This does a hemisphere but doesn’t map lng and lat. Gives an example of the kind of maths involved.
https://math.stackexchange.com/questions/1334241/interpolate...It’s not a simple linear x0, y0, z0 -> x1, y1, z1 because you kind of want an animation like a square blanket is wrapping around a ball, stretching the right amount in different places such that it becomes a sphere.
I tried to find a single magical equation that would give a x0, y0, 0 -> x1, y1, z1 based on param t that goes from 0 - 1. I’m sure there exists one but I tried all sorts of fancy math I knew for a day or two and still had major trouble writing it down as a single equation function.
Then I thought about it in a different way after I ate a fruit and realized I need an orange analogy. What if we cut slices of orange. The ends are crescent shapes. Now it’s a matter of animating the crescents into rectangular strips where the pointy ends stretches but the middle part remains the same.
Since in computer graphics, a sphere isn’t really a true sphere, but an approximation with triangles the strip analogy works well. We just need to animate the ends of triangles.
The gotcha is the pointy end of sphere expands from 0 to length of mercerator projection (flat square) so they cut a tiny bit of the top and bottom otherwise you have infinite stretching at the end
So you have three little functions. For an x,y in a square you figure out what slice they go into and what side of the slice. The further to the poles, the more you squeeze towards the center, so you get something like this https://blenderartists.org/t/producing-a-12-petal-gores-from...
Then you have a blanket wrapping function I.e morphing a line into circle and that’s a line -> arc -> circle animation. That’s not too hard. The arc of a very large circle is almost flat. So you divide the line into two and animate the left and right segments into two hemicircle arcs that then form a proper circle. Kind of like curving the lines from the end to bend it into a circle.
Now longitudes and latitudes both go through this line to circle animation. To get the z, we take the x arc position, the y arc position, the point on crescent strip and we do matrix rotate transforms to get final x,y,z
So rather than it being one single equation (which i’m guessing is possible using super crazy maths), we take a piecemeal approach and animate lots of little segments that add up as a plane to sphere animation using simpler maths and matrix transforms.
