|
In mathematics, curvature is 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'', or ''straight'' in the case of a line, 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 Riemannian manifold. This article deals primarily with the first concept. The canonical example of extrinsic curvature is that of a circle, which everywhere has curvature equal to the reciprocal of its radius. 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. More commonly this is a scalar quantity, but one may also define 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. == Curvature of plane curves == Cauchy defined the centre of curvature ''C'' as the intersection point of two infinitely close normals to the curve, the radius of curvature as the distance from the point to ''C'', and the curvature itself as the inverse of the radius of curvature.〔 *〕 Let ''C'' be a plane curve (the precise technical assumptions are given below). The curvature of ''C'' at a point is a measure of how sensitive its tangent line is to moving the point to other nearby points. There are a number of equivalent ways that this idea can be made precise. One way is geometrical. It is natural to define the curvature of a straight line to be identically zero. The curvature of a circle of radius ''R'' should be large if ''R'' is small and small if ''R'' is large. Thus the curvature of a circle is defined to be the reciprocal of the radius:〔Morris Kline, ''Calculus: an intuitive and physical approach'', 2nd edition, p. 458〕 : Given any curve ''C'' and a point ''P'' on it, there is a unique circle or line which most closely approximates the curve near ''P'', the osculating circle at ''P''. The curvature of ''C'' at ''P'' is then defined to be the curvature of that circle or line. The radius of curvature is defined as the reciprocal of the curvature. Another way to understand the curvature is physical. Suppose that a particle moves along the curve with unit speed. Taking the time ''s'' as the parameter for ''C'', this provides a natural parametrization for the curve. The unit tangent vector T (which is also the velocity vector, since the particle is moving with unit speed) also depends on time. The curvature is then the magnitude of the rate of change of T. Symbolically, : This is the magnitude of the acceleration of the particle and the vector is the acceleration vector. Geometrically, the curvature measures how fast the unit tangent vector to the curve rotates.〔Andrew Pressley, ''Elementary differential geometry'', 1st edition, p. 29〕 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. These two approaches to the curvature are related geometrically by the following observation. In the first definition, the curvature of a circle is equal to the ratio of the angle of an arc to its length. Likewise, the curvature of a plane curve at any point is the limiting ratio of ''dθ'', an infinitesimal angle (in radians) between tangents to that curve at the ends of an infinitesimal segment of the curve, to the length of that segment ''ds'', i.e., ''dθ/ds''.〔A. V. Pogorelov, ''Differential geometry'', 1st edition, p. 49〕 If the tangents at the ends of the segment are represented by unit vectors, it is easy to show that in this limit, the magnitude of the difference vector is equal to ''dθ'', which leads to the given expression in the second definition of curvature. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Curvature」の詳細全文を読む スポンサード リンク
|