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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 ...
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.
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:
The sagitta also has uses in physics where it is used, along with chord length, to calculate the radius of curvature of an accelerated particle. This is used especially in bubble chamber experiments where it is used to determine the momenta of decay particles. Likewise historically the sagitta is also utilised as a parameter in the calculation ...
The reciprocal of the curvature is called the radius of curvature. A circle with radius r has a constant curvature of κ ( t ) = 1 r {\displaystyle \kappa (t)={\frac {1}{r}}} whereas a line has a curvature of 0.
Curvature radius of lens/mirror r, R: m [L] Focal length f: m [L] Quantity (common name/s) (Common) symbol/s ... The Cambridge Handbook of Physics Formulas. Cambridge ...
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 both of this approximate formula and the exact formula can be found here .
Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve. [1] The locus of centers of curvature for each point on the curve comprise the evolute of the curve. This term is generally used in physics regarding the study of lenses and mirrors (see radius of curvature (optics)).