page_title: Log-polar coordinates
section_title: Discrete geometry
hierarchical_section_title: Log-polar coordinates / Discrete geometry
caption_attribution_description: English: Picture of the image under the complex exponential function of a grid of squares in the plane parallel to the coordinate axes. Svenska: Avbildning under den komplexa exponentialfunktionen av ett rutnät i planet parallellt med koordinataxlarna.
context_page_description: In mathematics, log-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.
context_section_description: 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/n, where n is a positive integer. 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.
caption: Discrete coordinate system in a circular disc given by log-polar coordinates (n = 25)
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