Geometry

 

A and B are two points on a sphere of radius 2. We know the space distance between A and B is 2, What is distance from A to B along the (minor) arc of a great circle?    

 

Consider the lines from the center of the sphere to A and to B.    

Each of these two lines is the radius of the sphere, namely, 2.    

Since the distance from A to B is 2, an equilateral triangle is formed.    

 

Every angle of an equilateral triangle is 60^o which is 1/6 of the great circle.    

The circumference of the great circle is π d  =  3.1416 • 4  =  12.5664.    

1/6 • 12.5664  =  2.0944 is the distance from A to B along the curvature of the sphere.    

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What is distance from A to B along the (minor) arc of a great circle?

 

\(s=2\\ r=2\\ b=2\cdot \frac{\alpha}{2}\cdot \frac{2\pi r}{360^{\circ}} \\sin\frac{\alpha}{2}=\frac{1}{2}\\ b=2\cdot arcsin \frac{1}{2}\cdot\frac{2\pi \cdot 2}{360^{\circ}}\\ \color{blue}b=2.0944\)