Geodesics

Differential Geometry · Axiom Academy

Understanding the straightest paths on curved surfaces 1. Definition: Zero Geodesic Curvature A geodesic is a curve on a surface whose geodesic curvature vanishes at every point. While the curve may have curvature in 3D space, it has zero curvature within the surface itself. 2. Geodesics as Locally Shortest Paths Geodesics are curves that minimize distance between nearby points. While not always globally shortest, they are locally the shortest path—meaning for any sufficiently small segment, no other curve connecting the endpoints is shorter. 3. Geodesics Parallel Transport Their Tangent A defining characteristic of geodesics is that the tangent vector is parallel transported along the curve. This means the tangent vector's direction (within the surface) remains constant as you move along the geodesic. Mathematically, this is expressed as the geodesic equation : 4. Great Circles on the Sphere On a sphere, geodesics are great circles —circles whose center coincides with the center of the sphere. These include the equator and all meridians (lines of longitude). 5. Geodesics on Cylinder and Cone Cylinder: Geodesics are either circular cross-sections, vertical lines (generators), or helices that wrap around the cylinder at a constant angle. Cone: When a cone is "unrolled" into a flat sector, geodesics become straight lines. Rolling it back up, these appear as curves wrapping around the cone.

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