Circular section

In geometry, a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid). It is a special plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the circle. Any plane section of a sphere is a circular section, if it contains at least 2 points. Any quadric of revolution contains circles as sections with planes that are orthogonal to its axis; it does not contain any other circles, if it is not a sphere. More hidden are circles on other quadrics, such as tri-axial ellipsoids, elliptic cylinders, etc. Nevertheless, it is true that: Any quadric surface which contains ellipses contains circles, too. Equivalently, all quadric surfaces contain circles except parabolic and hyperbolic cylinders and hyperbolic paraboloids. If a quadric contains a circle, then every intersection of the quadric with a plane parallel to this circle is also a circle, provided it contains at least two points. Except for spheres, the circles contained in a quadric, if any, are all parallel to one of two fixed planes (which are equal in the case of a quadric of revolution). Circular sections are used in crystallography. The circular sections of a quadric may be computed from the implicit equation of the quadric, as it is done in the following sections. They may also be characterised and studied by using synthetic projective geometry. Let C be the intersection of a quadric surface Q and a plane P. In this section, Q and C are surfaces in the three-dimensional Euclidean space, which are extended to the projective space over the complex numbers. Under these hypotheses, the curve C is a circle if and only if its intersection with the plane at infinity is included in the ombilic (the curve at infinity of equation ). The first case to be considered is when the intersection of Q with the plane at infinity consists of one or two real lines, that is when Q is either a hyperbolic paraboloid, a parabolic cylinder or a hyperbolic cylinder.