Paraboloid
A paraboloid is a quadric surface in three-dimensional Euclidean space defined by a quadratic equation, characterized by two primary types: the elliptic paraboloid, which forms a bowl-like shape, and the hyperbolic paraboloid, which exhibits a saddle-like curvature.[1] The elliptic paraboloid is generated by rotating a parabola about its axis of symmetry, resulting in a surface of revolution with the standard equation z = c ( \frac{x^2}{a^2} + \frac{y^2}{b^2} ), where a, b, and c > 0 determine its scaling and orientation; vertical cross-sections parallel to the z-axis yield parabolas, while horizontal cross-sections are ellipses (or circles if a = b).[1] This surface opens upward from the origin along the z-axis, resembling a paraboloidal cup, and can be shifted or rotated for various orientations.[1] In contrast, the hyperbolic paraboloid follows the equation z = c ( \frac{x^2}{a^2} - \frac{y^2}{b^2} ), producing a ruled surface (generatable by straight lines) with hyperbolic cross-sections in planes parallel to the xy-plane and parabolic sections in vertical planes; it flares upward in one direction and downward in the perpendicular direction, creating its distinctive saddle geometry.[1] Both types are unbounded and classified as non-degenerate quadrics, playing fundamental roles in multivariable calculus for studying surfaces and volumes.[1] Paraboloids find extensive applications in engineering and physics due to their unique reflective and structural properties. Elliptic paraboloids are commonly employed in optics for parabolic mirrors and lenses, such as in satellite dishes and radio telescopes, where they focus incoming parallel rays (e.g., electromagnetic waves) to a single focal point, enhancing signal collection efficiency.[2] Off-axis parabolic mirrors, derived from elliptic paraboloids, are particularly valued in astronomical instruments to avoid central obstructions and improve light gathering while being free from spherical aberration.[3] Hyperbolic paraboloids, meanwhile, are utilized in architecture for thin-shell roofs and bridges owing to their efficient load distribution and minimal material use, as well as in antenna design for dual-reflector systems that achieve wide bandwidths.[4] In advanced optics, tilted paraboloidal reflectors enable precise focusing of far-infrared rays in specialized lenses, supporting applications in spectroscopy and thermal imaging.[5]Definition and Classification
Mathematical Definition
A paraboloid is a quadric surface defined by a quadratic equation in three dimensions. The elliptic paraboloid can be geometrically defined as the locus of all points in three-dimensional space that are equidistant from a fixed point, known as the focus, and a fixed plane, known as the directrix plane.[6] This geometric property generalizes the definition of a parabola from two dimensions to three. Additionally, an elliptic paraboloid can be generated as a surface of revolution by rotating a parabola around its axis of symmetry.[7] The general equation for an elliptic paraboloid, oriented along the z-axis, is given by z = \frac{x^2}{a^2} + \frac{y^2}{b^2}, where the surface opens upward from the origin.[8] For a hyperbolic paraboloid, the equation takes the form z = \frac{x^2}{a^2} - \frac{y^2}{b^2}, resulting in a saddle-shaped surface.[8] These forms represent non-degenerate quadrics classified by their single axis of symmetry.[9] An elliptic paraboloid can be derived by rotating a parabolic cylinder, defined by z = x^2 / a^2 in the xz-plane, around the z-axis, which stretches the cross-sections into ellipses controlled by the parameter b.[10] The hyperbolic paraboloid is a ruled surface, composed of straight lines lying entirely on the surface.[11] The parameters a and b represent the semi-axes lengths that determine the scaling and eccentricity of the paraboloid's cross-sections; when a = b, the elliptic paraboloid becomes circular in symmetry.[8]Types of Paraboloids
Paraboloids, as a class of quadric surfaces, are categorized into elliptic and hyperbolic types based on the signature of the quadratic form in their defining equation, distinguishing their geometric and topological properties.[1] The elliptic paraboloid exhibits a bowl-shaped geometry, characterized by a positive definite quadratic form that causes the surface to open upward (or downward) in a single direction from its vertex.[12] Its standard equation in Cartesian coordinates is z = \frac{x^2}{a^2} + \frac{y^2}{b^2}, where a > 0 and b > 0 determine the scaling along the x- and y-axes, respectively.[1] This form ensures the surface is convex and simply connected, with cross-sections parallel to the xy-plane forming ellipses that expand as z increases.[13] In contrast, the hyperbolic paraboloid possesses a saddle-shaped structure, resulting from an indefinite quadratic form that allows the surface to open in two opposite directions along perpendicular axes.[12] Its equation is given by z = \frac{x^2}{a^2} - \frac{y^2}{b^2}, with a > 0 and b > 0, producing hyperbolic cross-sections in planes parallel to the xy-plane.[1] This configuration imparts negative Gaussian curvature throughout the surface, leading to a hyperbolic topology that distinguishes it from the elliptic case.[14] Degenerate cases of paraboloids occur when one of the coefficients in the quadratic form vanishes, such as in the equation z = x^2, which describes a parabolic cylinder as a limiting form where the surface extends infinitely in one direction without variation.[15] These degenerate forms represent transitional boundaries in the classification of quadric surfaces, blending paraboloidal and cylindrical characteristics.[16]Geometric Properties
Coordinate Representations
Paraboloids are quadric surfaces commonly expressed in Cartesian coordinates, where the equation involves quadratic terms in two variables and a linear term in the third. For an elliptic paraboloid, the standard form is z = x^2 + y^2, representing a surface that opens upward along the z-axis with circular cross-sections parallel to the xy-plane.[9] This form assumes unit scaling; more generally, it can be z = \frac{x^2}{a^2} + \frac{y^2}{b^2} to account for elliptical cross-sections, where a and b determine the semi-axes.[1] For a hyperbolic paraboloid, the standard form is z = x^2 - y^2, producing a saddle-shaped surface with hyperbolic cross-sections.[11] In general, these can be scaled as z = \frac{x^2}{a^2} - \frac{y^2}{b^2}, adjusting the curvature along each direction.[17] Parametric representations facilitate analysis and visualization by parameterizing the surface with two variables. For the elliptic paraboloid z = x^2 + y^2, a common parametrization is \begin{align*} x &= u \cos v, \\ y &= u \sin v, \\ z &= u^2, \end{align*} where u \geq 0 and v \in [0, 2\pi).[9] This uses polar-like parameters, with u scaling the radial distance and v the azimuthal angle. For the hyperbolic paraboloid z = x^2 - y^2, the parametrization is \begin{align*} x &= u, \\ y &= v, \\ z &= u^2 - v^2, \end{align*} with u, v \in \mathbb{R}.[11] This direct substitution highlights the ruled nature of the surface. For rotationally symmetric cases, such as the circular elliptic paraboloid, cylindrical coordinates (r, \theta, z) provide a natural adaptation, simplifying the equation to z = r^2, where r = \sqrt{x^2 + y^2} and \theta = \tan^{-1}(y/x).[18] This form is useful for integration or symmetry exploitation, as the surface depends only on r and z. Spherical coordinates (\rho, \phi, \theta) offer another adaptation for the same symmetric paraboloid z = x^2 + y^2, yielding \rho \cos \phi = \rho^2 \sin^2 \phi, or equivalently \rho = \cot \phi \csc \phi.[19] Hyperbolic paraboloids lack rotational symmetry, so these coordinate systems are less straightforward without additional transformations. The implicit quadric form accommodates paraboloids in arbitrary orientations, given by ax^2 + by^2 + cz = 2dx + 2ey + f, where the absence of squared terms in z (or the linear variable) distinguishes it from ellipsoids or hyperboloids. This equation represents a general paraboloid after translation and rotation to align with the principal axes, with coefficients a, b, c, d, e, f determining the type (elliptic if a and b have the same sign, hyperbolic if opposite) and position.Curvature and Focal Points
The elliptic paraboloid exhibits positive Gaussian curvature at every point, classifying all points as elliptic points where the surface bends in the same manner in all directions locally. For the surface defined by z = px^2 + qy^2 with p, q > 0, the Gaussian curvature is given byK = \frac{4pq}{(1 + 4p^2 x^2 + 4q^2 y^2)^2},
which is maximum at the vertex ($4pq) and decreases to zero as z \to \infty.[9] The mean curvature H is positive, reflecting the convex nature of the surface.[9] In contrast, the hyperbolic paraboloid has negative Gaussian curvature everywhere, resulting in hyperbolic points that form a saddle shape. For the surface z = \frac{x^2}{a^2} - \frac{y^2}{b^2}, the Gaussian curvature is
K = -\frac{4}{a^2 b^2 \left(1 + \frac{4x^2}{a^4} + \frac{4y^2}{b^4}\right)^2},
always negative and approaching zero far from the origin. The mean curvature H is zero for the standard form z = xy, indicating a minimal surface, though it varies in general parametrizations.[11] The principal curvatures \kappa_1 and \kappa_2 (with |\kappa_1| \geq |\kappa_2|) are the eigenvalues of the shape operator, satisfying K = \kappa_1 \kappa_2 and H = (\kappa_1 + \kappa_2)/2. For the elliptic paraboloid z = x^2 + 2y^2, at the vertex, \kappa_1 = 4 and \kappa_2 = 2, both positive, decreasing with distance from the vertex along principal directions aligned with the axes. For the hyperbolic paraboloid z = x^2 - y^2, at the vertex \kappa_1 = 2 and \kappa_2 = -2, with opposite signs; the hyperbolic paraboloid is doubly ruled, with straight-line generators (rulings) lying in principal directions where one curvature is non-zero while the other varies.[22] Focal points arise from the paraboloid's reflective geometry, derived via the law of reflection, which states that the incident ray, reflected ray, and surface normal are coplanar with equal angles to the normal. For the elliptic paraboloid z = \frac{x^2 + y^2}{4p}, parallel rays incident along the z-axis reflect to converge at the single focus (0, 0, p); this follows from parametrizing a ray at point (x, y, z) with unit incident vector \mathbf{V}_i = (0, 0, -1), computing the unit normal \mathbf{N} = \frac{(-x/z, -y/z, 1)}{\sqrt{1 + (x/z)^2 + (y/z)^2}}, and applying \mathbf{V}_r = \mathbf{V}_i - 2 (\mathbf{N} \cdot \mathbf{V}_i) \mathbf{N}, yielding trajectories intersecting at the focus. Unlike the elliptic paraboloid, the hyperbolic paraboloid lacks a single focal point but its saddle geometry allows for applications in optics involving beam deviation or wide-field reflection.[23]