Osculating circle

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. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for 'kissing circle') by Leibniz.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. 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. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for 'kissing circle') by Leibniz. 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: 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. Let γ {displaystyle gamma } (s) be a regular parametric plane curve, where s is the arc length, or natural parameter. This determines the unit tangent vector T(s), the unit normal vector N(s), the signed curvature k(s) and the radius of curvature R(s) at each point for which s is composed: Suppose that P is a point on γ 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 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 point P. The plane curve can also be given in a different regular parametrization γ ( t ) = ( x 1 ( t ) x 2 ( t ) ) {displaystyle gamma (t),=,{egin{pmatrix}x_{1}(t)\x_{2}(t)end{pmatrix}},} where regular means that γ ′ ( t ) ≠ 0 {displaystyle gamma '(t) eq 0} for all t {displaystyle t} . Then the formulas for the signed curvature k(t), the normal unit vector N(t), the radius of curvature R(t), and the center Q(t) of the osculating circle are We can obtain the center of the osculating circle in Cartesian coordinates if we substitute t = x {displaystyle t=x} and y = f ( x ) {displaystyle y=f(x)} for some f function. If we do the calculations the results for the X and Y coordinates of the center of the osculating circle are: 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. This is entirely analogous to the construction of the tangent to a curve as a limit of the secant lines through pairs of distinct points on C approaching P.