Let 
be a curve, let 
be a fixed point (the pole), and
let 
be a second fixed point. Let 
and 
be points on a line through 
meeting 
at 
such that 
.
The locus of 
and 
is called the strophoid of 
with respect to the pole 
and fixed point 
. Let 
be represented parametrically by 
, and let 
and 
. Then the equation of the strophoid is
 |  |  |
(1)
|
 |  |  |
(2)
|
where
 |
(3)
|
The name strophoid means "belt with a twist," and was proposed by Montucci in 1846 (MacTutor Archive). The polar form for a general strophoid is
 |
(4)
|
If 
, the curve is a right
strophoid. The following table gives the strophoids of some common curves.
| curve | pole | fixed point | strophoid |
| line | not on line | on line | oblique strophoid |
| line | not on line | foot of perpendicular origin to line | right strophoid |
| circle | center | on the circumference | Freeth's nephroid |
