PlanetMath: osculating curve

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osculating curve

(Definition)

Definition. From a family of plane curves, the osculating curve of the curve in a certain point is the curve of the family which has the highest order contact with the curve in that point.

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

$\displaystyle P_n(x) := c_0+c_1(x-x_0)+\ldots+c_n(x-x_0)^n,$

the osculating curve of

in $(x_0,\,f(x_0))$ is the Taylor polynomial of degree

of the function

Example 2. Determine the osculating hyperbola with axes parallel to the coordinate axes for the curve $y = \cos{x}$ in the point $(0,\,1)$ . What is the order of contact?

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

$\begin{align*}\begin{cases}b^2x^2-a^2(y-y_0)^2+ a^2b^2 = 0,\\ b^2x-a^2(y-y_0)y' ... ... 0,\\ 10y'''^2+15y''y''''+6y'y^{(5)}+(y-y_0)y^{(6)} = 0. \end{cases}\end{align*}$

(2)

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

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

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

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

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 in the point $(0,\,1)$ is

$\displaystyle 4x^2+y^2+4xy+14x-20y+19 = 0.$

The order of contact is only 3.

"osculating curve" is owned by pahio.

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See Also: order of vanishing, circle of curvature, cosine, quadratic curves

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Cross-references: parabola, cosine, derivatives, contact point, equations, denominators, coordinate, parallel, hyperbola, function, degree, Taylor polynomial, polynomial functions, graphs, order contact, point, curve, plane curves
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This is version 4 of osculating curve, born on 2008-03-28, modified 2008-03-29.
Object id is 10454, canonical name is OsculatingCurve.
Accessed 250 times total.

Classification:

AMS MSC:	51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry)
	53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

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