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.[1][2][3]
- ^ W. H. Westphal: Physikalisches Wörterbuch: Zwei Teile in Einem Band. Springer-Verlag, 1952, ISBN 978-3-662-12707-0, p. 350.
- ^ H. Tertsch: Die Festigkeitserscheinungen der Kristalle. Springer-Verlag, Wien, 1949, ISBN 978-3-211-80120-8, p. 87.
- ^ G. Masing: Lehrbuch der Allgemeinen Metallkunde. Springer-Verlag, Berlin, 1950, ISBN 978-3-642-52-993-1, p. 355.