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Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...
The curvature is the reciprocal of radius of curvature. That is, the curvature is =, where R is the radius of curvature [5] (the whole circle has this curvature, it can be read as turn 2π over the length 2π R). This definition is difficult to manipulate and to express in formulas.
where R is the radius of curvature of the optical surface. The sag S ( r ) is the displacement along the optic axis of the surface from the vertex, at distance r {\displaystyle r} from the axis. A good explanation of both this approximate formula and the exact formula can be found here .
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The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. A geometric construction was described by Isaac Newton in his Principia:
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). [1] [page needed] On the other hand, since a sphere of radius R has constant positive curvature R −2 and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar ...
Minimum railway curve radius, the shortest allowable design radius for the centerline of railway tracks Topics referred to by the same term This disambiguation page lists articles associated with the title Curve radius .