In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from is the same as the distance from to (i.e.). The locus of such points and is then the strophoid of with respect to the pole and fixed point . Note that and are at right angles in this construction.
In the special case where is a line, lies on, and is not on, then the curve is called an oblique strophoid. If, in addition, is perpendicular to 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 be given by
r=f(\theta),
K=(r\cos\theta, r\sin\theta)
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}.
f(\theta)\pmd
r=f(\theta)\pm\sqrt{(f(\theta)\cos\theta-a)2+(f(\theta)\sin\theta-b)2}
Let be given parametrically by . Let be the point and let be the point . Then, by a straightforward application of the polar formula, the strophoid is given parametrically by:
u(t)=p+(x(t)-p)(1\pmn(t)), v(t)=q+(y(t)-q)(1\pmn(t)),
n(t)=\sqrt{
| (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 is a sectrix of Maclaurin with poles and .
Let be the origin and be the point . Let be a point on the curve, the angle between and the -axis, and the angle between and the -axis. Suppose can be given as a function, say
\vartheta=l(\theta).
\psi=\vartheta-\theta.
{r\over\sin\vartheta}={a\over\sin\psi}, r=a
| \sin\vartheta | |
| \sin\psi |
=a
| \sinl(\theta) | |
| \sin(l(\theta)-\theta) |
.
Let and be the points on that are distance from, numbering so that
\psi=\angleP1KA
\pi-\psi=\angleAKP2.
\tfrac{\pi-\psi}{2}.
l1(\theta)=\vartheta+\angleKAP1=\vartheta+(\pi-\psi)/2=\vartheta+(\pi-\vartheta+\theta)/2=(\vartheta+\theta+\pi)/2.
By a similar argument, or simply using the fact that and are at right angles, the angle between and the -axis is then
l2(\theta)=(\vartheta+\theta)/2.
The polar equation for the strophoid can now be derived from and from the formula above:
\begin{align} &r1=a
| \sinl1(\theta) | |
| \sin(l1(\theta)-\theta) |
=a
| \sin((l(\theta)+\theta+\pi)/2) | |
| \sin((l(\theta)+\theta+\pi)/2-\theta) |
=a
| \cos((l(\theta)+\theta)/2) | |
| \cos((l(\theta)-\theta)/2) |
\\ &r2=a
| \sinl2(\theta) | |
| \sin(l2(\theta)-\theta) |
=a
| \sin((l(\theta)+\theta)/2) | |
| \sin((l(\theta)+\theta)/2-\theta) |
=a
| \sin((l(\theta)+\theta)/2) | |
| \sin((l(\theta)-\theta)/2) |
\end{align}
is a sectrix of Maclaurin with poles and when is of the form
q\theta+\theta0,
Let be a line through . Then, in the notation used above,
l(\theta)=\alpha
l1(\theta)=(\theta+\alpha+\pi)/2
l2(\theta)=(\theta+\alpha)/2.
r=a
| \cos((\alpha+\theta)/2) | |
| \cos((\alpha-\theta)/2) |
r=a
| \sin((\alpha+\theta)/2) | |
| \sin((\alpha-\theta)/2) |
.
Moving the origin to (again, see Sectrix of Maclaurin) and replacing with produces
| r=a | \sin(2\theta-\alpha) |
| \sin(\theta-\alpha) |
,
\alpha
| r=a | \sin(2\theta+\alpha) |
| \sin(\theta) |
.
In rectangular coordinates, with a change of constant parameters, this is
y(x2+y2)=b(x2-y2)+2cxy.
Putting
\alpha=\pi/2
| r=a | \sin(2\theta-\alpha) |
| \sin(\theta-\alpha) |
| r=a | \cos2\theta |
| \cos\theta |
=a(2\cos\theta-\sec\theta).
The Cartesian equation is
y2=x2(a-x)/(a+x).
The curve resembles the Folium of Descartes[1] and the line is an asymptote to two branches. The curve has two more asymptotes, in the plane with complex coordinates, given by
x\pmiy=-a.
Let be a circle through and, where is the origin and is the point . Then, in the notation used above,
l(\theta)=\alpha+\theta
\alpha
l1(\theta)=\theta+(\alpha+\pi)/2
l2(\theta)=\theta+\alpha/2.
r=a
| \cos(\theta+\alpha/2) | |
| \cos(\alpha/2) |
r=a
| \sin(\theta+\alpha/2) | |
| \sin(\alpha/2) |
.
\pi/4