# Geodesics on R^2: Exact Formula?

• Statis
In summary, the conversation is discussing whether a general exact formula for geodesics can be found using the provided metric and the geodesic equation. It is suggested that since the model is a reparametrization of the Hyperbolic space, the geodesics should be known. A link is provided as evidence.

#### Statis

Hello,

Suppose that $$R^2$$ is provided with the following metric

$$ds^2 = dx^2 + (\cosh(x))^2 dy^2$$
Can we find a general exact formula $$\alpha(t)$$ for the geodesics (starting at an arbitrary point) ?

The geodesic equation gives
$$x'' - \cosh(x)\sinh(x) (y')^2 = 0$$
$$y'' + 2 \tanh(x) x' y' = 0$$

I guess that since this model is simply a reparametrization of the Hyperbolic space on R^2 the geodesics should be known ?

Thank you

## 1. What is a geodesic on R^2?

A geodesic on R^2 is a curve that follows the shortest distance between two points on a flat plane. It can also be thought of as the path of a particle that is moving at a constant speed and is only affected by the gravity of the plane's surface.

## 2. How are geodesics on R^2 calculated?

The exact formula for calculating geodesics on R^2 involves using differential equations and the concept of parallel transport. This formula takes into account the curvature of the plane's surface and the initial conditions of the geodesic.

## 3. What are some real-world applications of geodesics on R^2?

Geodesics on R^2 have numerous applications in fields such as engineering, architecture, and physics. They are used in the design of roads, bridges, and pipelines to determine the most efficient route between two points. They are also used in satellite navigation systems and computer graphics.

## 4. How do geodesics on R^2 differ from geodesics on other surfaces?

Geodesics on R^2 are unique in that they are the straightest possible curves on a flat plane. On other surfaces, such as a sphere or a saddle-shaped surface, geodesics may appear curved or even spiral. This is due to the curvature of the surface and the varying gravitational forces acting on the particle.

## 5. Can geodesics on R^2 ever intersect?

No, geodesics on R^2 can never intersect. This is because they are always following the shortest distance between two points and cannot deviate from this path. If two geodesics appear to intersect, it is because they are on two different planes within R^2.