Information AboutCurvature |
| CATEGORIES ABOUT CURVATURE | |
| differential geometry | |
| riemannian geometry | |
| multivariable calculus | |
| curvature mathematics | |
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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. Further, 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 direction of the bend as well as its sharpness. The curvature of more complex objects (such as Surface s or even curved ''n''-dimensional Space s) are 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 For a Plane Curve ''C'', the curvature at a given point ''P'' has a Magnitude equal to the '' Reciprocal '' of the Radius of an Osculating Circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center. The smaller the radius ''r'' of the osculating circle, the larger the magnitude of the curvature (1/''r'') will be; so that where a curve is "nearly straight", the curvature will be close to zero, and where the curve undergoes a tight turn, the curvature will be large in magnitude. The magnitude of curvature at points on physical curves can be measured in Diopter s (also spelled dioptre); a diopter has the dimension ''one-per-meter''. A straight line has curvature 0 everywhere; a circle of radius ''r'' has curvature 1/''r'' everywhere. Local expressions For a plane curve given parametrically as the curvature is |
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