I'm trying to evaluate the arc length between two points on a 2-sphere.
The geodesic equation of a 2-sphere is:
$$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$
According to this article:
http://vixra.org/pdf/1404.0016v1.pdf
the arc lenght parameterization of the 2-sphere geodesics is given by:
$$\cos(\theta)=\sqrt{1-K^2}\cdot \sin(\frac{s}{R})$$
$$\tan(\phi-\phi_{0})=K \cdot \tan(\frac{s}{R})$$
However, when I evaluate the integral:
$$s=\int_{s_{1}}^{s_{2}}{ds}=s_{2}-s_{1}$$
I don't obtain the right solution.
Is there another way of parametrize the geodesics of a 2-sphere in terms of the arc length?