# order of contact

Suppose that $A$ and $B$ are smooth curves in $\mathbb{R}^{n}$ which pass through a common point $P$. We say that $A$ and $B$ have zeroth order contact if their tangents at $P$ are distinct.

Suppose that $A$ and $B$ are tangent at $P$. We may then set up a coordinate system in which $P$ is the origin and the $x_{1}$ axis is tangent to both curves. By the implicit function theorem, there will be a neighborhood of $P$ such that $A$ can be described parametrically as $x_{i}=f_{i}(x_{1})$ with $i=2,\ldots,n$ and $B$ can be described parametrically as $x_{i}=g_{i}(x_{1})$ with $i=2,\ldots,n$. We then define the order of contact of $A$ and $B$ at $P$ to be the largest integer $m$ such that all partial derivatives of $f_{i}$ and $g_{i}$ of order not greater than $m$ at $P$ are equal.

Title order of contact OrderOfContact 2013-03-22 16:59:49 2013-03-22 16:59:49 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Definition msc 53A04 order contact