a pair of complex conjugate points (which are double points). A real plane contains these two points if and only if it is perpendicular to the axis of revolution. Thus the circular sections are the plane sections by a plane perpendicular to the axis, that have at least two real points. In the other cases, the intersection of with the ombilic consists of two different pairs of complex conjugate points. As is a curve of degree two, its intersection with the plane at infinity consists of two points, possibly equal. The curve is thus a circle, if these two points are one of these two pairs of complex conjugate points on the ombilic. Each of these pairs defines a real line (passing through the points), which is the intersection of with the plane at infinity. Thus, one has a circular section if and only has at least two real points and contains one of these lines at infinity (that is if is parallel to one of two directions defined by these lines at infinity). Determination of circular sections of a quadric In order to find the planes, which contain circular sections of a given quadric, one uses the following statements: (S:) If the common points of a quadric with a sphere are contained in a pair of planes, then the intersection curve consists of two circles. (P:) If the intersection of a plane and a quadric is a circle, then any parallel plane, that contains at least two points of the quadric, intersects the quadric in a circle, too. Hence the strategy for the detection of circular sections is: 1) Find a sphere that intersects the quadric in a pair of planes and 2) The planes that are parallel to the detected ones deliver the remaining circular sections. Tri-axial ellipsoid For the ellipsoid with equation and the semi-axes one uses an auxiliary sphere with equation The sphere's radius has to be chosen such that the intersection with the ellipsoid is contained in two planes through the origin. Multiplication of the ellipsoid's equation by and subtracting the sphere's equation yields: This equation describes a pair of planes, if one of the 3 coefficients is zero. In case of or the equation is only fulfilled by either the x-axis or the z-axis. Only in case of one gets a pair of planes with equation because only in this case the remaining coefficients