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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 *P*_{1} and *P*_{2} 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 *P*_{1} and *P*_{2} is then the strophoid of C with respect to the pole *O* and fixed point *A*. Note that *AP*_{1} and *AP*_{2} 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 , where the origin is taken to be *O*. Let *A* be the point (*a*, *b*). If is a point on the curve the distance from *K* to *A* is

- .

The points on the line *OK* have polar angle , and the points at distance *d* from *K* on this line are distance from the origin. Therefore, the equation of the strophoid is given by

Let *C* be given parametrically by (*x*(*t*), *y*(*t*)). Let *A* be the point (a, b) and let *O* be the point (*p*, *q*). Then, by a straightforward application of the polar formula, the strophoid is given parametrically by:

- ,

where

- .

The complex nature of the formulas given above limits their usefulness in specific cases. There is an alternative form which is sometimes simpler to apply. This is particularly useful when *C* is a sectrix of Maclaurin with poles *O* and *A*.

Let *O* be the origin and *A* be the point (*a*, 0). Let *K* be a point on the curve, the angle between *OK* and the x-axis, and the angle between *AK* and the x-axis. Suppose can be given as a function , say . Let be the angle at *K* so . We can determine *r* in terms of *l* using the law of sines. Since

- .

Let *P*_{1} and *P*_{2} be the points on *OK* that are distance *AK* from *K*, numbering so that and . is isosceles with vertex angle , so the remaining angles, and , are . The angle between *AP*_{1} and the x-axis is then

- .

By a similar argument, or simply using the fact that *AP*_{1} and *AP*_{2} are at right angles, the angle between *AP*_{2} and the x-axis is then

- .

The polar equation for the strophoid can now be derived from *l*_{1} and *l*_{2} from the formula above:

*C* is a sectrix of Maclaurin with poles *O* and *A* when *l* is of the form , in that case *l*_{1} and *l*_{2} will have the same form so the strophoid is either another sectrix of Maclaurin or a pair of such curves. In this case there is also a simple polar equation for the polar equation if the origin is shifted to the right by *a*.

Let *C* be a line through *A*. Then, in the notation used above, where is a constant. Then and . The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at *O* are then

and

- .

It's easy to check that these equations describe the same curve.

Moving the origin to *A* (again, see Sectrix of Maclaurin) and replacing −*a* with *a* produces

- ,

and rotating by in turn produces

- .

In rectangular coordinates, with a change of constant parameters, this is

- .

This is a cubic curve and, by the expression in polar coordinates it is rational. It has a crunode at (0, 0) and the line *y*=*b* is an asymptote.

Putting in

gives

- .

This is called the **right strophoid** and corresponds to the case where *C* is the *y*-axis, *A* is the origin, and *O* is the point (*a*,0).

The Cartesian equation is

- .

The curve resembles the Folium of Descartes^{[1]} and the line *x* = −*a* is an asymptote to two branches. The curve has two more asymptotes, in the plane with complex coordinates, given by

- .

Let *C* be a circle through *O* and *A*, where *O* is the origin and *A* is the point (*a*, 0). Then, in the notation used above, where is a constant. Then and . The polar equations of the resulting strophoid, called an oblique strophoid, with the origin at *O* are then

and

- .

These are the equations of the two circles which also pass through *O* and *A* and form angles of with *C* at these points.

**^**Chisholm, Hugh, ed. (1911).*Encyclopædia Britannica*. Vol. 16 (11th ed.). Cambridge University Press. p. 919. .

- J. Dennis Lawrence (1972).
*A catalog of special plane curves*. Dover Publications. pp. 51–53, 95, 100–104, 175. ISBN 0-486-60288-5. - E. H. Lockwood (1961). "Strophoids".
*A Book of Curves*. Cambridge, England: Cambridge University Press. pp. 134–137. ISBN 0-521-05585-7. - R. C. Yates (1952). "Strophoids".
*A Handbook on Curves and Their Properties*. Ann Arbor, MI: J. W. Edwards. pp. 217–220. - Weisstein, Eric W. "Strophoid".
*MathWorld*. - Weisstein, Eric W. "Right Strophoid".
*MathWorld*. - Sokolov, D.D. (2001) [1994], "Strophoid",
*Encyclopedia of Mathematics*, EMS Press - O'Connor, John J.; Robertson, Edmund F., "Right Strophoid",
*MacTutor History of Mathematics archive*, University of St Andrews

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