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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 convex hull of every bounded rectifiable closed curve C has perimeter at most the length of C, with equality only when C is already a convex curve. Cauchy's surface area formula: Given any convex compact subset , let [| |] be the expected shadow area of (that is, is the orthogonal projection to a random hyperplane of ), then by integrating ...
Let R be the radius of the arc which forms part of the perimeter of the segment, θ the central angle subtending the arc in radians, c the chord length, s the arc length, h the sagitta of the segment, d the apothem of the segment, and a the area of the segment. Usually, chord length and height are given or measured, and sometimes the arc length ...
The circle is the plane curve enclosing the maximum area for a given arc length. ... the formula for the arc length s of a circular ... largest area for a given ...
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.
The equator of the unit sphere is a parametrized curve given by ((), ()) = (,) with t ranging from 0 to 2 π. The line element may be used to calculate the length of this curve. The line element may be used to calculate the length of this curve.
The radius of such a curve is 5729.57795. If the chord definition is used, each 100-unit chord length will sweep 1 degree with a radius of 5729.651 units, and the chord of the whole curve will be slightly shorter than 600 units.
has a length equal to one and is thus a unit tangent vector. If the curve is twice differentiable, that is, if the second derivatives of x and y exist, then the derivative of T(s) exists. This vector is normal to the curve, its length is the curvature κ(s), and it is oriented toward the center of curvature. That is,