ELLIPTICAL CONE

 Other name: degree-two cone (implying: non-decomposed).

 Reduced equation:  (with , cone of revolution if and only if a = b). Sections by the plane z = k are ellipses with half-axes 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 : . 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). Elliptic cone with its curvature lines, i.e. its straight lines and their orthogonal trajectories.

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

© Robert FERRÉOL  2020