Right Strophoid

Cartesian equation:
y2 = x2(a - x)/(a + x)
Polar equation:
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

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The strophoid first appears in work by Isaac Barrow in 1670. However Torricelli describes the curve in his letters around 1645 and Roberval found it as the locus of the focus of the conic obtained when the plane cutting the cone rotates about the tangent at its vertex.

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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JOC/EFR/BS January 1997

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