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Right Strophoid


RightStrophoid

A right strophoid is the strophoid of a line L with pole O not on L and fixed point O^' being the point where the perpendicular from O to L cuts L is called a right strophoid. It is therefore a general strophoid with a=pi/2.

The right strophoid is given by the Cartesian equation

 y^2=(c-x)/(c+x)x^2,
(1)

or the polar equation

 r=ccos(2theta)sectheta.
(2)

The parametric form of the strophoid is

x(t)=(c^2-t^2)/(t^2+c^2)c
(3)
y(t)=(t(t^2-c^2))/(t^2+c^2).
(4)

The right strophoid has curvature, arc length, and tangential angle given by

kappa(t)=-(4c^3(c^2+3t^2))/((c^4+6c^2t^2+t^4)^(3/2))
(5)
s(t)=ikc[(2sqrt(2)-3)E(phi_0,k^2)+2kF(phi_0,k^2)+4Pi(k^2,phi_0,k^2)]
(6)
phi(t)=-2tan^(-1)(t/c)+tan^(-1)[((sqrt(2)-1)t)/c]-tan^(-1)[((sqrt(2)+1)t)/c],
(7)

where

k=1+sqrt(2)
(8)
phi_0=isinh^(-1)((kt)/(|c|)),
(9)

E(phi,k), F(phi,k) and Pi(phi,z,k) are incomplete elliptic integrals of the first, second, and third kinds, respectively.

The right strophoid first appears in work by Isaac Barrow in 1670, although 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 (MacTutor Archive).

RightStrophoidLoop

The area of the loop, corresponding to t in [-c,c], is given by

A=1/2int(yx^'-xy^')dt
(10)
=1/2int_(-c)^c((c^2-t^2)/(c^2+t^2))^2dt
(11)
=c^2int_0^1((1-u^2)/(1+u^2))du
(12)
=1/2c^2(4-pi)
(13)

(MacTutor Archive). The arc length of the loop is given by

 s=2ikc[(2sqrt(2)-3)E(icsch^(-1)k,k^2)+2kF(icsch^(-1)k,k^2)+4Pi(k^2,icsch^(-1)k,k^2)],
(14)

where k is again defined as above.

Let C be the circle with center at the point where the right strophoid crosses the x-axis and radius the distance of that point from the origin. Then the right strophoid is invariant under inversion in the circle C and is therefore an anallagmatic curve.


See also

Conchoid of de Sluze, Conchoid of Nicomedes, Right Strophoid Inverse Curve, Strophoid, Trisectrix, Tschirnhausen Cubic

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 92, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 100-104, 1972.Lockwood, E. H. "The Right Strophoid." Ch. 10 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 90-97, 1967.MacTutor History of Mathematics Archive. "Right Strophoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Right.html.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 330, 1958.

Cite this as:

Weisstein, Eric W. "Right Strophoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RightStrophoid.html

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