osculating curve
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
the osculating curve of in 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 in the point . What is the order of contact?
We may seek the osculating hyperbola from the three-parametric family
(1) |
Removing the denominators and differentiating six times successively yield the equations
(2) |
Into these equations we can substitute the coordinates of the contact point and the values of the derivatives
of cosine in that point; the values are . The first, third and fifth of the equations (2) give the result , 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 exponential curve (http://planetmath.org/ExponentialFunction) in the point 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 |