Spiric section

In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form ( x 2 + y 2 ) 2 = d x 2 + e y 2 + f . {\displaystyle (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f.\,} Equivalently, spiric sections can be defined as bicircular quartic curves that are symmetric with respect to the x and y-axes. Spiric sections are included in the family of toric sections and include the family of hippopedes and the family of Cassini ovals. The name is from σπειρα meaning "torus" in ancient Greek. A spiric section is sometimes defined as the curve of intersection of a torus and a plane parallel to its rotational symmetry axis. However, this definition does not include all of the curves given by the previous definition unless imaginary planes are allowed. Spiric sections were first described by the ancient Greek geometer Perseus in roughly 150 BC, and are assumed to be the first toric sections to be described. The name spiric is due to the ancient notation spira of a torus.,