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In geometry, the center of curvature of a curve is a point located at a distance from the curve equal to the radius of curvature lying on the curve normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature C as the ...
The curvature of a differentiable curve was originally defined through osculating circles. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve. [3]
The corresponding center of curvature is the point Q at distance R along N, in the same direction if k is positive and in the opposite direction if k is negative. The circle with center at Q and with radius R is called the osculating circle to the curve γ at the point P.
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 ...
The locus of the centers of all the osculating circles (also called "centers of curvature") is the evolute of the curve. If the derivative of curvature κ'(t) is zero, then the osculating circle will have 3rd-order contact and the curve is said to have a vertex. The evolute will have a cusp at the center of the circle.
The evolute of a curve (blue parabola) is the locus of all its centers of curvature (red). The evolute of a curve (in this case, an ellipse) is the envelope of its normals. In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point ...
A center of curvature is defined at each position s located a distance ρ (the radius of curvature) from the curve on a line along the normal u n (s). The required distance ρ(s) at arc length s is defined in terms of the rate of rotation of the tangent to the curve, which in turn is determined by the path itself.
The center of the osculating sphere is offset from the center of the ellipsoid, but is at the center of curvature for the given point on the ellipsoid surface. This concept aids the interpretation of terrestrial and planetary radio occultation refraction measurements and in some navigation and surveillance applications.