Strophoid

In geometry, a strophoid is a curve generated from a given curve C and points A (the fixed point) and O (the pole) as follows: Let L be a variable line passing through O and intersecting C at K. Now let P₁ and P₂ be the two points on L whose distance from K is the same as the distance from A to K (i.e. KP₁ = KP₂ = AK). The locus of such points P₁ and P₂ is then the strophoid of C with respect to the pole O and fixed point A. Note that AP₁ and AP₂ are at right angles in this construction.

In the special case where C is a line, A lies on C, and O is not on C, then the curve is called an oblique strophoid. If, in addition, OA is perpendicular to C then the curve is called a right strophoid, or simply strophoid by some authors. The right strophoid is also called the logocyclic curve or foliate.

Let the curve C be given by ${\displaystyle r=f(\theta ),}$ where the origin is taken to be O. Let A be the point (a, b). If ${\displaystyle K=(r\cos \theta ,\ r\sin \theta )}$ is a point on the curve the distance from K to A is

${\displaystyle d={\sqrt {(r\cos \theta -a)^{2}+(r\sin \theta -b)^{2}}}={\sqrt {(f(\theta )\cos \theta -a)^{2}+(f(\theta )\sin \theta -b)^{2}}}.}$

The points on the line OK have polar angle θ, and the points at distance d from K on this line are distance ${\displaystyle f(\theta )\pm d}$ from the origin. Therefore, the equation of the strophoid is given by

${\displaystyle r=f(\theta )\pm {\sqrt {(f(\theta )\cos \theta -a)^{2}+(f(\theta )\sin \theta -b)^{2}}}}$

Let C be given parametrically by (x(t), y(t)). Let A be the point (a, b) and let O be the point (p, q). Then, by a straightforward application of the polar formula, the strophoid is given parametrically by:

${\displaystyle u(t)=p+(x(t)-p)(1\pm n(t)),\ v(t)=q+(y(t)-q)(1\pm n(t)),}$

where

${\displaystyle n(t)={\sqrt {\frac {(x(t)-a)^{2}+(y(t)-b)^{2}}{(x(t)-p)^{2}+(y(t)-q)^{2}}}}.}$

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 ⁠${\displaystyle \vartheta }$⁠ the angle between AK and the x-axis. Suppose ⁠${\displaystyle \vartheta }$⁠ can be given as a function θ, say ${\displaystyle \vartheta =l(\theta ).}$ Let ψ be the angle at K so ${\displaystyle \psi =\vartheta -\theta .}$ We can determine r in terms of l using the law of sines. Since

${\displaystyle {r \over \sin \vartheta }={a \over \sin \psi },\ r=a{\frac {\sin \vartheta }{\sin \psi }}=a{\frac {\sin l(\theta )}{\sin(l(\theta )-\theta )}}.}$

Let P₁ and P₂ be the points on OK that are distance AK from K, numbering so that ${\displaystyle \psi =\angle P_{1}KA}$ and ${\displaystyle \pi -\psi =\angle AKP_{2}.}$ △P₁KA is isosceles with vertex angle ψ, so the remaining angles, ⁠${\displaystyle \angle AP_{1}K}$⁠ and ⁠${\displaystyle \angle KAP_{1},}$⁠ are ${\displaystyle {\tfrac {\pi -\psi }{2}}.}$ The angle between AP₁ and the x-axis is then

${\displaystyle l_{1}(\theta )=\vartheta +\angle KAP_{1}=\vartheta +(\pi -\psi )/2=\vartheta +(\pi -\vartheta +\theta )/2=(\vartheta +\theta +\pi )/2.}$

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

${\displaystyle l_{2}(\theta )=(\vartheta +\theta )/2.}$

The polar equation for the strophoid can now be derived from l₁ and l₂ from the formula above:

${\displaystyle {\begin{aligned}&r_{1}=a{\frac {\sin l_{1}(\theta )}{\sin(l_{1}(\theta )-\theta )}}=a{\frac {\sin((l(\theta )+\theta +\pi )/2)}{\sin((l(\theta )+\theta +\pi )/2-\theta )}}=a{\frac {\cos((l(\theta )+\theta )/2)}{\cos((l(\theta )-\theta )/2)}}\\&r_{2}=a{\frac {\sin l_{2}(\theta )}{\sin(l_{2}(\theta )-\theta )}}=a{\frac {\sin((l(\theta )+\theta )/2)}{\sin((l(\theta )+\theta )/2-\theta )}}=a{\frac {\sin((l(\theta )+\theta )/2)}{\sin((l(\theta )-\theta )/2)}}\end{aligned}}}$

C is a sectrix of Maclaurin with poles O and A when l is of the form ${\displaystyle q\theta +\theta _{0},}$ 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.

Let C be a line through A. Then, in the notation used above, ${\displaystyle l(\theta )=\alpha }$ where α is a constant. Then ${\displaystyle l_{1}(\theta )=(\theta +\alpha +\pi )/2}$ and ${\displaystyle l_{2}(\theta )=(\theta +\alpha )/2.}$ The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at O are then

${\displaystyle r=a{\frac {\cos((\alpha +\theta )/2)}{\cos((\alpha -\theta )/2)}}}$

and

${\displaystyle r=a{\frac {\sin((\alpha +\theta )/2)}{\sin((\alpha -\theta )/2)}}.}$

It's easy to check that these equations describe the same curve.

Moving the origin to A (again, see Sectrix of Maclaurin) and replacing −a with a produces

${\displaystyle r=a{\frac {\sin(2\theta -\alpha )}{\sin(\theta -\alpha )}},}$

and rotating by ${\displaystyle \alpha }$ in turn produces

${\displaystyle r=a{\frac {\sin(2\theta +\alpha )}{\sin(\theta )}}.}$

In rectangular coordinates, with a change of constant parameters, this is

${\displaystyle y(x^{2}+y^{2})=b(x^{2}-y^{2})+2cxy.}$

This is a cubic curve and, by the expression in polar coordinates it is rational. It has a crunode at (0, 0) and the line y = b is an asymptote.

Putting ${\displaystyle \alpha =\pi /2}$ in

${\displaystyle r=a{\frac {\sin(2\theta -\alpha )}{\sin(\theta -\alpha )}}}$

gives

${\displaystyle r=a{\frac {\cos 2\theta }{\cos \theta }}=a(2\cos \theta -\sec \theta ).}$

This is called the right strophoid and corresponds to the case where C is the y-axis, A is the origin, and O is the point (a, 0).

The Cartesian equation is

${\displaystyle y^{2}=x^{2}(a-x)/(a+x).}$

The curve resembles the Folium of Descartes^[1] and the line x = –a is an asymptote to two branches. The curve has two more asymptotes, in the plane with complex coordinates, given by

${\displaystyle x\pm iy=-a.}$

Let C be a circle through O and A, where O is the origin and A is the point (a, 0). Then, in the notation used above, ${\displaystyle l(\theta )=\alpha +\theta }$ where ${\displaystyle \alpha }$ is a constant. Then ${\displaystyle l_{1}(\theta )=\theta +(\alpha +\pi )/2}$ and ${\displaystyle l_{2}(\theta )=\theta +\alpha /2.}$ The polar equations of the resulting strophoid, called an oblique strophoid, with the origin at O are then

${\displaystyle r=a{\frac {\cos(\theta +\alpha /2)}{\cos(\alpha /2)}}}$

and

${\displaystyle r=a{\frac {\sin(\theta +\alpha /2)}{\sin(\alpha /2)}}.}$

These are the equations of the two circles which also pass through O and A and form angles of ${\displaystyle \pi /4}$ with C at these points.

• Conchoid
• Cissoid

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 51–53, 95, 100–104, 175. ISBN 0-486-60288-5.
• E. H. Lockwood (1961). "Strophoids". A Book of Curves. Cambridge, England: Cambridge University Press. pp. 134–137. ISBN 0-521-05585-7.
• R. C. Yates (1952). "Strophoids". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 217–220.
• Weisstein, Eric W. "Strophoid". MathWorld.
• Weisstein, Eric W. "Right Strophoid". MathWorld.
• Sokolov, D.D. (2001) [1994], "Strophoid", Encyclopedia of Mathematics, EMS Press
• O'Connor, John J.; Robertson, Edmund F., "Right Strophoid", MacTutor History of Mathematics Archive, University of St Andrews

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