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- Summary:
- Parametrize the geodesics of a 2-sphere in terms of the arc lenght.

I'm trying to evaluate the arc lenght 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 lenght?

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 lenght?