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Concept
Rayon de courbure
Résumé
In differential geometry, the radius of curvature (Rc), 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 combinations thereof.
In the case of a space curve, the radius of curvature is the length of the curvature vector.
In the case of a plane curve, then R is the absolute value of
where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature.
If the curve is given in Cartesian coordinates as y(x), i.e., as the graph of a function, then the radius of curvature is (assuming the curve is differentiable up to order 2):
and denotes the absolute value of z.
Also in Classical mechanics branch of Physics Radius of curvature is given by (Net Velocity)2/Acceleration Perpendicular
If the curve is given parametrically by functions x(t) and y(t), then the radius of curvature is
Heuristically, this result can be interpreted as
If γ : R → Rn is a parametrized curve in Rn then the radius of curvature at each point of the curve, ρ : R → R, is given by
As a special case, if f(t) is a function from R to R, then the radius of curvature of its graph, γ(t) = (t, f(t)), is
Let γ be as above, and fix t. We want to find the radius ρ of a parametrized circle which matches γ in its zeroth, first, and second derivatives at t. Clearly the radius will not depend on the position γ(t), only on the velocity γ′(t) and acceleration γ′′(t). There are only three independent scalars that can be obtained from two vectors v and w, namely v · v, v · w, and w · w. Thus the radius of curvature must be a function of the three scalars ^2, ^2 and γ′(t) · γ′′(t).
The general equation for a parametrized circle in Rn is
where c ∈ Rn is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ Rn are perpendicular vectors of length ρ (that is, a · a = b · b = ρ^2 and a · b = 0), and h : R → R is an arbitrary function which is twice differentiable at t.
Source officielle
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