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

In the special case where  is a line,  lies on , and  is not on , then the curve is called an oblique strophoid. If, in addition,  is perpendicular to  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.

Equations

Polar coordinates
Let the curve  be given by  where the origin is taken to be . Let  be the point . If  is a point on the curve the distance from  to  is 

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

Cartesian coordinates
Let  be given parametrically by . Let  be the point  and let  be the point . Then, by a straightforward application of the polar formula, the strophoid is given parametrically by:

where

An alternative polar formula
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  is a sectrix of Maclaurin with poles  and .

Let  be the origin and  be the point . Let  be a point on the curve,  the angle between  and the -axis, and  the angle between  and the -axis. Suppose  can be given as a function , say  Let  be the angle at  so  We can determine  in terms of  using the law of sines. Since

Let  and  be the points on  that are distance  from , numbering so that  and   is isosceles with vertex angle , so the remaining angles,  and  are  The angle between  and the -axis is then 

By a similar argument, or simply using the fact that  and  are at right angles, the angle between  and the -axis is then 

The polar equation for the strophoid can now be derived from  and  from the formula above:

 is a sectrix of Maclaurin with poles  and  when  is of the form  in that case  and  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 .

Specific cases

Oblique strophoids
Let  be a line through . 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  are then

and

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

Moving the origin to  (again, see Sectrix of Maclaurin) and replacing  with  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  and the line  is an asymptote.

The right strophoid

Putting  in

gives

This is called the right strophoid and corresponds to the case where  is the -axis,  is the origin, and  is the point .

The Cartesian equation is 

The curve resembles the Folium of Descartes and the line  is an asymptote to two branches. The curve has two more asymptotes, in the plane with complex coordinates, given by

Circles
Let  be a circle through  and , where  is the origin and  is the point . 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  are then

and

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

See also
Conchoid
Cissoid

References

External links

Curves