I am trying to figure out the proper definition of a small circle on a biaxial ellipsoid of revolution. One definition is the intersection of the ellipsoid with a cone emanating from the center of the ellipsoid.
The other way I can imagine to define it is a plane intersecting the ellipsoid in which the plane does not also intersect the center of the ellipsoid (or else it would be a great circle).
This Wikipedia article also discusses sphere-intersection, but it is limited to spheres, and I am interested in ellipsoids.
Does anyone know if these two methods, cone intersection and plane intersection, result in the same curve? If not, which one is a small circle, and what would be the name of the other resulting curve?
Best Answer
The intersection with a plane is, in general, an ellipse. The intersection with a cone is a more complicated curve, in general not lying on a plane. If what you want is a planar circle all lying on the ellipsoid, then I'm afraid that is not possible, unless particular conditions are met (e.g. a rotational symmetric ellipsoid, intersected by a plane perpendicular to the axis of symmetry).
EDIT.
If what you want is a closed line whose points have all the same distance $r$ from a given point $P$ on the surface of the ellipsoid, you can obtain it from the intersection of the ellipsoid with a sphere of radius $r$ centred at $P$.