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 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: Equivalently, all quadric surfaces contain circles except parabolic and hyperbolic cylinders and hyperbolic paraboloids.
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).
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.
A real plane contains these two points if and only if it is perpendicular to the axis of revolution.
In the other cases, the intersection of Q with the ombilic consists of two different pairs of complex conjugate points.
As C is a curve of degree two, its intersection with the plane at infinity consists of two points, possibly equal.
Each of these pairs defines a real line (passing through the points), which is the intersection of P with the plane at infinity.
Thus, one has a circular section if and only C has at least two real points and P contains one of these lines at infinity (that is if P is parallel to one of two directions defined by these lines at infinity).
In order to find the planes, which contain circular sections of a given quadric, one uses the following statements: Hence the strategy for the detection of circular sections is: 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.
one gets a pair of planes with equation because only in this case the remaining coefficients 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).
all the planes are orthogonal to the z-axis (rotation axis).
Center and radius of the circle can be found be completing the square.
For the hyperboloid of one sheet with equation analogously one gets for the intersection with the sphere
one gets a pair of planes: 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 : 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: The elliptical cone with equation is shifted such that the vertex is not the origin (see diagram): Now a sphere with center at the origin is suitable: Elimination 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
