Equiangular spiral

                         i                         

                         i                         

                         i                         

                         i                         

                 ::::::::t:::.                     

              :::        i    ::::                 

            ::           i       .::               

         .:.         .:::t:.        ,              

        ::        ::::   i.:::       :.            

       ,        :.    .::t:   .:      ,            

       ,      .,    ::   i :,   ::     ,           

      ,       ,.   ,.    i   ,.  ,     ,           

::::::i:::::::i:::i::::::j:::;;::i:::::;:::::::::  

      ,       ,   :.     i      ::    .:           

      ,       ,    ::    i     ::     ,         .L 

      ,        ,    .::  i  .::.     ,          ,  

       ,        ,      .:t::       .:          ,   

        ,        .::     i      .::           ,    

       ::          :::::t:::::::           .:.    

          ::             i                 :.      

           .:            i               ::        

             .::.        i            :::          

                 ::::.   i      .::::.             

                     ::::t:::::::                  

If P is any point on the spiral then the length of the spiral from P to the origin is finite. In fact, from the point P which is at distance d from the origin measured along a radius vector, the distance from P to the pole is d sec b. Jacob Bernoulli in 1692 called it spira mirabilis and it is carved on his tomb in Basel.

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.

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.

Polar Equation:
r = a exp(θ cot b)