refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat,
but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature
, which is defined for objects embedded in another space (usually a Euclidean space
) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature
, which is defined at each point in a differential manifold. This article deals primarily with the first concept.
The primordial example of extrinsic curvature is that of a circle, which has curvature equal to the inverse of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.
In a plane, this is a scalar quantity, but in three or more dimensions it is described by a curvature vector that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor.
The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.
See the links below for further reading.
One dimension in two dimensions: Curvature of plane curves
For a plane curve C, the mathematical definition of curvature uses a parametric representation of C with respect to the arc length parametrization. It can be computed given any regular parametrization by a more complicated formula given below. Let γ(s) be a regular parametric curve, where s is the arc length, or natural parameter. This determines the unit tangent vector T, the unit normal vector N, the curvature κ(s), the oriented or signed curvature k(s), and the radius of curvature at each point:
The curvature of a straight line is identically zero. The curvature of a circle of radius R is constant, i.e. it does not depend on the point and is equal to the reciprocal of the radius:
Thus for a circle, the radius of curvature is simply its radius. Straight lines and circles are the only plane curves whose curvature is constant. Given any curve C and a point P on it where the curvature is non-zero, there is a unique circle which most closely approximates the curve near P, the osculating circle at P. The radius of the osculating circle is the radius of curvature of C at this point.
The meaning of curvature
Suppose that a particle moves on the plane with unit speed. Then the trajectory of the particle will trace out a curve C in the plane. Moreover, taking the time as the parameter, this provides a natural parametrization for C. The instanteneous direction of motion is given by the unit tangent vector T and the curvature measures how fast this vector rotates. If a curve keeps close to the same direction, the unit tangent vector changes very little and the curvature is small; where the curve undergoes a tight turn, the curvature is large.
The magnitude of curvature at points on physical curves can be measured in diopters (also spelled dioptre) — this is the convention in optics. A diopter has the dimension
The signed of the signed curvature k
indicates the direction in which the unit tangent vector rotates as a function of the parameter along the curve. If the unit tangent rotates counterclockwise, then k
> 0. If it rotates clockwise, then k
The signed curvature depends on the particular parametrization chosen for a curve. For example the unit circle can be parametrised by (counterclockwise, with k > 0), or by (clockwise, with k < 0). More precisely, the signed curvature depends only on the choice of orientation of an immersed curve. Every immersed curve in the plane admits two possible orientations.
For a plane curve given parametrically as
the curvature is
and the signed curvature k is
For the less general case of a plane curve given explicitly as the curvature is