Log-polar coordinates
Log-polar coordinates are closely connected to polar coordinates, which are usually used to describe domains in the plane with some sort of rotational symmetry.
Log-polar coordinates in the plane consist of a pair of real numbers (ρ,θ), where ρ is the logarithm of the distance between a given point and the origin and θ is the angle between a line of reference (the x-axis) and the line through the origin and the point.
The formulas for transformation from Cartesian coordinates to log-polar coordinates are given by and the formulas for transformation from log-polar to Cartesian coordinates are By using complex numbers (x, y) = x + iy, the latter transformation can be written as i.e. the complex exponential function.
From this follows that basic equations in harmonic and complex analysis will have the same simple form as in Cartesian coordinates.
Laplace's equation in two dimensions is given by in Cartesian coordinates.
Thus, when considering Laplace's equation for a part of the plane with rotational symmetry, e.g. a circular disk, log-polar coordinates is the natural choice.
A similar situation arises when considering analytical functions.
written in Cartesian coordinates satisfies the Cauchy–Riemann equations: If the function instead is expressed in polar form
, the Cauchy–Riemann equations take the more complicated form Just as in the case with Laplace's equation, the simple form of Cartesian coordinates is recovered by changing polar into log-polar coordinates (let
this equation can be written in the equivalent form When one wants to solve the Dirichlet problem in a domain with rotational symmetry, the usual thing to do is to use the method of separation of variables for partial differential equations for Laplace's equation in polar form.
takes the simple form When solving the Dirichlet problem in Cartesian coordinates, these are exactly the equations for
Thus, once again the natural choice for a domain with rotational symmetry is not polar, but rather log-polar, coordinates.
If the domain has rotational symmetry and you want a grid consisting of rectangles, polar coordinates are a poor choice, since in the center of the circle it gives rise to triangles rather than rectangles.
Divide the plane into a grid of squares with side length 2
Use the complex exponential function to create a log-polar grid in the plane.
The left half-plane is then mapped onto the unit disc, with the number of radii equal to n. It can be even more advantageous to instead map the diagonals in these squares, which gives a discrete coordinate system in the unit disc consisting of spirals, see the figure to the right.
The latter coordinate system is for instance suitable for dealing with Dirichlet and Neumann problems.
If the discrete coordinate system is interpreted as an undirected graph in the unit disc, it can be considered as a model for an electrical network.
To every line segment in the graph is associated a conductance given by a function
The electrical network will then serve as a discrete model for the Dirichlet problem in the unit disc, where the Laplace equation takes the form of Kirchhoff's law.
On the nodes on the boundary of the circle, an electrical potential (Dirichlet data) is defined, which induces an electric current (Neumann data) through the boundary nodes.
In the case with the continuous disc, it follows that if the conductance is homogeneous, let's say
everywhere, then the Dirichlet-to-Neumann operator satisfies the following equation Already at the end of the 1970s, applications for the discrete spiral coordinate system were given in image analysis ( image registration ) .
Also, the photo receptors in the retina in the human eye are distributed in a way that has big similarities with the spiral coordinate system.
Log-polar coordinates can also be used to construct fast methods for the Radon transform and its inverse.
