How do we define touching lines?

Question

If two curves are touching at one point and intersect one another, how do we define it?

If two lines are touching at a point then $L\cap K=\{q\}$ for two lines L and K and q is the touching point.

If the two curves are touching at q and intersect at $p (\neq q)$ then how do we define it?

$L\cap K\setminus \{p\}=\{q\}$. Is that the way?

Answer 1

You're trying to capture the notion of "touching but not intersecting"? If the curves are parameterized by $\gamma_1,\gamma_2$ respectively, you could define touching but not intersecting as $\gamma_1(t) = \gamma_2(t)$ and $\gamma_1'(t) = \pm\gamma_2'(t)$.

Answer 2

If I understand your question correctly: One way you could do this is as follows. Suppose that $\gamma_1(t)$ is the first curve, and $\gamma_2(t)$ is the second curve, and they are both such that $\gamma_1(0) = \gamma_2(0) = q$ is their point of intersection.

Now, let $f(x,y)$ be a function which vanishes identically along $\gamma_2$ (i.e. $f(\gamma_2(t)) = 0$ for all $t$). Then the composition $f \circ \gamma_1(t)$ is a function from $\mathbb{R} \to \mathbb{R}$, and in particular, it vanishes at 0 (since $\gamma_1(0) = q$, which lies on $\gamma_2$).

So examine the order of vanishing of $f \circ \gamma$ at zero. That is, write out this function as a power series which will look like $$ f \circ \gamma_1 = \sum_{n=k}^\infty a_nt^n $$ for some $k \geq 1$. This integer $k$ is the order of vanishing. If it is greater than 1, then we have that the curves are touching in the sense that your picture shows.

The key is that this is a purely local phenomena; it only depends on both of the curves near the point that they intersect, and so we can understand it by looking at a local description of the curves. Assuming that everything is "sufficiently nice", this means that we can work with power series, which aren't too bad to compute.

Let's look at a quick example. Consider the point where the $x$-axis meets the parabola $y = x^2$. Then the function $f(x,y)$ that we are interested in is $f(x,y) = y - x^2$. Furthermore, a parameterization of the $x$-axis is given by $\gamma(t) = (t,0)$.

Thus, the composition is given by $f(\gamma(t)) = f(t,0) = t^2$. As the first non-zero coefficient is on the $t^2$ term, it follows that the order of vanishing is 2, which is greater than 1. Hence the curves only just "touch".