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The circle with center at Q and with radius R is called the osculating circle to the curve γ at the point P. If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N. It lies in the osculating plane, the plane spanned by the tangent and principal normal vectors T and N at the ...
Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip, also known as an osculating circle.. To travel along a circular path, an object needs to be subject to a centripetal acceleration (for example: the Moon circles around the Earth because of gravity; a car turns its front wheels inward to generate a centripetal force).
The osculating circle shares both its first and second derivatives (equivalently, its slope and curvature) with C. [1] [2] [4] The osculating parabola to C at p, the osculating curve from the family of parabolas, has third order contact with C. [2] [4] The osculating conic to C at p, the osculating curve from the family of conic sections, has ...
Osculating circles of the Archimedean spiral, tangent to the spiral and having the same curvature at the tangent point. The spiral itself is not drawn, but can be seen as the points where the circles are especially close to each other.
The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve. [1] The locus of centers of curvature for each point on the curve comprise the evolute of the curve.
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
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where c ∈ ℝ n is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝ n 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.