Strophoid
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Allgemeine_strophoide5.svg/280px-Allgemeine_strophoide5.svg.png)
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.
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 θ, 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, θ the angle between OK and the x-axis, and the angle between AK and the x-axis. Suppose can be given as a function θ, say Let ψ 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 △P1KA is isosceles with vertex angle ψ, 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 α 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]![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/08/Gerade_strophoide_rechts.svg/220px-Gerade_strophoide_rechts.svg.png)
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[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
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 strophoid, 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]- ^ Chisholm, Hugh, ed. (1911). . Encyclopædia Britannica. Vol. 16 (11th ed.). Cambridge University Press. p. 919.
- 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
External links
[edit] Media related to Strophoid at Wikimedia Commons