# 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

 $P_{n}(x):=c_{0}+c_{1}(x-x_{0})+\ldots+c_{n}(x-x_{0})^{n},$

the osculating curve of  $y=f(x)$  in  $(x_{0},\,f(x_{0}))$  is the Taylor polynomial  of degree $n$ of the function  $f$.

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

 $\displaystyle\frac{x^{2}}{a^{2}}-\frac{(y-y_{0})^{2}}{b^{2}}=-1.$ (1)

Removing the denominators and differentiating six times successively yield the equations

 $\displaystyle\begin{cases}b^{2}x^{2}-a^{2}(y-y_{0})^{2}+a^{2}b^{2}=0,\\ b^{2}x-a^{2}(y-y_{0})y^{\prime}=0,\\ b^{2}-a^{2}y^{\prime 2}-a^{2}(y-y_{0})y^{\prime\prime}=0,\\ 3y^{\prime}y^{\prime\prime}+(y-y_{0})y^{\prime\prime\prime}=0,\\ 3y^{\prime\prime 2}+4y^{\prime}y^{\prime\prime\prime}+(y-y_{0})y^{\prime\prime% \prime\prime}=0,\\ 10y^{\prime\prime}y^{\prime\prime\prime}+5y^{\prime}y^{\prime\prime\prime% \prime}+(y-y_{0})y^{(5)}=0,\\ 10y^{\prime\prime\prime 2}+15y^{\prime\prime}y^{\prime\prime\prime\prime}+6y^{% \prime}y^{(5)}+(y-y_{0})y^{(6)}=0.\end{cases}$ (2)

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

 $y^{\prime}=-\sin{x},\;y^{\prime\prime}=-\cos{x},\;y^{\prime\prime\prime}=\sin{% x},\;y^{\prime\prime\prime\prime}=\cos{x},\;y^{(5)}=-\sin{x},\;y^{(6)}=-\cos{x}$

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  $y_{0}=4,\;\,b^{2}=9,\;\,a^{2}=3$, whence the osculating hyperbola is

 $\frac{x^{2}}{3}-\frac{(y-4)^{2}}{9}=-1.$

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 exponential   curve (http://planetmath.org/ExponentialFunction)  $y=e^{x}$  in the point $(0,\,1)$  is

 $4x^{2}+y^{2}+4xy+14x-20y+19=0.$

The order of contact is only 3.

Title osculating curve OsculatingCurve 2013-03-22 17:57:17 2013-03-22 17:57:17 pahio (2872) pahio (2872) 7 pahio (2872) Definition msc 51N05 msc 53A04 OrderOfVanishing CircleOfCurvature Cosine QuadraticCurves