have different signs (due to: ). The diagram gives an impression of more common intersections between a sphere and an ellipsoid and highlights the exceptional circular case (blue). If the values of the semi-axes are approaching, the two pencils of planes (and circles) approach either. For all the planes are orthogonal to the z-axis (rotation axis). Proof of property (P): Turning the ellipsoid around the y-axis such that one of the two circles (blue) lies in the x-y-plane results in a new equation of the ellipsoid: For one gets , which has to be the equation of a circle. This is only true, if . The intersection of the ellipsoid by a plane with equation , (parallel to the x-y-plane) has the equation . This equation describes a circle or a point or the empty set. Center and radius of the circle can be found be completing the square. Elliptical hyperboloid of one sheet For the hyperboloid of one sheet with equation analogously one gets for the intersection with the sphere the equation Only for one gets a pair of planes: Elliptical cylinder For the elliptical cylinder with equation one gets the equation Only for one gets a pair of planes: Elliptical paraboloid For the elliptical paraboloid with equation one chooses a sphere containing the vertex (origin) and with center on the axis (z-axis) : After elimination of the linear parts one gets the equation Only for one gets a pair of planes : Elliptical hyperboloid of two sheets The hyperboloid of two sheets with equation is shifted at first such that one vertex is the origin (s. diagram): Analogously to the paraboloid case one chooses a sphere containing the origin with center on the z-axis: After elimination of the linear parts one gets the equation Only for one gets a pair of planes: Elliptical cone The elliptical cone with equation is shifted such that the vertex is not the origin (s. diagram): Now a sphere with center at the origin is suitable: Eliminatiion of yields: In this case completing the square gives: In order to get the equation of a pair of planes, the right part of the equation has to be zero, which is true for The solution for z gives: References H. F. Baker: Principles of Geometry, Volume 3, Cambridge University Press, 2010, . D. M. Y. Sommerville: Analytical Geometry of Three Dimensions, Cambridge University Press, 1959,