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I am trying to calculate distortion so I can distort overlaying text and forms to precisely match an image of an equirectangular projection.

So, how does one calculate the distortion at a given latitude on an equirectangular projection 1:45,000,000 (say, 2000 pixels wide x 1000 pixels high)?

I've been trying to figure out this post and its links to no avail: How to create an accurate Tissot Indicatrix?

I am not a professional, just a very interested amateur, so please dumb it down for me!

Many thanks!

Thanks for the prompt replies! Here's the long story; I hope it is clearer.

I am visualizing/mapping data using the Processing programming language and would like to have the 2D mapped data (different sized fonts and circles) appear undistorted when wrapped to a 3D globe. The data is mapped using equirectangular x, y's and the maps I want to use as backdrops are all this projection, so I'm assuming I want to "match" this distortion (e.g. by calculating distortion via latitude using Tissot equations?). Using the programming language I can precisely distort both the text and the circles. I think all I need are the equations to do it correctly.

Here is the original 2D data map:

enter image description here

When wrapped it looks distorted, like this:

enter image description here

The $10,000 Question: How can I make my 2D image look undistorted when wrapped to the 3D sphere?

For reference, here's the same question asked differently on the Processing forum.

Thanks again!

If I understand you correctly I'm not sure I want to reproject to an orthographic projection. I want my 2D data map to wrap to a 3D sphere model that can be interacted with (i.e. spun).

I am using a 3D modeling program (Cinema 4D) to wrap a sphere with a 2MB "Blue Marble" image (equirectangular projection) from NASA.

When wrapped it appears undistorted from all hemispheres (not just one hemisphere, as an orthographic projection would be?), see: still from 3D model above. (The modeling program is doing the orthographic projection for me as I rotate the object, I suppose.) Therefore, I think that if I distort my 2D data map in a similar way it too will appear undistorted on the 3D sphere. Here's a shot I took with an equation that approximates equirectangular distortion. You'll notice the egg shaped ellipses from the 2D image look like a circle when wrapped to the 3D sphere. Similarly, the Tissot ellipses also appear as circles on the 3D sphere.

Tissot Indicatrix with distorted circles Distorted circles wrapped to 3D sphere

This is why I was looking at the Tissot equations...to more precisely figure out the distortion of the equirectangular projection at different latitudes so I could distort my overlay accordingly.

Hope this all makes sense.

Perhaps you're right that I should use a GIS program. I just downloaded Cartographica and will see if I can figure it out. Any Mac software suggestions for a newbie undertaking this task?

Thanks again.

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Do you really mean calculating the distortion or do you really want to know how to calculate the projection itself? Perhaps you could make an image available on the Web to illustrate what you're trying to accomplish. Your use of "match" suggests you would like to determine how to transform one image to another, indicating you need to specify what you're starting with as well as what you want to end up with. – whuber♦ Jun 27 '12 at 22:27
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We understand it is hard to describe what you want to do without knowing the jargon, but it sounds like you're trying to describe the process rather than the result. Try starting with a problem you want to solve, then the results you would like, and we'll try to fill in the gaps :) – MerseyViking Jun 27 '12 at 23:37
In technical terms: you want to reproject from the equirectangular projection to an orthographic ("world from space") projection. What software can you use? If you have GIS software or are willing to code against a projection library, the work is basically done for you. Otherwise, you need to implement the equations for unprojecting the equirectangular projection (easy) and projecting the orthographic projection (not too hard, but requires some skills in coding numerical routines). – whuber♦ Jun 28 '12 at 14:07
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I saw this post and I'm trying to basically do the exact same thing. I want to draw 2D circles that distort correctly when projected onto a 3D sphere. I was wondering if you would be willing to share the algorithm you used for the distortion of the 2D circles? Should really have been a comment not an answer, but I wrote it in the wrong spot. Sorry. – HankTurbo Jul 25 '12 at 3:37
You should draw your data in 3D space then project it back to the sphere. – AngelLeliel Nov 19 '12 at 8:59
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2 Answers

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Assuming the shapes being drawn cover a small portion of the sphere, you should be able to get by with scaling the width by 1/cos(lat) and leaving height alone.

The larger the shape, the less well this will work though.

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Could you please explain why this would work? It looks like it would fail dramatically even at small portions of the sphere near the edge of its rendering in the example images. – whuber♦ Dec 17 '12 at 22:00
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I can't figure out how to add a comment so I will put this in the solution and let the moderators scramble to figure out why I can't comment.

My first impression when reading your question was "Why are you not designing your circles in a conformal projection like Mercator". You could project this map into a Mercator projection and see your circle and text distortion, fix everything to look nice and when you project it to your globe, the shapes should stay correct (that is the definition of a conformal projection).

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This reads like an answer to me, so I'll leave it. But it's incorrect, being based on a misunderstanding: conformal projections do not project all circles to circles. They do so only infinitesimally. The distinction is huge: consider what the Mercator does to any circle that goes around the Earth's axis, for instance. It cannot possibly map it as a circle--it must break it somewhere. For more discussion of this, please search our site for Tissot. – whuber♦ 22 hours ago

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