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ELLIPTIC CONE
Other name: degreetwo cone (implying: nondecomposed). 
Reduced equation:
(with ,
cone of revolution
if and only if a = b).
Developable ruled quadric. Cartesian parametrization: . Parametrization for which the coordinate lines are the curvature lines (case ): (see opposite) Half major angle at the vertex: , Half minor angle at the vertex: . Another reduced equation in the case where one of the 2 angles at the vertex is right: ; the other angle then is (cone of revolution for k = 2). 

An elliptic cone is a cone a directrix of which is an ellipse; it is defined up to isometry by its two angles at the vertex.
Characterization: cone of degree two not decomposed into two planes.
See the level and slope lines of the cone here.
See also focal circular cubic.
Triple orthogonal system two families of which are composed of elliptic cones
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© Robert FERRÉOL 2017