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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 ...
The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...
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 ...
The nested osculating circles of an Archimedean spiral.The spiral itself is not shown, but is visible where the circles are more dense. In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other. [1]
One method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to trisect an angle. Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs. [3] Mechanism of a scroll compressor
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.
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This fact leads to an easy proof of the Tait–Kneser theorem on nesting of osculating circles. [ 4 ] The normals of the given curve at points of nonzero curvature are tangents to the evolute, and the normals of the curve at points of zero curvature are asymptotes to the evolute.