"For problems involving directions from a fixed origin (or 'pole') O, we often find it convenient to specify a point P by its polar coordinates
where is the distance OP and is the angle that the direction OP makes with a given initial line OX, which may be identified with the x-axis of rectangular Cartesian coordinates. Of course, the point is the same as for any integrer n " (Coxeter, p.110)
We can use Polar coordinates for describing spirals, for example.
Coxeter introduce the Equiangular Spiral (or 'Logaritmic spiral') as the result of a continuous dilative rotation.
The polar equation of the Equiangular Spiral is
It receives the name of equiangular because the angle between the radius vector and the tangent is constant.
If is the angle between the position vector OP and the tangent at P then (in general, in polar coordinates)
In the Equiangular spiral case, we can write
Calculating the derivative
The angle between the position vector OP and the tangent at P is a constant and we can write the Equiangular spiral equation
This spiral was called "spiral mirabilis" by Jacob Bernouilli.
We can see that the angle formed by the radius vector and the tangent is constant.
In the next video you can see a particular case: the Golden Equiangular Spiral. This spiral is related with the Golden Rectangle.
This example is the golden equiangular spiral.
Coxeter - Introduction to Geometry (John Whiley and sons)
Steinhaus - Mathematical Snapshots.
D'Arcy Thompson - On Growth and Form. (Cambridge University Press)