Calc III, Parametric Equations, 3D Arc Length. How Do.
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I'm trying to get advanced for a math project involving satellites. The normal project requires I get the amount of satellites it'd take to cover a 2D representation of the Earth, assuming it's a perfect circle. The extra extra advanced version would be doing it in 3D, and I could do an approximation, I want to do it the hardcore exact way.
But getting onto my problem, one of the required steps is calculating the arc length/arc measurement of the sphere. This is easy enough in two dimensions, but trying to get the exacts in 3D involves a lot more work.
If I'm not mistaken I have to use this formula, integral b..a ||vector: r'(t)||dt, I haven't really used ||..|| too often, but I hope that I'm right in saying it means to normalize the vector: r'(t), which expanded is sqrt((dx/dt)^2 + (dy/dt)^2 + dz/dt)^2). I was told by my math teacher that I would need to parametrize the area that the satellite covers, how would I go about doing that and then using that information in the formula I had before to get the 3D Arc Length of that sector of the sphere/earth?
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