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Would a large, I mean large, impact crater on a spherical world be noticeably distorted on maps using different projections? Eg mercator, hanmer-aitoff, equidistant Etc.

Side question - Would there be a difference if it was a head-on circular crater or a side-on deeper on one end crater?

I assume if there was distortion, it would depend on the latitude? Further north/south resulting in more skewed depictions? Is there a handy calculation I can do to figure out what this distortion is, so I can draw the crater correctly using different projections.

My scenario has a large (haven't actually measured it yet) crater between the tropics and about about 50 degrees south. Rough guesstimate using 69miles to a degree latitude equals about 1380 to 2070miles wide. Let's make it nice and easy and say, 1500mile wide crater.

I'm trying to figure out how to draw my map compared to what you would see from space...how I wish I could just have a 3D model!

Which projection would result in a depiction of a recognisably circle crater with little to no distortion?

Edit: This question is not (and answers should not be) opinion based, but rather an answer that provides the least visual distortion. For the close and down voters, I have removed the request for the 'best' which would be opinion based, and clarified it to simply the one that would result in the least distortion. No matter how much you prefer a particular projection, distortion is pretty hard to put down as opinion. It is either distorted or not! FYI, I can't see close votes on my phone, so some comments next time would be very helpful! oops, triple checked. Apologies. I can see close votes on my phone, but no notifications pop up so it isn't drawn to my attention at the time. Please, still leave some comments so that I can get a heads-up.

I have accepted an answer for the extreme helpfulness of it but feel free to answer still if a better projection is out there! Thank you to Durakken for the very helpful Link that has helped me enormously to visualise my test map as a sphere. I now just have to scale down the landmass and play around with the different projections (I may have had the measurements wrong for my crater size!) test map as a globe

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Here is a equirectangular projection of how the hit would look. red lines are 0 and 45 degrees, black line is 50 degrees. The circle is the rough size of your crator on an earth map.

enter image description here

You can DL this and see how it projects on a 3d globe as well as a number of other projections Here

Basically, all you have to do this is draw a circle and squash 60-70% on the north and south side. I could have made it perfect by getting the exact number from wiki, but I just thought of that this instant and don't feel like editing so... blah

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    $\begingroup$ @EveryBitHelps This is a 1000x2000px image. The roughly 12miles per pixels. And there is indeed a calculation you can use for mercator maps, though I've never used it cuz I suck with that stuff. But anyways the link I put in the answer should help you immensely... Also I'm thinking about it and I probably did it wrong, not sure I probably should have stretched the east west rather than squashed the north south. $\endgroup$ – Durakken Oct 4 '16 at 15:30
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    $\begingroup$ @EveryBitHelps Yup I found that some time ago and I need to figure out how to get the source code for it just in case the maker ever decides to delete it ^.^ It's just 1 of many tools for map making that there are. $\endgroup$ – Durakken Oct 4 '16 at 15:47
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    $\begingroup$ @EveryBitHelps The 3d height map bit kinda sucks, but if you know what you're doing you can do some pretty cool things with it. It also has a distance calculator on it so you can measure from 1 spot to another. $\endgroup$ – Durakken Oct 4 '16 at 19:55
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    $\begingroup$ @Durakken, The source code can be found on the Bitbucket repository (the "readme" link in the Help/About menu). And since hosting it from Bitbucket is free, I don't have any plans to delete it :) $\endgroup$ – Braffin Oct 6 '16 at 19:41
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    $\begingroup$ @Durakken, well you made me realize that I really do a terrible job at explaining how to actually make the Night Image work. You have to enable the atmosphere option to make it work. Then, if you upload a night map image (something like this) along with a normal map image, the result looks like this. $\endgroup$ – Braffin Oct 6 '16 at 20:53
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Quoting Wikipedia:

The stereographic is the only projection that maps all circles of a sphere to circles.

(Some links added by me.) So if you want to make sure that your crater appears circular no matter where on the planet it is, then I'd go for this projection. You can project half the planet's suface reasonably well (albeit already with considerable length distortion), but you'd need the whole infinite plane to show the whole planet which is obviously not feasible.

If you can choose the projection after you know the crater, you can choose an azimuthal projection with the center of projection (which need not be the center of the map) incident with the center of the crater. Due to the radial symmetry of an azimuthal projection this ensures that the crater will be represented as a circle.

Of course, most azimutal projections tend to have their center located at a pole, or perhaps at the equator. So using such a map on earth (for one of the “tiny” craters we have) would likely be perceived as strange, and putting undue emphasis on such a feature. But if your crater is as pronounced as the size suggests, then it might well be “the obvious thing” for any civilization to use this prominent feature of their planet as the center for all world maps.

All of this is assuming a circular crater, as from a head-on impact. If the crater is not circular, then these projections would represent that fact by not creating a circular image of it. Since I don't recall seeing any pictures of non-circular craters on the moon or similar, I'm not sure this can actually happen. Perhaps the main force behind the formation of the crater is the shockwave which radiates out from the point of impact, and the horizontal velocity of the object has negligible effect on this? Perhaps should ask on Physics SE…

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    $\begingroup$ Excellent answer based on fact and reference (even if it is Wikipedia, I'll take it!). You even answered the side question. Thank you! Now. Do I change the accepted answer or not? I'll see what the voters say! $\endgroup$ – EveryBitHelps Oct 4 '16 at 22:52
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I've actually created a simple interactive plot of how circles appear in a Mercator projection:

https://www.desmos.com/calculator/0k3bojxx6k

You can vary the two sliders. R is the radius of the circle (expressed in degrees, 90 corresponds to a circle that covers half of the sphere) and φ is the latitude of the center of the circle. I've set the parameters roughly equal to what your crater would look like if the planet is similar in size to earth.

Here's a quick image of what that looks like:

enter image description here

The orange curves show an orthographic projection of the sphere and the circle (what the planet would look like from really far away). The red curves show the Mercator projection.

So at the proposed size, Mercator leaves the crater looking mostly circular, but it will distort the position of the center of the crater.

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  • $\begingroup$ A very helpful link. Thanks for sharing your work! I'm going to save this link! $\endgroup$ – EveryBitHelps Oct 4 '16 at 20:14
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    $\begingroup$ Mercator is a navigational map, it's almost certainly the wrong map to use for this task. The purpose of the Mercator is to plot compass bearings as straight lines, distortion of geographical features is not a consideration, only the paths between them. $\endgroup$ – Separatrix Oct 14 '16 at 7:28
  • $\begingroup$ @Separatrix Indeed. My answer only exists for completeness to show exactly how Mercator projects (and distorts) circles. $\endgroup$ – Kyle Oct 14 '16 at 13:34
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I've already had this problem. To solve it, I drew a circle on a globe (an old globe, may I add, which I reused before chucking it for good, and I used a pin, a string and a thin-tipped felt pen to produce the circle) with lines for latitude and longitude, then I got a mercator map with only longitude and latitude lines and drew a point at each one of those lines. Then I drew a line connecting all the dots.

To be fair, I wanted an extremely large crater (hundreds of km) and, for the end result, I needed the margins of the crater to have eroded quite a bit, which meant the lines connecting the dots could zig-zag some (not to mention I further distorted the resulting crater edge later on).

Nevertheless, the end result can be applied as a mask on Google Earth and looks rather round, while it looks plain funny on the mercator map.

EDIT
There's another way, which should be easy with the latitudes, but not so much with longitudes. This will only work well for large craters... like your own.

First, find out how many miles wide you want your crater to be and draw a circle on a milimeter paper (or a very finely scaled layer on GIMP/Photoshop/etc) where you can have an easy scale. Now you know that 1500 miles equals 150in and you can easily add a layer for latitude where you draw a line every 69 miles. Then you decide which one is for the Equator and number them all accordingly.

Next, find out how many miles the crater spans at each latitude. Now add another layer with lines for latitude and longitude.

Next you need to go to a site like http://www.nhc.noaa.gov/gccalc.shtml and find out how many miles span a 1º longitude at each 1º latitude your crater spans. So, at 0 latitude, each 1º longitude encompasses 69 miles too; but at 50 S latitude, 1º longitude encompasses 44 miles.

So, if at latitude 46 S your crater spans, say 56 miles (while 1º longitude spans 48 miles) you know you must plot the edges of the crater a bit wider than a 1º longitude.

Or you can do this only for the radius, rather than the diameter, and then just copy onto a new layer, flip and voila.

It will be more accurate than my first suggestion, and a whole lot more labour intensive, but it should work.

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  • $\begingroup$ I was thinking of buying an old globe, painting over it and then drawing my map onto it. But I knew there must be a easier, less 'arts and crafts' sort of answer! Although I may still do that one day for fun :) $\endgroup$ – EveryBitHelps Oct 4 '16 at 20:15
  • $\begingroup$ @EveryBitHelps: I first started by drawing circles, slightly eschewing them and then trying them on Google Earth. But it was taking forever getting to the right shape. But although, I suppose you could... wait, I'll just edit. $\endgroup$ – SC for reinstatement of Monica Oct 5 '16 at 8:59
  • $\begingroup$ I wish I could +1 again. That is very helpful. $\endgroup$ – EveryBitHelps Oct 5 '16 at 10:11
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If you're concerned about distortion, you could always use a Dymaxion map. It'll keep you from having any distortion, though placement of the crater might get tricky at that size if you want it all on the same map segment.

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    $\begingroup$ I was kinda hoping to avoid those sort of maps, though to be fair I hadn't seen that particular one before! $\endgroup$ – EveryBitHelps Oct 4 '16 at 16:24
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    $\begingroup$ "It'll keep you from having any distortion" - this is patently false. It will minimize distortion compared to many other models, but you cannot map a 3-D sphere onto 2-D space without distortion. $\endgroup$ – akirilov Oct 4 '16 at 16:27
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    $\begingroup$ @akirilov - that's fair, perhaps I overstated it. But to the best of my knowledge, the Dymaxion projection is the least distorted 2D representation. It is pretty remarkably low in distortion, at least those types typically found in flat projections. $\endgroup$ – Jesse Williams Oct 4 '16 at 16:46
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    $\begingroup$ @JesseWilliams I would believe that. If you wanted to go super crazy you could use even bigger geodesic grids: en.wikipedia.org/wiki/Geodesic_grid $\endgroup$ – akirilov Oct 4 '16 at 17:07
  • $\begingroup$ Yeah, the Dymaxion map was the work of Buckminster Fuller. He did a considerable amount to popularize the geodesic shapes. $\endgroup$ – Jesse Williams Oct 4 '16 at 17:17
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Three map projections provide good, minimally distorted maps of areas similar in size and shape to North America. Each of these projections can be optimized for the latitudes you are mapping:

The USGS has a graphical comparison of how some of the most common map projections are projected, and how they make North America look.

J.P. Snyder of the USGS wrote an excellent book titled Map Projections -- A Working Manual. It explains (and proves mathematically) the properties of the common map projections. It discusses how the projections are related, how to measure distortion, and how to optimize the projections to minimize distortion.

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