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 , where the origin is taken to be O. Let A be the point (a, b). If is a point on the curve the distance from K to A is

.

The points on the line OK have polar angle , and the points at distance d from K on this line are distance from the origin. Therefore the equation of the strophoid is given by

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:

,

where

.

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

.

Let P1 and P2 be the points on OK that are distance AK from K, numbering so that and . is isosceles with vertex angle , so the remaining angles, and , are . The angle between AP1 and the x-axis is then

.

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

.

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

C is a sectrix of Maclaurin with poles O and A when l is of the form , 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, where is a constant. Then and . The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at O are then

and

.

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

,

and rotating by in turn produces

.

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

.

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]

A right strophoid

Putting in

gives

.

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

.

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

.

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, where is a constant. Then and . The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at O are then

and

.

These are the equations of the two circles which also pass through O and A and form angles of 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). "article name needed". Encyclopædia Britannica (11th ed.). Cambridge University Press.