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ELLIPTICAL CONE
Other name: degreetwo cone (implying: nondecomposed). 
Reduced equation:
(with ,
cone
of revolution if and only if a = b).
Sections by the plane z = k are ellipses with halfaxes ak/h and bk/h. 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: . Volume of the solid between the planes z=0 and
z=h
: .


An elliptical 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.
Contrary to appearances, every elliptical cone contains
circles.
If we turn the plane z = h by an angle with respect to the horizontal around the major axis of the ellipse, the intersection with the cone is a circle as well as all the intersections by parallel planes. Thus, the elliptical cone is also an oblique circular cone. 
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 2020