Right strophoid

In geometry, a strophoid is a curve generated from a given curve C and points A (the fixed point) and O (the pole) as follows: Let L be a variable line passing through O and intersecting C at K. Now let P₁ and P₂ be the two points on L whose distance from K is the same as the distance from A to K. The locus of such points P₁ and P₂ is then the strophoid of C with respect to the pole O and fixed point A. Note that AP₁ and AP₂ are at right angles in this construction.

In the special case where C is a line, A lies on C, and O is not on C, then the curve is called an oblique strophoid. If, in addition, OA is perpendicular to C then the curve is called a right strophoid, or simply strophoid by some authors. The right strophoid is also called the logocyclic curve or foliate.

Let the curve C be given by ${\displaystyle r=f(\theta )}$, where the origin is taken to be O. Let A be the point (a, b). If ${\displaystyle K=(r\cos \theta ,\ r\sin \theta )}$ is a point on the curve the distance from K to A is

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