The three coordinate surfaces of prolate spheroidal coordinates. The red prolate spheroid (stretched sphere) corresponds to μ = 1, and the blue two-sheet hyperboloid corresponds to ν = 45°. The yellow half-plane corresponds to φ = −60°, which is measured relative to the x-axis (highlighted in green). The black sphere represents the intersection point of the three surfaces, which has Cartesian coordinates of roughly (0.831, −1.439, 2.182).
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.
Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the z-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H2+. Another example is solving for the electric field generated by two small electrode tips. Other limiting cases include areas generated by a line segment (μ = 0) or a line with a missing segment (ν=0). The electronic structure of general diatomic molecules with many electrons can also be solved to excellent precision in the prolate spheroidal coordinate system.[1]
^Lehtola, Susi (21 May 2019). "A review on non-relativistic, fully numerical electronic structure calculations on atoms and diatomic molecules". Int. J. Quantum Chem. 119: e25968. arXiv:1902.01431. doi:10.1002/qua.25968.
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produces oblate spheroidalcoordinates. Prolatespheroidalcoordinates can also be considered as a limiting case of ellipsoidal coordinates in which the...
spherical coordinates lead to spherical harmonics. They are called oblate spheroidal wave functions if oblate spheroidalcoordinates are used and prolate spheroidal...
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots...
{\displaystyle z} -direction. The prolatespheroidalcoordinates are produced by rotating the elliptic coordinates about the x {\displaystyle x} -axis...
In applied mathematics, oblate spheroidal wave functions (like also prolatespheroidal wave functions and other related functions) are involved in the...
triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry (Euler 1755). If the Earth is treated as a sphere, the geodesics...
ordinary differential equations when using prolatespheroidalcoordinates instead of cartesian coordinates. The analytical solution of the equation, the...
third axis is shorter, the ellipsoid is an oblate spheroid; if it is longer, it is a prolatespheroid. If the three axes have the same length, the ellipsoid...
equivalent. For example, Io is better modeled by triaxial ellipsoid or prolatedspheroid with small eccentricities. Haumea's shape is a Jacobi ellipsoid, with...
There are two types of ellipsoidal domes: prolate ellipsoidal domes and oblate ellipsoidal domes. A prolate ellipsoidal dome is derived by rotating an...
original on 2011-07-18. Retrieved 2009-03-25. Fukushima, Toshio (2014). "Prolatespheroidal harmonic expansion of gravitational field". Astrophys. J. 147 (6):...
the early surveys of Jacques Cassini (1720) led him to believe Earth was prolate (wider at the poles). The subsequent French geodesic missions (1735-1739)...
prolatespheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to the polar radius of the oblate spheroidal...
Earth was closer to an oblate spheroid than to a sphere. However, around 1730, Jacques Cassini argued for a prolatespheroid instead, due to different interpretations...
about its major axis, the shape becomes a prolatespheroid; rotated about the minor axis, an oblate spheroid. A sphere is uniquely determined by four points...
along its major axis to produce an ellipsoidal mirror (specifically, a prolatespheroid), this property holds for all rays out of the source. Alternatively...
although it is often called a degenerate quadric or a reducible quadric. In coordinates x1, x2, ..., xD+1, the general quadric is thus defined by the algebraic...
the degree diminishing with increasing latitude, the Earth must be a prolatespheroid. This conclusion was totally opposed to the theoretical investigations...
periodic differential equations, as for Lamé functions and prolate and oblate spheroidal wave functions. Mathieu's differential equations appear in a...
"What Do a Submarine, a Rocket and a Football Have in Common? Why the prolatespheroid is the shape for success" (Scientific American, November 8, 2010) Conservation...
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