Epispiral

The epispiral is a plane curve with polar equation

\ r=a\sec {n\theta }

.

There are n sections if n is odd and 2n if n is even.

It is the polar or circle inversion of the rose curve.

In astronomy the epispiral is related to the equations that explain planets' orbits.

Alternative definition[edit]

There is another definition of the epispiral that has to do with tangents to circles:^[1]

Begin with a circle.

Rotate some single point on the circle around the circle by some angle $\theta$ and at the same time by an angle in constant proportion to $\theta$ , say $c\theta$ for some constant $c$ .

The intersections of the tangent lines to the circle at these new points rotated from that single point for every $\theta$ would trace out an epispiral.

The polar equation can be derived through simple geometry as follows:

To determine the polar coordinates $(\rho ,\phi )$ of the intersection of the tangent lines in question for some $\theta$ and $-1<c<1$ , note that $\phi$ is halfway between $\theta$ and $c\theta$ by congruence of triangles, so it is ${\frac {(c+1)\theta }{2}}$ . Moreover, if the radius of the circle generating the curve is $r$ , then since there is a right-angled triangle (it's right-angled as a tangent to a circle meets the radius at a right angle at the point of tangency) with hypotenuse $\rho$ and an angle ${\frac {(1-c)\theta }{2}}$ to which the adjacent leg of the triangle is $r$ , the radius $\rho$ at the intersection point of the relevant tangents is $r\sec({\frac {(1-c)\theta }{2}})$ . This gives the polar equation of the curve, $\rho =r\sec({\frac {(1-c)\phi }{c+1}})$ for all points $(\rho ,\phi )$ on it.