What projection could be used for a very large mountain?

[Thurlor]

The title says it all.

What projection could I use to 'unwrap' a large mountain to a flat top down view? I hope that makes sense. By very large mountain I mean a mountain much like Castle Mount in the Majipoor Series by Robert Silverberg.

[ManOfSteel]

I'm not familiar with the Majipoor series but I would think that anythingn as "small" as a mountain could be most accurately depicted with something simple like an equirectangular projection. After all, you're not dealing with the problem of trying to flatten a sphere. If depicting the different elevations was imperative than a topographical map (in equirectangular projection) would be the ticket. Remember that all maps warp a sphere, but you''re not dealing with a sphere, so applying another projection would distort the mountain.

[Thurlor]

@ ManOfSteel

Imagine unwrapping the surface of the mountain then laying it flat. If the mountain were a simple cone it would be easy.

I realise none of this is really needed, but the idea I have is of a civilization living on the slopes of a giant mountain/volcano (probably as large as Olympus Mons on Mars). Anyway, they flatten all of there maps so as there is no distortion from height.

[vorropohaiah]

Anyway, they flatten all of there maps so as there is no distortion from height.

I'm sure there's people more knowledgeable than me in this field, though your statement there is somewhat counter productive, as, given your other example of a cone, flattening such an object has to leave distortion - if not height, then with width. I'd say try and come up with something we dont have in the real world, which also serves to give their culture something distinct, making them seem more real. dont ask what though i have enough on my plate coming up with thing for my world! :p

and i like the idea you're aiming for, makes sense to me.

[Hai-Etlik]

@ ManOfSteel

Imagine unwrapping the surface of the mountain then laying it flat. If the mountain were a simple cone it would be easy.

I realise none of this is really needed, but the idea I have is of a civilization living on the slopes of a giant mountain/volcano (probably as large as Olympus Mons on Mars). Anyway, they flatten all of there maps so as there is no distortion from height.

Put simply, there's no such projection for pretty much the same reason there are none for a globe, both have complex curvature. I'd draw such a map the same way I would any other similarly large scale map. Any projection tangent or secant to the globe near the mountain would work. The easiest to understand would be an orthographic projection centred on the mountain (Roughly, the view of the mountain from directly above it in space.) If there are any really steep areas with significant details, try covering them with side view insets.

You could also make a conical/pyramidal simplification of the mountain which does have simple curvature. Then project the surface of the mountain onto that, then unwrap. You will still end up with a discontinuity this way where you "cut" the cone but you will only need the one. This is roughly what is done on the globe in using a ellipsoid to approximate the more correct geoid (An equipotential surface of the Earth's gravitational field.). Defining the mountain to cone projection would be a bit of a pain though as would picking a particular cone (We have a lot of different ellipsoids for approximating the geoid)

[Redrobes]

You can project any shape with no distortion if that shape could be printed on a ( 2D ) piece of paper and then folded smoothly into the desired 3D shape. If you can fold it into the 3D shape there exists a projection that can leave no distortion. If you cant without stretching or cutting the paper then you will always have distortion and its more of a choice of what kind of distortion do you prefer or where do you want it to dominate.

For a perfect cone you could map it as a sector of a disk where the ends of the segment wrap back on themselves. If the mountain is not perfectly conical then it cannot be made into a perfect sector of a disk but I would think a map based on a sector of disk is a good starting point.

You could make something like a Mercator projection for a cone by using cylindrical coords. The top of the map could be at the center of the cone and at the highest elevation. Then the bottom of the map would be the base of the cone. The map would be stretched so that the higher elevations would have much larger map area than the lower but it would fit on a rectangular piece of paper.

The first is almost equal area but not quite when the mountain is far from a perfect cone. Also, when the code is not perfect then neither of these two would produce a 1:1 mapping between mountain and map tho this is true for all maps of irregular objects.

[Thurlor]

Thanks for the replies.

I've decided to just use a conical projection. The end result will most likely be an irregular circle with a wedge missing.