Curvature Center
The point on the positive ray of the normal vector at a distance rho(s), where rho is the radius of curvature. It is given by
z = x+rhoN
(1)
= [画像:x+rho^2(dT)/(ds),]
(2)
where N is the normal vector and T is the tangent vector. It can be written in terms of x explicitly as
| [画像: z=x+(x^('')(x^'·x^')^2-x^'(x^'·x^')(x^'·x^('')))/((x^'·x^')(x^('')·x^(''))-(x^'·x^(''))^2). ] |
(3)
|
For a curve represented parametrically by (f(t),g(t)),
alpha = [画像:f-((f^('2)+g^('2))g^')/(f^'g^('')-f^('')g^')]
(4)
beta = [画像:g+((f^('2)+g^('2))f^')/(f^'g^('')-f^('')g^')]
(5)
(Lawrence 1972, p. 25).
See also
Curvature, Osculating CircleExplore with Wolfram|Alpha
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References
Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, 1972.Referenced on Wolfram|Alpha
Curvature CenterCite this as:
Weisstein, Eric W. "Curvature Center." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CurvatureCenter.html