title: Hippopede
text: # Hippopede
  
  In geometry, a hippopede (from Ancient Greek ἱπποπέδη (hippopédē) 'horse fetter') is a plane curve determined by an equation of the form
  
      $(x^{2}+y^{2})^{2}=cx^{2}+dy^{2},$
  
  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.
  
  ## Special cases
  
  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 19th-century mathematician James Booth 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.
  
  ## Definition as spiric sections
  
  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
  
      $r^{2}=4b(a-b\sin ^{2}\!\theta )$
  
  or in Cartesian coordinates
  
      $(x^{2}+y^{2})^{2}+4b(b-a)(x^{2}+y^{2})=4b^{2}x^{2}$.
  
  Note that when a > b the torus intersects itself, so it does not resemble the usual picture of a torus.
  
  