Equiangular Spiral

Polar equation:
r=aexp(θcotb)r = a \exp( \theta \cot b)


The equiangular (or logarithmic) spiral was invented by Descartes in 1638. Torricelli worked on it independently and found the length of the curve.

If PP is any point on the spiral then the length of the spiral from PP to the origin is finite. In fact, from the point PP which is at distance dd from the origin measured along a radius vector, the distance from PP to the pole is dsecbd \sec b. Jacob Bernoulli in 1692 called it spira mirabilis and he wanted it carved on his tomb in Basel. See THIS LINK.

It occurs naturally in many places like sea-shells where the growth of an organism is proportional to the size of the organism. In his book Growth and Form, D'Arcy Thompson devoted a whole chapter to this curve and describes its occurence in Nature as the result of coiling a cone upon itself, contrasting it with the Spiral of Archimedes which is formed by coiling a cylinder as a sailor coils a rope upon the deck.

You can see this shell of the many-chambered Nautilus or the shell with the curve super-imposed.

The spiral make a constant angle bb with any radius vector. Of course in the special case where b=π2b = \large\frac{\pi}{2}\normalsize one obtains a circle. For the curve displayed above b=7π16b = \large\frac{7\pi}{16}\normalsize. Therefore the length of the curve from a point at distance dd from the origin along a radius vector is about 5.126d5.126 d.

Any radius from the origin meets the spiral at distances which are in geometric progression.

The pedal of an equiangular spiral, when the pedal point is the pole, is an identical equiangular spiral.

The evolute and the involute of an equiangular spiral is an identical equiangular spiral. This was shown by Johann Bernoulli.

The caustic of the equiangular spiral, where the pole is taken as the radiant, is an equal equiangular spiral.

The equiangular spiral is "self-similar". That is any expansion of the curve will produce the same curve.

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Xah Lee