Calculating Area on Sphere: Unit Sphere & Rings

[Vrbic]

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 $\sin(\theta) d \theta d \phi$. But how to define (describe) a ring? I guess I need some function $\theta(\phi)$ and than integrate over ...what, all or some separate region of $\phi$? Or what would you suggest?

Thank you for your advice and help.

 

[HallsofIvy]

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 $\phi= \frac{2\pi r}{R}$. The area of that circle is
$$\int_{0}^{2\pi}\int_0^{\frac{2\pi r}{R}} sin(\theta)d\phi d\theta$$.

 

[Vrbic]

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 $\phi= \frac{2\pi r}{R}$. The area of that circle is
$$\int_{0}^{2\pi}\int_0^{\frac{2\pi r}{R}} sin(\theta)d\phi d\theta$$.

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.

 

[tommyxu3]

For the general case, because the boundary of the area may be strange, I thing numerical approximation is a feasible that can be considered.

 

[Vrbic]

For the general case, because the boundary of the area may be strange, I thing numerical approximation is a feasible that can be considered.

I agree with numerical aprox., but I have to know some theoretical base.