osculating curve

Definition.  From a family of plane curves, the osculating curve of the curve  y=f(x)  in a certain point is the curve of the family which has the highest order contact with the curve  y=f(x)  in that point.

Example 1.  From the family of the graphs of the polynomial functions


the osculating curve of  y=f(x)  in  (x0,f(x0))  is the Taylor polynomialMathworldPlanetmath of degree n of the functionMathworldPlanetmath f.

Example 2.  Determine the osculating hyperbolaPlanetmathPlanetmath with axes parallelMathworldPlanetmathPlanetmath to the coordinate axes for the curve  y=cosx  in the point  (0, 1).  What is the order of contact?

We may seek the osculating hyperbola from the three-parametric family

x2a2-(y-y0)2b2=-1. (1)

Removing the denominators and differentiating six times successively yield the equations

{b2x2-a2(y-y0)2+a2b2=0,b2x-a2(y-y0)y=0,b2-a2y2-a2(y-y0)y′′=0,3yy′′+(y-y0)y′′′=0,3y′′2+4yy′′′+(y-y0)y′′′′=0,10y′′y′′′+5yy′′′′+(y-y0)y(5)=0,10y′′′2+15y′′y′′′′+6yy(5)+(y-y0)y(6)=0. (2)

Into these equations we can substitute the coordinates  x=0,y=1  of the contact point and the values of the derivatives


of cosine in that point; the values are  0,-1, 0, 1, 0,-1.  The first, third and fifth of the equations (2) give the result  y0=4,b2=9,a2=3, whence the osculating hyperbola is


When we substitute the pertinent values of the cosine derivatives into the two last equations (2), we see that only the former of them is satisfied.  It means that the order of contact between the cosine curve and the hyperbola is 5.

Example 3.  The osculating parabola of the exponentialMathworldPlanetmathPlanetmath curve (http://planetmath.org/ExponentialFunction)  y=ex  in the point (0, 1)  is


The order of contact is only 3.

Title osculating curve
Canonical name OsculatingCurve
Date of creation 2013-03-22 17:57:17
Last modified on 2013-03-22 17:57:17
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition
Classification msc 51N05
Classification msc 53A04
Related topic OrderOfVanishing
Related topic CircleOfCurvature
Related topic Cosine
Related topic QuadraticCurves