Geodesic Calculations

Differential Geometry · Axiom Academy

Computing shortest paths on different surfaces in differential geometry Excellent work! You've completed the geodesic calculations. Here's what we learned: Flat spaces: Geodesics are straight lines in Euclidean space, minimizing the standard distance. Spherical geometry: Great circles are geodesics on spheres, representing the shortest path between two points on a curved surface. Cylindrical surfaces: Geodesics can be straight lines, circles, or helices depending on initial conditions and the surface parameterization. General surfaces: The geodesic equations involve Christoffel symbols derived from the metric tensor, encoding how the surface curves in space. Variational principle: All geodesics minimize arc length locally and satisfy the Euler-Lagrange equations from the calculus of variations. Understanding geodesics is fundamental to differential geometry and general relativity, where they represent the natural motion of free particles!

This is the written version of the interactive lesson above. See the full Differential Geometry course.