Epispiral

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 ${\displaystyle \theta }$ and at the same time by an angle in constant proportion to ${\displaystyle \theta }$, say ${\displaystyle c\theta }$ for some constant ${\displaystyle c}$.

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

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

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

• Logarithmic spiral
• Rose (mathematics)

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 192. ISBN 0-486-60288-5.
• https://www.mathcurve.com/courbes2d.gb/epi/epi.shtml

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