Cissoid
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A cissoid is a curve derived from a fixed point O and two other curves α and β. Every line through O cutting α at A and β at B cuts the cissoid at the midpoint of <math>\overline{AB}</math>.
The simplest expression uses polar coordinates with O at the origin. If <math>r=\alpha(\theta)</math> and <math>r=\beta(\theta)</math> express the two curves then <math>r=\frac12(\beta(\theta)+\alpha(\theta))</math> expresses the cissoid.
Sometimes this cissoid is described as a sum <math>r=\beta(\theta)+\alpha(\theta)</math> or difference <math>r=\beta(\theta)-\alpha(\theta)</math>; these are basically equivalent except for doubling the size and possibly needing one curve reflected through O.
Every conchoid is a cissoid with the other curve a circle centered on O.
The cissoid of Diocles was the prototype for this general construction.
A cissoid of Zahradnik replaced Diocles' circle with a conic section.
The often-so-called conchoid of de Sluze has α a circle passing through O less O itself and β a line parallel to α's tangent at O. It is, in fact, not a conchoid.