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, θ the angle between OK and the x-axis, and the angle between AK and the x-axis. Suppose can be given as a function θ, say Let ψ be the angle at K so We can determine r in terms of l using the law of sines. Since
Let P1 and P2 be the points on OK that are distance AK from K, numbering so that and △P1KA is isosceles with vertex angle ψ, so the remaining angles, and are The angle between AP1 and the x-axis is then
By a similar argument, or simply using the fact that AP1 and AP2 are at right angles, the angle between AP2 and the x-axis is then
The polar equation for the strophoid can now be derived from l1 and l2 from the formula above:
C is a sectrix of Maclaurin with poles O and A when l is of the form in that case l1 and l2 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.