y2 = x2(a - x)/(a + x)
r = a cos(2θ)/cos(θ)
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 - 2θ)/sin(a - θ).
The particular case of a right strophoid in where a = π/2 and the equation, in cartesians and polars, is that given above.
The area of the loop of the right strophoid is a2(4 - π)/2 and the area between the curve and its asymptote is a2(4 - π)/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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