Curvature

Curvature measures how sharply a curve or surface departs from being straight or flat. Its precise definition depends on the geometric setting and the object being measured.

Radius of Curvature: The reciprocal of curvature for a curve at a point where curvature is nonzero. It expresses local bending as the radius of the circle that best approximates a curve.
Differential Geometry: The study of smooth shapes using calculus and linear algebra. Curvature is a central quantity in this framework for describing shape.
Curve Shortening Flow: An evolution of plane curves in which each point moves in its normal direction at speed proportional to curvature. Curvature directly sets the speed at which each point of the curve moves.
Torsion: A measure of how a space curve twists out of its osculating plane. Curvature measures bending, while torsion measures three-dimensional twisting.
Curvature Vector: A vector describing a curve’s curvature magnitude and the direction in which it bends. It extends scalar curvature by recording the direction of bending.
Tangent Vector: A vector indicating the instantaneous direction of a curve at a point. Curvature tracks how the tangent direction changes along a curve.
Geodesic: A locally length-minimizing or straightest path in a curved space. Surface curvature influences how geodesics behave and converge.
Intrinsic Geometry: Geometry determined by distances measured within a space, independent of its embedding. Gaussian curvature is intrinsic, unlike bending measures that depend on an embedding.
Frenet–Serret Formulas: Differential equations describing how a space curve’s tangent, normal, and binormal vectors change along its length. They show how curvature governs changes in a space curve’s tangent.
Arc Length: The distance measured along a curve between points. Curve curvature is defined using change in direction per unit arc length.

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