In mathematics, log-polar coordinates (or logarithmic polar coordinates) is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle. 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. In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates.
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 angular coordinate is the same as for polar coordinates, while the radial coordinate is transformed according to the rule
r=e\rho
where
r
\begin{cases}\rho=ln\left(\sqrt{x2+y2}\right),\ \theta=\operatorname{atan2}(y,x).\end{cases}
and the formulas for transformation from log-polar to Cartesian coordinates are
\begin{cases}x=e\rho\cos\theta,\ y=e\rho\sin\theta.\end{cases}
By using complex numbers (x, y) = x + iy, the latter transformation can be written as
x+iy=e\rho+i\theta
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. This is not the case for polar coordinates.
Laplace's equation in two dimensions is given by
| \partial2u | |
| \partialx2 |
+
| \partial2u | |
| \partialy2 |
=0
in Cartesian coordinates. Writing the same equation in polar coordinates gives the more complicated equation
| r | \partial | \left(r |
| \partialr |
| \partialu | |
| \partialr |
\right)+
| \partial2u | |
| \partial\theta2 |
=0
or equivalently
| \left(r | \partial |
| \partialr |
\right)2u+
| \partial2u | |
| \partial\theta2 |
=0
However, from the relation
r=e\rho
| r | \partial |
| \partialr |
=
| \partial | |
| \partial\rho |
| \partial2u | |
| \partial\rho2 |
+
| \partial2u | |
| \partial\theta2 |
=0
has the same simple expression as in Cartesian coordinates. This is true for all coordinate systems where the transformation to Cartesian coordinates is given by a conformal mapping. 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. An analytical function
f(x,y)=u(x,y)+iv(x,y)
| \partialu | |
| \partialx |
=
| \partialv | , | |
| \partialy |
| \partialu | |
| \partialy |
=-
| \partialv | |
| \partialx |
If the function instead is expressed in polar form
f(rei\theta)=Rei\Phi
| r | \partiallogR |
| \partialr |
=
| \partial\Phi | , | |
| \partial\theta |
| \partiallogR | |
| \partial\theta |
=-r
| \partial\Phi | |
| \partialr |
,
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
P=logR
| \partialP | |
| \partial\rho |
=
| \partial\Phi | , | |
| \partial\theta |
| \partialP | |
| \partial\theta |
=-
| \partial\Phi | |
| \partial\rho |
The Cauchy - Riemann equations can also be written in one single equation as
| \left( | \partial |
| \partialx |
+i
| \partial | |
| \partialy |
\right)f(x+iy)=0
By expressing
| \partial | |
| \partialx |
| \partial | |
| \partialy |
| \partial | |
| \partial\rho |
| \partial | |
| \partial\theta |
| \left( | \partial |
| \partial\rho |
+i
| \partial | |
| \partial\theta |
\right)f(e\rho)=0
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. This means that you write
u(r,\theta)=R(r)\Theta(\theta)
\begin{cases}\Theta''(\theta)+\nu2\Theta(\theta)=0\ r2R''(r)+rR'(r)-\nu2R(r)=0\end{cases}
where
\nu
r2R''(r)+crR'(r)+dR(r)=0
where
c,d
R(r)=rλ
P''(\rho)+(c-1)P'(\rho)+dP(\rho)=0
When considering Laplace's equation,
c=1
d=-\nu2
r
P''(\rho)-\nu2P(\rho)=0
When solving the Dirichlet problem in Cartesian coordinates, these are exactly the equations for
x
y
In order to solve a PDE numerically in a domain, a discrete coordinate system must be introduced in this domain. 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. However, this can be remedied by introducing log-polar coordinates in the following way. Divide the plane into a grid of squares with side length 2
\pi
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
\gamma
Λ\gamma
In the case with the continuous disc, it follows that if the conductance is homogeneous, let's say
\gamma=1
| 2 | |
| Λ | |
| \gamma |
+
| \partial2 | |
| \partial\theta2 |
=0
Already at the end of the 1970s, applications for the discrete spiral coordinate system were given in image analysis (image registration) . To represent an image in this coordinate system rather than in Cartesian coordinates, gives computational advantages when rotating or zooming in an image. 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.[1] It can also be found in the Mandelbrot fractal (see picture to the right).
Log-polar coordinates can also be used to construct fast methods for the Radon transform and its inverse.[2]