Cartesian equation: y
= x(a-x)/(a+x)
Polar equation: r = a cos(2
)/cos()
Click below to see one of the Associated curves.
Definitions of the Associated curves Evolute Involute 1 Involute 2 Inverse curve wrt origin Inverse wrt another circle Pedal curve wrt origin Pedal wrt another point Negative pedal curve wrt origin Negative pedal wrt another point Caustic wrt horizontal rays Caustic curve wrt another point
The name (meaning a 'belt with a twist') was proposed by Montucci in 1846. The general strophoid has equation
r = b sin(a-2)/sin(a-).
The particular case of a right strophoid in where a = 
/2 and the equation, in cartesians and polars, is that given above.
The area of the loop of the right strophoid is a(4-)/2 and the area between the curve and its asymptote is a(4-)/2.
Let C be the circle with centre at the point where the right strophoid crosses the x-axis and radius the distance of that point from the origin. Then the strophoid is invariant under inversion in the circle C. Hence the strophoid is an anallagmatic curve.
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