{a^2} + \frac{y^2}{b^2}.[11] Degenerate hyperboloids arise when the matrix becomes singular or the equation factors linearly, such as when one eigenvalue is zero, leading to a hyperbolic cylinder with hyperbolic cross-sections extending infinitely along one axis.[10] Further degeneration, where the quadratic factors into two linear terms (e.g., (x - y)(x + y) = 0), results in a pair of intersecting planes.[12]Mathematical Representations
Cartesian Equations
The general equation of a quadric surface in three-dimensional Cartesian coordinates is given byAx^2 + By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + Gx + Hy + Iz + J = 0,
where A, B, C, D, E, F, G, H, I, J are constants.[13] This represents a hyperboloid when the associated quadratic form has a signature of either two positive and one negative eigenvalue (hyperboloid of one sheet) or one positive and two negative eigenvalues (hyperboloid of two sheets), provided the surface is non-degenerate and the constant term after reduction satisfies specific sign conditions.[10] To derive the general Cartesian equation for a hyperboloid from its canonical form, one first applies a translation to shift the coordinate origin to the center of the surface, eliminating the linear terms Gx + Hy + Iz. The center (x_0, y_0, z_0) is found by solving the system of partial derivatives set to zero:
$2Ax + 2D y + 2E z + G = 0,
$2D x + 2B y + 2F z + H = 0,
$2E x + 2F y + 2C z + I = 0.
This translation yields a centered equation of the form \mathbf{x'}^T A \mathbf{x'} + J' = 0, where \mathbf{x'} = (x - x_0, y - y_0, z - z_0)^T and A is the symmetric matrix
A = \begin{pmatrix} A & D & E \\ D & B & F \\ E & F & C \end{pmatrix}. [14] Next, an orthogonal transformation (rotation) is applied to diagonalize the quadratic form \mathbf{x'}^T A \mathbf{x'}, using the eigenvectors of A to determine the principal axes and orientation. The eigenvalues \lambda_1, \lambda_2, \lambda_3 of A provide the coefficients in the rotated coordinates (x'', y'', z''), resulting in \lambda_1 (x'')^2 + \lambda_2 (y'')^2 + \lambda_3 (z'')^2 + J' = 0. Normalizing by dividing through by -J' (assuming J' \neq 0) and scaling yields the canonical form for the one-sheet hyperboloid:
\frac{(x')^2}{a^2} + \frac{(y')^2}{b^2} - \frac{(z')^2}{c^2} = 1,
where the primed coordinates are aligned with the principal axes, and a, b, c > 0 are determined from the reciprocals of the eigenvalues adjusted for the right-hand side sign. For the two-sheet hyperboloid, the equation is
\frac{(x')^2}{a^2} + \frac{(y')^2}{b^2} - \frac{(z')^2}{c^2} = -1,
or equivalently,
-\frac{(x')^2}{a^2} - \frac{(y')^2}{b^2} + \frac{(z')^2}{c^2} = 1,
distinguished by the opposite sign configuration in the eigenvalue scaling to ensure the hyperbolic type./12%3A_Vectors_in_Space/12.06%3A_Quadric_Surfaces) The orientation is given by the rotation matrix whose columns are the normalized eigenvectors of A, and the semi-axes lengths a, b, c are identified from the diagonalized coefficients.[14]