Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Prolate spheroidal coordinates
open-in-new
<p>Prolate spheroidal coordinates are a three-dimensional <a href="/facts/Orthogonal_coordinates/LLoa2lAJ">orthogonal</a> <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a> that results from rotating the two-dimensional <a href="/facts/Elliptic_coordinates/aGfvboJP">elliptic coordinate system</a> about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces <a href="/facts/Oblate_spheroidal_coordinates/AqznWJGK">oblate spheroidal coordinates</a>. Prolate spheroidal coordinates can also be considered as a <a href="/facts/Limiting_case_(mathematics)/UEehGCom">limiting case</a> of <a href="/facts/Ellipsoidal_coordinates/7U3CW9hI">ellipsoidal coordinates</a> in which the two smallest <a href="/facts/Ellipsoid/pe9vm0Sh">principal axes</a> are equal in length. </p><p>Prolate spheroidal coordinates can be used to solve various <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a> 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 <i>z</i>-axis. One example is solving for the <a href="/facts/Wavefunction/KgM5ykjY">wavefunction</a> of an <a href="/facts/Electron/L0KBo5cW">electron</a> moving in the <a href="/facts/Electromagnetic_field/MacraS8y">electromagnetic field</a> of two positively charged <a href="/facts/Atomic_nucleus/orpIm5Xl">nuclei</a>, as in the <a href="/facts/Hydrogen_molecular_ion/MWh6WYVG">hydrogen molecular ion</a>, H2+. Another example is solving for the <a href="/facts/Electric_field/tt4MooBs">electric field</a> generated by two small <a href="/facts/Electrode/251PaKX8">electrode</a> tips. Other limiting cases include areas generated by a line segment (<i>μ</i> = 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. </p>