Geodesics

The main aim of the next pictures is to illustrate the notion of a geodesic. For this, it good to know that a geodesic is completely determined by a starting point and a tangent vector that will determine the direction of the geodesic.

Geometric intuition:  "In a manifold, a geodesic is a kind of curve with null-acceleration, it is the path of least resistance that a particle will follow in this space..."

Geodesics: on the Catenoid

On the Helix

On the Torus

Here, we can appreciate a fascinating phenomenon that occurs in the torus, namely, "the existence of curves of irrational slope in the  torus".