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. 