Editing Ellipsoid (section)

=== Properties of the focal hyperbola ===
[[File:Ellipsoid-pk-zk.svg|thumb|upright=1.5|'''Top:''' 3-axial Ellipsoid with its focal hyperbola.<br>
'''Bottom:''' parallel and central projection of the ellipsoid such that it looks like a sphere, i.e. its apparent shape is a circle]]
; True curve
: If one views an ellipsoid from an external point {{mvar|V}} of its focal hyperbola, then it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing point {{mvar|V}} are the lines of a circular [[cone]], whose axis of rotation is the [[Tangent (geometry)|tangent line]] of the hyperbola at {{mvar|V}}.<ref>D. Hilbert & S Cohn-Vossen: ''Geometry and the Imagination'', p. 24</ref><ref>O. Hesse: ''Analytische Geometrie des Raumes'', p.&nbsp;301</ref> If one allows the center {{mvar|V}} to disappear into infinity, one gets an [[Orthogonal projection|orthogonal]] [[parallel projection]] with the corresponding [[asymptote]] of the focal hyperbola as its direction. The ''true curve of shape'' (tangent points) on the ellipsoid is not a circle.{{paragraph}} The lower part of the diagram shows on the left a parallel projection of an ellipsoid (with semi-axes 60, 40, 30) along an asymptote and on the right a central projection with center {{mvar|V}} and main point {{mvar|H}} on the tangent of the hyperbola at point {{mvar|V}}. ({{mvar|H}} is the foot of the perpendicular from {{mvar|V}} onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin {{mvar|O}} is the circle's center; in the central case main point {{mvar|H}} is the center.
; Umbilical points
: The focal hyperbola intersects the ellipsoid at its four [[umbilical point]]s.<ref>W. Blaschke: ''Analytische Geometrie'', p. 125</ref>