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The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. (See ellipse.) [2] A prolate spheroid with c > a has surface area
Prolate spheroidal coordinates μ and ν for a = 1.The lines of equal values of μ and ν are shown on the xz-plane, i.e. for φ = 0.The surfaces of constant μ and ν are obtained by rotation about the z-axis, so that the diagram is valid for any plane containing the z-axis: i.e. for any φ.
As before, the oblate spheroid corresponding to σ is shown in red, and φ measures the azimuthal angle between the green and yellow half-planes. However, the surface of constant τ is a full one-sheet hyperboloid, shown in blue. This produces a two-fold degeneracy, shown by the two black spheres located at (x, y, ±z).
The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius r {\displaystyle r} , and caps with heights h 1 {\displaystyle h_{1}} and h 2 {\displaystyle h_{2}} , the area is
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid 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, an ellipsoid is ...
3.1 Spherical volume and area elements. 3.2 Polyspherical coordinates. ... Let be the surface area of the unit -sphere of ...
Better approximations can be made by modeling the entire surface as an oblate spheroid, using spherical harmonics to approximate the geoid, or modeling a region with a best-fit reference ellipsoid. For surveys of small areas, a planar (flat) model of Earth's surface suffices because the local topography overwhelms the curvature.
The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is =. It is also A = Ω r 2 {\displaystyle A=\Omega r^{2}} where Ω is the solid angle of the spherical sector in steradians , the SI unit of solid angle.