## Example for spherical coordinates

One possible transformation from spherical coordinates (r,φ,ϑ) to cartesian coordinates (x,y,z) is:

where:

where:

## Geodetic distance

My current try is:

Length is defined to be:

So if , then:

The usual definition of "distance" is "minimum length", so try to find the minimum:

-> min

where:

And x, y, z as above, but with (r,φ,ϑ) being functions of t.

So the extremum is determined by:

I'm not sure how to stay ON the sphere with the curve.

And so:

After dropping the square root (is that safe?), one gets:

How to stay on the surface of the sphere?

## Spherical Angles

The *Spatial Angle Ω* on a sphere with radius R is defined to be:

Where A is the area on the surface of the sphere enclosed by beams from the center to the object.

This is so that: