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Area on the sphere

  1. Jun 6, 2015 #1
    How to calculate some general area on a sphere for simplicity the unit sphere. Let's say I have a ball and I draw a ring on it. What is its area? I guess I need some initial point (some coordinate). Let's take a spherical coordinates with r=1. Element of area is [itex] \sin(\theta) d \theta d \phi [/itex]. But how to define (describe) a ring? I guess I need some function [itex]\theta(\phi) [/itex] and than integrate over ...what, all or some separate region of [itex]\phi[/itex]? Or what would you suggest?

    Thank you for your advice and help.
     
  2. jcsd
  3. Jun 6, 2015 #2

    HallsofIvy

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    You can always construct a new coordinate system with the z-axis passing through the center of the sphere and the center of the circle on the surface of the sphere. Given that the sphere has radius R, a "circle" of radius r with center on the z-axis is given by [itex]\phi= \frac{2\pi r}{R}[/itex]. The area of that circle is
    [tex]\int_{0}^{2\pi}\int_0^{\frac{2\pi r}{R}} sin(\theta)d\phi d\theta[/tex].
     
  4. Jun 6, 2015 #3
    Thank you for your post, I understand , but it was just example, the ring on the ball. I would like to have some procedure for general case of area.
     
  5. Jun 6, 2015 #4
    For the general case, because the boundary of the area may be strange, I thing numerical approximation is a feasible that can be considered.
     
  6. Jun 6, 2015 #5
    I agree with numerical aprox., but I have to know some theoretical base.
     
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