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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 astrodynamics, an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time.Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a conic section (i.e. circular ...
The following can be used to find the versine of a given constant radius curve: [2] The Hallade method is to use the chord to continuously measure the versine in an overlapping pattern along the curve. The versine values for the perfect circular curve would have the same number. [3]
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 ...
A curve may have nonzero curvature and zero torsion. For example, the circle of radius R given by r(t) = (R cos t, R sin t, 0) in the z = 0 plane has zero torsion and curvature equal to 1/R. The converse, however, is false. That is, a regular curve with nonzero torsion must have nonzero curvature.
In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. Therefore an intrinsic equation defines the shape of the curve without specifying its position relative ...
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 .)
Specifically, if the mass-density at time t=0 is given by a Dirac delta, which essentially means that the mass is initially concentrated in a single point, then the mass-distribution at time t will be given by a Gaussian function, with the parameter a being linearly related to 1/ √ t and c being linearly related to √ t; this time-varying ...