The radius of curvature at ? ( " s " ) is, in magnitude, the radius of the circle which forms the best approximation of the curve to second order at the point : that is, it is the radius of the circle making second order contact with the curve, the osculating circle.
22.
For a curve " C " given by a sufficiently smooth parametric equations ( twice continuously differentiable ), the osculating circle may be obtained by a limiting procedure : it is the limit of the circles passing through three distinct points on " C " as these points approach " P ".
23.
In differential geometry of curves, the "'osculating circle "'of a sufficiently smooth plane curve at a given point " p " on the curve has been traditionally defined as the circle passing through " p " and a pair of additional points on the curve infinitesimally close to " p ".
24.
This frame has a continuously changing origin, which at time " t " is the center of curvature ( the center of the osculating circle in Figure 1 ) of the path at time " t ", and whose rate of rotation is the angular rate of motion of the particle about that origin at time " t ".
25.
One speaks also of curves and geometric objects having " k "-th order contact at a point : this is also called " osculation " ( i . e . kissing ), generalising the property of being tangent . ( Here the derivatives are considered with respect to arc length . ) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact ( same tangent angle and curvature ), etc.
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