Curvature

In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined extrinsically relative to the ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined intrinsically without reference to a larger space. For curves, curvature describes how sharply the curve bends. The canonical examples are circles: smaller circles bend more sharply and hence have higher curvature. For a point on a general curve, the direction of the curve is described by its tangent line. How sharply the curve is bending at that point can be measured by how much that tangent line changes direction per unit distance along the curve. Curvature measures the angular rate of change of the direction of the tangent line, or the unit tangent vector, of the curve per unit distance along the curve. Curvature is expressed in units of radians per unit distance. For a circle, that rate of change is the same at all points on the circle and is equal to the reciprocal of the circle’s radius. Straight lines don’t change direction and have zero curvature. The curvature at a point on a twice differentiable curve is the magnitude of its curvature vector at that point and is also the curvature of its osculating circle, which is the circle that best approximates the curve near that point. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.