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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 ...
Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure 𝜃 is =, and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is A = 1 2 θ r 2 . {\displaystyle A={\frac {1}{2}}\theta r^{2}.}
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
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 ...
as one would expect. This is equivalent to the above definition of the 2D mean diameter. However, for historical reasons, the hydraulic radius is defined as the cross-sectional area of a pipe A, divided by its wetted perimeter P, which leads to =, and the hydraulic radius is half of the 2D mean radius. [3]
The length of the curve is given by the formula = | ′ | where | ′ | is the Euclidean norm of the tangent vector ′ to the curve. To justify this formula, define the arc length as limit of the sum of linear segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity.
The longest diameter is called the major axis. Conjugate diameters are a pair of diameters where one is parallel to a tangent to the ellipse at the endpoint of the other diameter. The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the Euclidean metric.
A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric. The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected. The eccentricity ϵ(v) of a vertex v is the greatest distance between v and any other vertex; in ...