What is the point of the idea of "contact" in differential geometry? I've been having introductory lectures on differential geometry and we came to the idea of "contact". There are two definitions:

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*Let $\alpha: I \to \Bbb{R}^3$ and $\beta: \overline{I} \to \Bbb{R}^3$ be regular curves such that $\alpha(t_0)=\beta (t_0)$ with $t_0 \in I \cap \overline{I}$. We say $\alpha,\beta$ have contact of order $n$ in $t_0$ if all derivatives of order $\leq n$ of $\alpha,\beta$ coincide in $t_0$ and the derivatives of order $n+1$ in $t_0$ are distinct.


*Let $\alpha:I\to \Bbb{R}^3$ a regular curve and $\pi$ a plane in $\Bbb{R}^3$ with a point $p=\alpha(t_0)$ for some $t_0\in I$. We say $\alpha$ and $\pi$ has contact of order $n$ in $p$ if there exists a regular curve $\beta: \overline{I} \to \Bbb{R}^3$ such that $\beta(\overline{I})\subset \pi$ and $\alpha,\beta$ haver contact of order $n$ in $t_0$.
And two theorems asserting that:

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*The only straight line with contact $1$ in a point of a curve is the tangent line.


*The only plane with contact $2$ in a point in a curve is the osculating plane.
But at least in the book I am reading, it seems it stops there. We just define "contact" and check that there are special lines and planes with a certain degrees of "contact". So I am curious: What is the point of the idea of "contact"?
 A: Order of contact is really a classical notion in projective geometry. You can think of it as a geometric view of Taylor polynomials of functions. The tangent line at $p$ has order of contact $1$ with the curve at $p$; the osculating circle at $p$ has order of contact $2$, the osculating sphere at $p$ to a space curve has order of contact $3$, etc. A line has order of contact $2$ with a curve at $p$ if it is the tangent line at an inflection point.
It becomes more complex if you talk about order of contact of a line in space with a surface. Not surprisingly, a line $\ell$ will have order of contact $1$ (or $2$-point contact) with a surface $S$ at $p$ if $\ell$ is tangent to $S$ at $p$. It will have order of contact $2$ (or $3$-point contact) with $S$ at $p$ if $\ell$ is an asymptotic direction of $S$ at $p$. And what it means to have order of contact $3$ (or $4$-point contact) is really quite nice: This means that $\ell$ is the tangent line at $p$ to an asymptotic curve passing through $p$ with an inflection point at $p$.
