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

  • 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:

  • 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"?

Red Banana
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1 Answers1

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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 $p$.

Ted Shifrin
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