Spheroid
A spheroid is a quadric surface formed by rotating an ellipse about one of its principal axes, resulting in a shape with two equal semi-axes and one distinct semi-axis, distinguishing it from a general ellipsoid with three unequal axes.[1] This surface of revolution approximates a sphere but exhibits elongation or flattening depending on the axis of rotation.[2] Spheroids are classified into two primary types: oblate and prolate. An oblate spheroid arises from rotating an ellipse about its minor axis, producing a flattened shape at the poles with an equatorial radius greater than the polar radius.[3] In contrast, a prolate spheroid results from rotation about the major axis, yielding an elongated form where the polar radius exceeds the equatorial radius.[4] In geodesy and planetary science, the oblate spheroid serves as a fundamental model for Earth's shape, accounting for the planet's equatorial bulge due to rotational forces, with the equatorial radius approximately 21 kilometers larger than the polar radius.[5] This approximation underpins reference ellipsoids used in global positioning systems (GPS) and cartographic projections, enabling precise mapping and navigation.[6] Prolate spheroids, while less common in natural contexts, appear in models of certain asteroids and in theoretical physics for symmetric potentials.[7]Fundamentals
Definition
A spheroid is a quadric surface formed by rotating an ellipse about one of its principal axes, generating a surface of revolution that extends the two-dimensional ellipse into three dimensions.[8] This rotation preserves the elliptical cross-sections while creating a symmetric shape around the axis of rotation, distinguishing it from a general ellipsoid, which has three unequal axes, as a spheroid specifically has two equal semi-axes.[1] Visually, a spheroid resembles a sphere that has been stretched or compressed along one direction, resulting in a more flattened or elongated profile depending on the axis of rotation. For instance, rotating an ellipse about its minor axis produces a shape widened at the equator and narrowed at the poles, akin to spinning a flattened circle to form a disc-like solid.[1] The key dimensions are the equatorial radius a, which measures the distance from the center to the equator along the plane perpendicular to the rotation axis, and the polar radius c, which measures along the rotation axis itself.[9] Spheroids are classified into two types based on the relative sizes of these radii: an oblate spheroid occurs when the equatorial radius exceeds the polar radius (a > c), creating a flattened appearance at the poles, while a prolate spheroid has a longer polar radius than equatorial (a < c), resulting in an elongated, rugby-ball-like form.[9] A sphere represents the special case of a spheroid where the equatorial and polar radii are equal (a = c).[1]Historical Context
The concept of the Earth's shape as a sphere was recognized by ancient Greek philosophers as early as the 5th century BCE, with Aristotle providing empirical evidence around 330 BCE through observations of lunar eclipses and the varying positions of stars, establishing a qualitative understanding of a rounded planet.[10] However, the notion of a spheroid—an ellipsoid of revolution deviated from a perfect sphere—emerged much later, with formal mathematical modeling beginning in the 17th century amid advances in mechanics and astronomy. In his Philosophiæ Naturalis Principia Mathematica published in 1687, Isaac Newton theorized that the Earth's rotation would cause centrifugal forces to flatten it at the poles and bulge it at the equator, predicting an oblate spheroid shape for a rotating fluid body in equilibrium.[11] This marked a pivotal shift from qualitative descriptions to quantitative predictions based on universal gravitation, influencing subsequent geodetic inquiries. During the 18th century, mathematicians refined Newton's model for practical geodetic applications. Colin Maclaurin provided a rigorous proof in 1740 for the equilibrium figure of a homogeneous rotating fluid, deriving the oblate spheroid as the stable form and enabling calculations of gravitational variations.[12] Independently, Alexis-Claude Clairaut developed a more general theory in his 1743 work Théorie de la figure de la Terre, accounting for density variations and confirming the oblate shape through differential equations that linked ellipticity to rotational effects, which supported expeditions measuring meridional arcs.[13] Advancements in the 19th and 20th centuries focused on precise ellipsoid approximations for global mapping and surveying, transitioning from theoretical models to standardized reference surfaces. Efforts culminated in the development of reference ellipsoids, such as the World Geodetic System 1984 (WGS84), adopted for international consistency in positioning and adopted by organizations like the U.S. Department of Defense.[14] This evolution in astronomy and geodesy progressed from Newton's qualitative insights to quantitative frameworks essential for accurate planetary modeling.Mathematical Formulation
Cartesian Equation
The Cartesian equation of a spheroid, which describes its surface in three-dimensional Cartesian coordinates, is \frac{x^2 + y^2}{a^2} + \frac{z^2}{c^2} = 1, where a denotes the semi-axis length in the equatorial plane (spanned by the x- and y-axes) and c denotes the semi-axis length along the polar z-axis.[1] This equation represents the canonical form for a spheroid aligned with the coordinate axes, distinguishing it as a special case of a quadric surface defined by a second-degree polynomial equation in x, y, and z./04%3A_Coordinate_Geometry_in_Three_Dimensions/4.03%3A_The_Ellipsoid) This form arises from the rotation of a two-dimensional ellipse about its symmetry axis. Consider the ellipse equation in the x z-plane given by \frac{x^2}{a^2} + \frac{z^2}{c^2} = 1; rotating this curve about the z-axis generates the spheroid, replacing x^2 with x^2 + y^2 to account for the circular symmetry in the equatorial plane./04%3A_Coordinate_Geometry_in_Three_Dimensions/4.03%3A_The_Ellipsoid)[15] The resulting surface is oblate if a > c (flattened at the poles, like Earth) or prolate if a < c (elongated along the polar axis, like a rugby ball).[1] In this convention, a is the equatorial semi-axis, reflecting the radius of the circular cross-section in the xy-plane at z = 0, while c is the polar semi-axis, corresponding to the extent along the axis of rotation.[16] This alignment ensures the equation captures the spheroid's rotational symmetry without loss of generality for axisymmetric cases.[15]Parametric Equations
The parametric equations provide an explicit representation of points on the surface of a spheroid, facilitating computations and visualizations in three-dimensional space. For a spheroid aligned with the z-axis, where a is the equatorial semi-axis and c is the polar semi-axis, the coordinates are given by \begin{align*} x &= a \sin \theta \cos \phi, \\ y &= a \sin \theta \sin \phi, \\ z &= c \cos \theta, \end{align*} with the polar angle \theta ranging from 0 to \pi and the azimuthal angle \phi ranging from 0 to $2\pi.[1] These parameters \theta and \phi parameterize the surface in a manner analogous to spherical coordinates for a sphere, but adjusted for the spheroid's eccentricity: \theta measures the colatitude from the positive z-pole, while \phi describes the longitude around the axis of rotation, ensuring full coverage of the surface without overlap except at the poles.[1] The equations derive from rotating a parametric ellipse in the xz-plane around the z-axis. The ellipse (x/a)^2 + (z/c)^2 = 1 is parameterized as x = a \sin \theta, z = c \cos \theta for \theta \in [0, \pi], and rotation by angle \phi yields the x and y components via the cylindrical transformation x' = x \cos \phi, y' = x \sin \phi, with z unchanged.[1] This parameterization offers advantages in numerical methods, such as generating surface plots by evaluating at discrete \theta and \phi grids, or performing surface integrals by leveraging the metric tensor derived from partial derivatives with respect to \theta and \phi.[1]Geometric Properties
Surface Area
The surface area of a spheroid is obtained by evaluating the surface integral over its parametric representation. The parametric equations are given by \mathbf{r}(\theta, \phi) = (a \sin \theta \cos \phi, a \sin \theta \sin \phi, c \cos \theta), where a is the equatorial semi-axis, c is the polar semi-axis, $0 \leq \theta \leq \pi, and $0 \leq \phi < 2\pi. The partial derivatives are \mathbf{r}_\theta = (a \cos \theta \cos \phi, a \cos \theta \sin \phi, -c \sin \theta) and \mathbf{r}_\phi = (-a \sin \theta \sin \phi, a \sin \theta \cos \phi, 0). The magnitude of their cross product is ||\mathbf{r}_\theta \times \mathbf{r}_\phi|| = a \sin \theta \sqrt{a^2 \cos^2 \theta + c^2 \sin^2 \theta}. Thus, the surface area S is S = \int_0^{2\pi} \int_0^\pi a \sin \theta \sqrt{a^2 \cos^2 \theta + c^2 \sin^2 \theta} \, d\theta \, d\phi = 2\pi a \int_0^\pi \sin \theta \sqrt{a^2 \cos^2 \theta + c^2 \sin^2 \theta} \, d\theta. This integral evaluates to distinct closed-form expressions for oblate and prolate spheroids, derived by substitution u = \cos \theta and recognizing the resulting form as solvable via hyperbolic or trigonometric functions, equivalent to certain elliptic integrals that simplify for the axisymmetric case.[17][18] For an oblate spheroid (a > c), the eccentricity is e = \sqrt{1 - (c/a)^2}. The exact surface area is S = 2\pi a^2 \left[ 1 + \frac{1 - e^2}{e} \tanh^{-1} e \right] = 2\pi a^2 + \frac{\pi c^2}{e} \ln \left( \frac{1 + e}{1 - e} \right), where \tanh^{-1} e = \frac{1}{2} \ln \left( \frac{1 + e}{1 - e} \right). This can also be expressed using the complete elliptic integral of the second kind E(e) = \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2 \psi} \, d\psi, though the elementary form is preferred for computation.[16] For a prolate spheroid (c > a), the eccentricity is e = \sqrt{1 - (a/c)^2}. The exact surface area is S = 2\pi a^2 \left[ 1 + \frac{1}{e} \sqrt{1 - e^2} \sin^{-1} e \right] = 2\pi a^2 + \frac{2\pi a c}{e} \sin^{-1} e. As with the oblate case, this arises from evaluating the parametric integral, reducing the elliptic form to elementary functions.[4] For small eccentricity (e \ll 1), applicable to near-spherical spheroids like Earth's oblate shape, the surface area approximates that of a sphere of radius a with corrections: S \approx 4\pi a^2 \left( 1 - \frac{1}{3} e^2 - \frac{1}{30} e^4 \right). Higher-order terms include -\frac{1}{840} e^6, but the e^2 term provides the primary deviation from sphericity. A similar expansion holds for prolate spheroids.[19]Volume
The volume V of a spheroid, defined by the equation \frac{x^2 + y^2}{a^2} + \frac{z^2}{c^2} = 1 where a is the equatorial semi-axis and c is the polar semi-axis, is given by the formula V = \frac{4}{3} \pi a^2 c. This expression arises as a special case of the general ellipsoid volume V = \frac{4}{3} \pi a b c when the intermediate axis equals the equatorial axis (b = a).[20][1] A direct derivation can be obtained using the disk method, integrating the cross-sectional areas perpendicular to the polar (z) axis. At a fixed height z between -c and c, the cross-section is a disk with radius r(z) = a \sqrt{1 - \frac{z^2}{c^2}}, so the area is \pi [r(z)]^2 = \pi a^2 \left(1 - \frac{z^2}{c^2}\right). The volume is then the integral V = \int_{-c}^{c} \pi a^2 \left(1 - \frac{z^2}{c^2}\right) \, dz = 2 \pi a^2 \int_{0}^{c} \left(1 - \frac{z^2}{c^2}\right) \, dz = 2 \pi a^2 \left[ z - \frac{z^3}{3 c^2} \right]_{0}^{c} = 2 \pi a^2 \left( c - \frac{c}{3} \right) = \frac{4}{3} \pi a^2 c. This approach works identically for both oblate (c < a) and prolate (c > a) spheroids. An alternative derivation employs triple integration in cylindrical coordinates or a change of variables scaling from the unit ball, yielding the same closed-form result.[21][22] When a = c = r, the spheroid degenerates to a sphere, and the volume simplifies to the familiar V = \frac{4}{3} \pi r^3. For a non-spherical spheroid, flattening (deviation of c from a) scales the volume relative to a sphere of equivalent "average" radius by the factor \frac{a^2 c}{r^3}, reducing it for oblate forms and increasing it for prolate ones compared to a sphere of radius a or c. Unlike the surface area, which requires elliptic integrals, the volume formula is elementary and does not involve special functions.[23]Circumference
The equatorial circumference of a spheroid, which lies in the plane perpendicular to the axis of rotation, forms a great circle of radius a, the semi-major axis. Thus, its length is given by C_e = 2\pi a. This expression follows directly from the geometry of a circle and applies to both oblate and prolate spheroids, where a is the equatorial radius.[24] The meridional circumference traces a closed meridian ellipse in a plane containing the axis of rotation, with semi-axes a (equatorial) and c (polar). Its total length is the perimeter of this ellipse. For an oblate spheroid (a > c), it is expressed as C_m = 4a E(e), where e = \sqrt{1 - (c/a)^2} is the eccentricity and E(e) denotes the complete elliptic integral of the second kind, defined by E(e) = \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2 \theta} \, d\theta. For a prolate spheroid (c > a), C_m = 4c E(e'), where e' = \sqrt{1 - (a/c)^2}.This integral arises in geodesy for computing distances on ellipsoidal models of Earth. To derive the meridional circumference for the oblate case, consider the parametric equations of the meridian ellipse in the x-z plane: x(\theta) = a \cos \theta, z(\theta) = c \sin \theta, where \theta is the parametric angle ranging from 0 to $2\pi. The arc length element is ds = \sqrt{(dx/d\theta)^2 + (dz/d\theta)^2} \, d\theta = \sqrt{a^2 \sin^2 \theta + c^2 \cos^2 \theta} \, d\theta. The quarter arc from equator to pole ( \theta = 0 to \pi/2 ) is a E(e), so the full circumference is four times this value. This formulation holds for the reference ellipsoid in coordinate systems like WGS84.[24] Circumferences of parallels at other latitudes \phi (measured from the equator) are circles parallel to the equator, with radius \nu(\phi) \cos \phi, where \nu(\phi) = a / \sqrt{1 - e^2 \sin^2 \phi} is the prime vertical radius of curvature for oblate spheroids. The length is thus C(\phi) = 2\pi \frac{a \cos \phi}{\sqrt{1 - e^2 \sin^2 \phi}}. For prolate spheroids, the formula requires adjustment using prolate coordinates. Unlike the meridian, this does not involve elliptic integrals, as each parallel is a true circle. For an oblate spheroid like Earth's, C(\phi) decreases from the equator toward the poles.[24]