Strophoid

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StrophoidConstruction.svg

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. 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.

Equations[edit]

Polar coordinates[edit]

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

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 \theta, and the points at distance d from K on this line are distance f(\theta) \pm d from the origin. Therefore the equation of the strophoid is given by

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

Cartesian coordinates[edit]

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:

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

where

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

An alternative polar formula[edit]

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

{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 \psi = \angle P_1KA and \pi-\psi = \angle AKP_2. \triangle P_1KA is isosceles with vertex angle \psi, so the remaining angles, \angle AP_1K and \angle KAP_1, are (\pi-\psi)/2. The angle between AP1 and the x-axis is then

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

l_2(\theta) = (\vartheta+\theta)/2.

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

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)}

C is a sectrix of Maclaurin with poles O and A when l is of the form 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.

Specific cases[edit]

Oblique strophoids[edit]

Let C be a line through A. Then, in the notation used above, l(\theta) = \alpha where \alpha is a constant. Then l_1(\theta) = (\theta + \alpha + \pi)/2 and l_2(\theta) = (\theta + \alpha)/2. The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at O are then

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

and

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

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

and rotating by \alpha in turn produces

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

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

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.

The right strophoid[edit]

Putting \alpha = \pi/2 in

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

gives

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, O is the origin, and A is the point (a,0).

The Cartesian equation is

y^2 = x^2(a-x)/(a+x).

The curve resembles the Folium of Descartes 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

x\pm iy = -a.

Circles[edit]

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, l(\theta) = \alpha+\theta where \alpha is a constant. Then l_1(\theta) = \theta + (\alpha + \pi)/2 and l_2(\theta) = \theta + \alpha/2. The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at O are then

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

and

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 \pi/4 with C at these points.

See also[edit]

References[edit]

External links[edit]

Media related to Strophoid at Wikimedia Commons

Public Domain This article incorporates text from a publication now in the public domainChisholm, Hugh, ed. (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press.