in Math

My own Map

I wanted to make my own map, and plotting system. Here’s what I imagined.

Zoomed In - Logarithmic Spiral on Azimuthal Projection

Zoomed Out - Logarithmic Spiral on Azimuthal Projection

Note: This post is a summary of all that I’ve read, and found to be related to this specific idea of mine – I don’t really know what I can do that would be counted as entirely ingenuous.

How the images above were made:

    • plots for the logarithmic spiral plotted, were obtained from Wolfram|Alpha, with a=0.5, and b=0.02 randomly put into the r=a*e^(b*theta)formula for a logarithmic spiral, so that the arms were relatively tight.
    • the “zoomed-in” Gnomonic projection of the globe, if from Wikipedia’s article on the same.
    • the “zoomed-out” one, below, is from this random … thinige’s site, taken and used without permission (though, with apparent, attribution).

I got the “inspiration”, from Escher’s Spiral Globe, the image of which, I first saw in The Infinite Book.

Escher Spiral Sphere

On a bit of looking around, I found out that that thing is called a Loxodrome, and it a logarithmic spiral projected onto a sphere. The cool thing, and why I want to use this map system, is that on a specific plot, I can use just one angle to specify where I am, as the distance from the centre, is only a function of the angle. It obviously reduces the fidelity two pieces of information provide, as in the case of a longitude and latitude angle, but I can add the piece of information about which longitude I am using as my X-axis, and then cover all locations.

What’s the benefit? By my cursory reading about this, the Rhumb line is similar to something called a Great Circle Route, (though this might be severely wrong, and just some sort of name confusion, because the wiki article about the Rhumb Line specifically talks about the severe difference between this Rhumb Line, and the Great Circle Route) which is the straightest point between two points, used by aircraft routes, and looks like a latitude  curve joining the two points. I don’t know how, yet, but somehow I could specify the translation of the origin, such that the source and destination lie on the same arm of the spiral, and again two angles would be adequate to specify along which course the plane should be navigating.

Understanding an azimuthal projection:

Stereographic projection of a sphere onto a plane

Rhumb Line Coolness:

Loxodrome -  45Loxodrome - from the top

  • * clueless *
    iLike the pictures, though 😛

    Ha ha humanities… 😉