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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 geometry, the Cesàro equation of a plane curve is an equation relating the curvature (κ) at a point of the curve to the arc length (s) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature (R) to arc length. (These are equivalent because R = 1 / κ .)
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 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.
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 .
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 ...
Arc measurement of Eratosthenes. Arc measurement, [1] sometimes called degree measurement [2] (German: Gradmessung), [3] is the astrogeodetic technique of determining the radius of Earth and, by extension, its circumference.
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