Geodesic Curvature

Differential Geometry · Axiom Academy

Understanding how curves bend within surfaces in differential geometry 1. Definition of Geodesic Curvature Consider a curve γ(t) on a surface S. The geodesic curvature κ g measures how much the curve deviates from being a geodesic (straightest possible path) on the surface. Unlike regular curvature which measures bending in 3D space, geodesic curvature is an intrinsic property - it only depends on measurements within the surface itself. 2. The Geodesic Curvature Formula The geodesic curvature can be computed using the following relationship: κ is the regular curvature of the curve in 3D space n is the principal normal to the curve (points toward center of curvature) N is the surface normal (perpendicular to the surface) T is the unit tangent vector to the curve The cross product n × N gives a vector in the tangent plane perpendicular to the curve, and dotting with T extracts the tangential component. 3. Geodesics Have Zero Geodesic Curvature A geodesic is a curve on a surface that is "as straight as possible" - it doesn't turn left or right within the surface. On a plane: straight lines (κ g = 0) On a sphere: great circles like the equator or meridians (κ g = 0) On a cylinder: helices at the right angle, horizontal circles, and vertical lines Even though a great circle on a sphere has curvature κ ≠ 0 in 3D space, its geodesic curvature is zero because it doesn't turn within the surface.

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