Great circles, helical geodesics on a torus, and straight lines on curved space
About
A geodesic is the straightest possible path on a surface — locally length-minimizing. On a sphere they are great circles; on a torus they can be helical or closed.
Choose a surface and add geodesics
Key Facts
Sphere: all geodesics are great circles; any two meet at antipodal points Torus: geodesics are (p,q)-curves; rational slope → closed, irrational → dense Hyperboloid: ruled surface — straight lines ARE geodesics Jacobi fields: geodesic deviation — spread of nearby geodesics encodes curvature
Geodesic Equation
ẍᵏ + Γᵏᵢⱼẋⁱẋʲ = 0
where Γ = Christoffel symbols (from metric g). Curvature bends the path.