# 卡西尼卵形線

$(x^{2}+y^{2})^{2}=cx^{2}+dy^{2}$ ,

where it is assumed that c>0 and c>d since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are bicircular rational algebraic curves of degree 4 and symmetric with respect to both the x and y axes. When d>0 the curve has an oval form and is often known as an oval of Booth, and when d<0 the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after James Booth (1810–1878) who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For d = −c, the hippopede corresponds to the lemniscate of Bernoulli.

## 作為環面嘅截面

Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the plane is parallel to the axis of the torus and tangent to it on the interior circle. Thus it is a spiric section which in turn is a type of toric section.

If a circle with radius a is rotated about an axis at distance b from its center, then the equation of the resulting hippopede in polar coordinates

$r^{2}=4b(a-b\sin ^{2}\theta )\,$ $(x^{2}+y^{2})^{2}+4b(b-a)(x^{2}+y^{2})=4b^{2}x^{2}$ .

Note that when a>b the torus intersects itself, so it does not resemble the usual picture of a torus.