Log Polar Plane Stencil
Smorgasbord / Kirt's Cogitations™ #332

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Canadian website visitor Richard F. sent me this photo of his "Log Polar Plane" acetate stencil, circa 1958. As a collector of vintage of science / technical paraphernalia, he ran across this as part of one of his acquisitions. "Computing Aids" is printed on it. I had never heard of the log polar plane, but according to the Wikipedia entry, "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."

The David Young, on the University of Edinburgh website, explains, "Log-polar sampling is a spatially-variant image representation in which pixel separation increases linearly with distance from a central point. It provides a way of concentrating computational resources on regions of interest, whilst retaining low-resolution information from a wider field of view. Foveal image representations like this are most useful in the context of active vision systems, where the densely sampled central region can be directed to pick up the most salient information."

Other sources say the log polar plane is convenient for plotting data with rotating or spiral path, such as a hurricane wind pattern. Also, cone and rod light retina receptor cells in the eye exhibit a logarithmic spiral pattern, so researchers use the log polar plane for mapping images in computer vision applications. This page from the Institute for Systems and Robotics (ISR|Lisboa) has an interesting representation of an image plotted in standard Cartesian coordinates, and the equivalent points plotted on the log polar plane and in retinal coordinates.

As for what Richard's log polar plane stencil would have been used for back in 1958, there was evidently applications in target tracking and targeting for military purposes that might have been facilitated by such a device. A lot of work was being done on satellite orbits and rocket trajectories that might have benefitted from log polar plane plots. As with plotting data on other special coordinate systems like polar, spherical, and Smith chart planes, deviations from the expected shape are usually more easily spotted than when represented in Cartesian coordinates.

If you have more information on the log polar plane and why such a stencil would have been available in 1958, please send me a note. 

Posted September 16, 2021