Cissoid

In geometry, a cissoid (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped') is a plane curve generated from two given curves $C 1$ , $C 2$ and a point $O$ (the pole). Let $L$ be a variable line passing through $O$ and intersecting $C 1$ at $P 1$ and $C 2$ at $P 2$ . Let $P$ be the point on $L$ so that ${\overline {OP}}={\overline {P_{1}P_{2}}}.$ (There are actually two such points but $P$ is chosen so that $P$ is in the same direction from $O$ as $P 2$ is from $P 1$ .) Then the locus of such points $P$ is defined to be the cissoid of the curves $C 1$ , $C 2$ relative to $O$ .

Slightly different but essentially equivalent definitions are used by different authors. For example, $P$ may be defined to be the point so that ${\overline {OP}}={\overline {OP_{1}}}+{\overline {OP_{2}}}.$ This is equivalent to the other definition if $C 1$ is replaced by its reflection through $O$ . Or $P$ may be defined as the midpoint of $P 1$ and $P 2$ ; this produces the curve generated by the previous curve scaled by a factor of 1/2.