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Um círculo osculador

Em Geometria diferencial, o círculo osculador de uma curva planar suave num dado ponto da curva é o círculo cujo centro se encontra na linha normal interior à curva, e cuja curvatura é igual à da curva nesse ponto. Este círculo, que de todos os círculos tangentes é o que melhor aproxima a curva no ponto dado, foi denomidado de circulum osculans (Latim para "círculo osculante") por Leibniz.

O centro e o raio do círculo osculador num dado ponto são chamados de centro de curvatura e raio de curvatura da curva nesse ponto.

The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. A geometric construction was described by Isaac Newton in his Principia:

There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre.

– Isaac Newton, Principia; PROPOSITION V. PROBLEM I.

Description in lay terms

Imagine a car moving along a curved road on a vast flat plane. Suddenly, at one point along the road, the steering wheel locks in its present position. Thereafter, the car moves in a circle that "kisses" the road at the point of locking. The curvature of the circle is equal to that of the road at that point. That circle is the osculating circle of the road curve at that point.

Definicao Matemática

Seja γ(s) um curva paramétrica regular|, em que s é o parametro. Os versores normais e tangentes, T e N respectivamente, assim como a curvatura k(s) e o raio de curvatura R(s) são dados por:

$T(s)=\gamma '(s),\quad T'(s)=k(s)N(s),\quad R(s)={\frac {1}{\left|k(s)\right|}}.$

Suppose that P is a point on C where k ? 0. 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 C 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 point P.