Longitude lines in perspective, what shape are they?
July 26, 2013 11:49 AM   Subscribe

I'm drawing/painting a globe (in time lapse, so the sun appears to rise) and trying to figure out what shape, exactly, the longitude (not latitude) lines are when a globe is flattened into a circle (I know I can just trace them, but I've gotten curious). So, the outermost longitude lines are tangent to the circle, they are half-circles, and the centermost line is a straight line from pole to pole, it's the progressively flatter curves in between that I'm trying to figure out. Do they have a name? Are they catenaries? Are they the curves of progressively larger circles? How do I construct them (easily)? Example image: HERE
posted by sexyrobot to Science & Nature (11 answers total) 2 users marked this as a favorite
Best answer: They are ellipses, or more precisely, halves of ellipses. Just like the outermost lines that form a circle (describing them as tangent to the circle doesn't make much sense) the remaining longitude lines are simply circles that have been rotated and projected into 2d (each one representing a slice through the middle of a sphere, it was originally a circle). When you project a circle into an orthographic 2d view, you get an ellipse.
posted by tylerkaraszewski at 12:07 PM on July 26, 2013 [1 favorite]

Response by poster: ah, of course (they looked a little 'flattened' to me)...ok...BONUS QUESTION: If I'm using the 'string tied to two pushpins' method of drawing these ellipses, where, between the center and the poles, am I putting the pins? Do they need to get progressively further apart? (like: .._.__.___.____._____.) Or can I just divide by six? (assuming 12 visible longitude lines)
posted by sexyrobot at 12:23 PM on July 26, 2013

Lines of common longitude are called meridians, and relate to half of the great circle passing through the North and South poles.

How meridians and parallels are shaped when you draw them is all a matter of projection.

There are numerous map projections, and you might find it useful to tinker with some of them programmatically before you physically draw them or paint them. You might find the finalist maps from this recent Dymaxion map projection contest from the Buckminster Fuller Institute to be of interest and inspiration.
posted by oceanjesse at 12:37 PM on July 26, 2013

If your globe has a diameter D, and you want an ellipse with minor axis W (i.e. the whole width of the ellipse), then your foci (where the pins go) should each be the following distance north and south from the center of your globe:

f = sqrt((D/2)^2 - (W/2)^2)

For example, if your globe has a diameter D=20 cm, and you want an ellipse half that width (W = 10 cm), then your foci should be f = 8.66 cm above and below the center of your globe.
posted by Salvor Hardin at 12:59 PM on July 26, 2013

Best answer: Also, the amount of string you use will be determined by how much string it takes to reach the north or south pole with your pencil.
posted by Salvor Hardin at 1:00 PM on July 26, 2013 [1 favorite]

Also, in line with oceanjesse's comment, it would be pretty easy to generate these ellipses using computers, so unless handmade ellipses are part of the fun, I would use something like this. Full disclosure, a friend of mine developed that online calculator, but it's awesome and free.
posted by Salvor Hardin at 1:02 PM on July 26, 2013

Response by poster: handmade ellipses are definitely part of the fun...but are they actually ellipses? Wouldn't an ellipse become tangent at the poles? All the images I see they converge to a point at the pole...what am I missing? (I realize that matching curves on either side of center are from 2 different longitude circles, but still..) or is every globe just drawn wrong?
posted by sexyrobot at 1:36 PM on July 26, 2013

Best answer: Imagine this roll of tape is a slice out of the earth, and one of it's edges is a meridian line. Note what it looks like at the top and bottom as the angle at which it's viewed changes. It never gets pointy.
posted by tylerkaraszewski at 2:00 PM on July 26, 2013

Best answer: maybe you are confusing how longitudinal and latitudinal lines work.

Latitudes never meet, so if you drew them on a 2d surface the're just straight horizontal lines across the circle. Longitudinal lines meet at the north and south poles, so you draw ellipses on a 2d surface to reflect that, and add the illusion of depth.

If you were looking at a 2d rendering of a globe tilted on its axis the latitudinal lines would be elispes as well but still not meet in a point.

Good example here.
posted by Max Power at 2:20 PM on July 26, 2013

Response by poster: OMG you filmed a roll of tape! :D
Ok, that's what I thought (my experiment, looking at a 6" mars globe without longitude lines and seams that dont go all the way to the pole, was somewhat less illuminating) and max powers illustration (on left) better reflects the real world than my example which is probably some other kind of projection...thanks!
posted by sexyrobot at 2:29 PM on July 26, 2013

This might help you: gprojector.
posted by DelusionsofGrandeur at 5:15 PM on July 26, 2013

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