Spheroids and Ellipsoids
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The standard equation of an ellipsoid in the coordinate frame depicted above is
The points (a,0,0), (0,b,0) and (0,0,c) lie on the surface and the line segments from the origin to these points are called the semi-principal axes of length a, b, c. There are four distinct cases:
- a > b > c : scalene ellipsoid
- a = b > c : oblate ellipsoid of revolution (oblate spheroid, aka "bun")
- a = b < c : prolate ellipsoid of revolution (prolate spheroid, aka "cigar")
- a = b = c : circle of revolution (sphere)
The oblate and prolate spheroids are symmetrical around the z-axis, and can be respectively likened to spheres that have been flattened or elongated along the +ve and –ve z-axes.
More precisely, the oblate or prolate ellipsoids can be generated by rotating the (x,z) ellipse through 180° around the z-axis, after first choosing c<a in the oblate case, or c>a in the prolate case. In both cases, b becomes necessarily equal to a.
In the context of domes, of course, we're really only interested in the upper halves (hemispheres or semi-ellipsoids) of these shapes.