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[Shades]

wrap a sheet of paper around a sphere. OMG! you can't!

Sure you can, just depends on the shape of the sheet.

[Rose]

flat sheet.

[Kogut]

Funny, I just started doing sth similar (but for now I have flat world and I am making animal interactions).

[Duke 2.0]

It's a rather strange a sheet, but still a flat sheet.

[Nadaka]

It will still be distorted when wrapped on a sphere, less than a rectangle, but still distorted.

[Akhier the Dragon hearted]

   Actually maps like that are made to not distort though
that one might. Its a weirder shape then it has to be
because whoever made it wanted to keep from cutting up
any continents.

[Kogut]

maps are made in ways to reduce distortion - it is impossible to keep accurate area, shape etc in the same type of map projection [ http://en.wikipedia.org/wiki/Map_projection#Choosing_a_model_for_the_shape_of_the_Earth ].

EDIT: from enwiki: NOTE: Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.

[Eagleon]

Guys, you're missing the simple solution here - reduce the paper to a soggy mass and spray the map over the sphere in a fine mist.

Problem, cartographers?

[Virex]

Guys, you're missing the simple solution here - reduce the paper to a soggy mass and spray the map over the sphere in a fine mist.

Problem, cartographers?

I think that would make a lot of Topologists cringe. If you're not careful you may also end up with 2 maps instead of one.

This model depends on rigid plates that don't have significant deformations or velocity deviations relative to their total velocity. I think I can fix for the deformations by adding in a deformation factor somewhere in there, but I'm currently stuck on the actual process of separate plates forming. Since the rift has not completely separated the two plates yet, a flood-fill algorithm will overstep this boundary. However, the velocities of the plates on either side of the forming rift are bound to be in complete opposite directions.

You could make your flood fill algorithm look at the plate's direction. If it hits a discontinuity in the direction of movement, you've hit a future rift. Of course this would mean plates moving parallel to each other are considered as one plate so if you need to you may want to catch that.

[Shades]

maps are made in ways to reduce distortion - it is impossible to keep accurate area, shape etc in the same type of map projection

It's not impossible, you just don't make a conformal one. Unwrapping a sphere is easy, and a map on a sphere would also be perfectly fine on the unwrap. It's just north would not be the same direction at every point and so would be a fairly unhelpful as a map, but a still flat sheet and fit perfectly onto the sphere. And if you still don't think unwrapping a sphere is possible, try peeling an orange in one connected layer and laying the skin flat.

If your talking about conformal, rectangular maps then yes you have a point but when your talking surface map generation there is no need to stick that that.

[Nadaka]

An orange peel is elastic. when you lay it flat after peeling it, it distorts. To place translate a surface curved in more than one dimension onto a flat surface you must distort it. You can never slice it to 0 wide stripes, because then the flat map would have no area. Any non-0 slice of that curved surface will itself be curved and have to be distorted to lay flat.

[Shades]

An orange peel is elastic. when you lay it flat after peeling it, it distorts. To place translate a surface curved in more than one dimension onto a flat surface you must distort it. You can never slice it to 0 wide stripes, because then the flat map would have no area. Any non-0 slice of that curved surface will itself be curved and have to be distorted to lay flat.

Your right, I'm being an idiot and not thinking straight. Sorry

[Nadaka]

Your right, I'm being an idiot and not thinking straight. Sorry

no need to apologize!  Did my text have a mean tone of voice? It does that some times. Oh, and you should totally ignore my ridiculously bad grammar. Sometimes when I rewrite a sentence, it forget to do the whole thing.

I should think of something I can say that is on topic...

This is an interesting idea, though I generally use midpoint displacement to produce fractal maps that just look nice, instead of being physically accurate.

[Virex]

I just realized that you want the edges of plates to be where the stress exceeds a certain threshold, for which you could just use vector math. Since stress is proportional to the difference between a plate's general movement direction and the direction that the underlying magma is moving in, the general stress at any one point can be defined as:

S = V_p-V_m
With S the stress, V_p the plate velocity and V_m the magma velocity at the given point, all vectors.


This stress consists of 2 parts, compressive stress, S_c and shear stress S_s. You'd probably only want to get plate boundaries there where either |S_s| (the magnitude of S_s, defined as sqrt(S_s.X² + S_s.Y²)) is sufficiently large, or where |S_c| is sufficiently large and in the same direction as the plate's movement. In the case where S_c is in the opposite direction, the plate is being compressed which does not cause a plate rupture (but it may have other effects).


S_c = S * cos(a),
S_s = S * sin(a),
With a the angle between S and V_p


So for a given initial V_p the plate boundaries can be calculated. The problem is now that while the rupture zones can be found, there is no guarantee that these zones will actually connect to form new plates. To ensure this, it'd probably be best not to blindly calculate where S_c or S_s crosses a certain limit. Instead it should be observed that a rupture at one spot will induce stress in the rest of the rock. This stress, S_r, is in addition to the already existing stress, which means that a small rupture can propagate trough the rock until it hits another rupture.


To solve this, observe that at a rupture, there are essentially 2 plates being formed. The direction and magnitude of S_r are proportional to the stress causing the rupture, but on either end of the rupture the direction of S_r is directly opposite to the direction of S_r at the other side. So a rupture due to a high S_c will lead to S_r1 and S_r2 pointing away from the rupture, while a rupture due to a high S_s leads to S_r1 and S_r2 that are perpendicular to the rupture but in opposite directions.


To then calculate how the crack will propagate, for the points around the rupture, evaluate S' = S+S_r, obtain S'_c and S'_s and see if new cracks are created. This will probably cause a lot of small cracks around existing cracks. To combat that, make a rupture only propagate along the line where S'*cos(b) is the highest, where b is the angle between the local direction of the rupture and S'.


Now, this doesn't completely solve the problem yet. A rupture may hit a "dead zone" trough which it can not propagate, leading to ruptures that start and end in nothing. This can be avoided by scaling S_r with a value L, where L is a function of the length of the rupture it connects to (I'd expect a logarithmic function to work well, but you'll need to experiment with this). This way, a long rupture will have the tendency to continue even if the local bedrock is not under large stresses. Finally, terminate a rupture as soon as it hits an existing rupture.
After running the rupture algorithm, obtain directions for the newly created plates using a flood-fill algorithm and optionally rerun the rupture algorithm, with 1 modification. In the calculation of S, also take into account the stress caused by one plate pushing against another.


Edit: I see you've already got compressive and shear stress worked out, so feel free to ignore the trigonometry if you don't need it.

[Max White]

Your strange multidimensional sphere is actually a simple torus. Spheres in higher dimensions are much weirder than that.

Thank you   

EDIT

It's not even a simple torus, actually. It's toroidal, but with an odd distortion going on.

Well, thank you even more so.