| rdfs:comment | - where ax and ay are the transverse, equatorial radii (along the x and y axes) and b is the conjugate, polar radius (along the z-axis), all of which are fixed positive real numbers determining the shape of the ellipsoid (traditionally, ax and ay are denoted as a and b, respectively——thereby defining the equator as an ellipse——and b as c; however, in most cases——such as Earth——the equator is considered spherical, with the equatorial radius being defined simply as a and the polar as b——like an ellipse——thus a,b,c notation can be needlessly confusing).
|
| abstract | - where ax and ay are the transverse, equatorial radii (along the x and y axes) and b is the conjugate, polar radius (along the z-axis), all of which are fixed positive real numbers determining the shape of the ellipsoid (traditionally, ax and ay are denoted as a and b, respectively——thereby defining the equator as an ellipse——and b as c; however, in most cases——such as Earth——the equator is considered spherical, with the equatorial radius being defined simply as a and the polar as b——like an ellipse——thus a,b,c notation can be needlessly confusing). More generally, a not-necessarily-axis-aligned ellipsoid is defined by the equation where A is a symmetric positive definite matrix and x is a vector. In that case, the eigenvectors of A define the principal directions of the ellipsoid and the inverse of the square root of the eigenvalues are the corresponding equatorial radii. If all three radii are equal, the solid body is a sphere; if two radii are equal, the ellipsoid is a spheroid:
* Sphere;
* Oblate spheroid, or oblatum (disk-shaped);
* Prolate spheroid, or prolatum (like a rugby ball);
* Scalene ellipsoid ("three unequal sides"). The points (ax,0,0), (0,ay,0) and (0,0,b) lie on the surface and the line segments from the origin to these points are called the semi-principal axes. These correspond to the semi-major axis and semi-minor axis of the appropriate ellipses. Scalene ellipsoids are frequently called "triaxial ellipsoids", the implication being that all three axes need to be specified to define the shape.
|
