Definition of Tangent between Two Curves

Question

Consider the two curves $y=e^x\cdot x^2$ and $y=e^{-x}\cdot x^2$ both have $y=0$ as their tangent line at $x=0$.

Another example is the curve pair $y=e^{-ex}$ and $y=\log_{e^{-e}} x$.

Image Link 1: $y=e^x\cdot x^2$ and $y=e^{-x}\cdot x^2$

Image Link 2: $y=e^{(x/e)}$ and $y=\log_{e^{1/e}}x$

Do we consider these two curves tangent to each other? If not, what is the best mathematical keyword to describe their behavior? If yes, how do we separate this behavior from touch of the two curves? Is there a better way to describe 'touch' mathematically?

I learn from Wikipedia that the tangent line does not necessarily touch the curve without crossing it. A point where the tangent (at this point) crosses the curve is called an inflection point (e.g. $y=x^3$ at $x=0$). I am not sure how to generalize this to tangency between two curves.

Answer 1

Let $f:S\subseteq\mathbb{R}\to\mathbb{R}^2$ be a curve differentiable at $t_0\in S$. The tangent line of $f$ at point $f(t_0)$ is defined as the best linear approximation of $f$ at $t_0$, namely $$L(t)=f(t_0)+\frac{df}{dt}(t_0)(t-t_0)$$ for all $t\in\mathbb{R}$.

If two curves $f,g$ have the same tangent line at point $(x,y)$, then we say that $f,g$ are tangent to each other at $(x,y)$.

For example, if $f(t)=(t, e^tt^2)$ and $g(t)=(t, e^{-t}t^2)$ as given by OP, then $\frac{df}{dt}(t)=(1,e^t(t^2+2t))$ and $\frac{dg}{dt}(t)=(1, e^{-t}(-t^2+2t))$. We have that $f(0)=(0,0)=g(0)$ and $\frac{df}{dt}(0)=(1,0)=\frac{dg}{dt}(0)$. Thus, the tangent line $L$ at $(0,0)$ of both curves is the same: $$L(t)=(0,0)+(1,0)t$$ for all $t\in\mathbb{R}$. We say that the $f,g$ are tangent to each other at point $(0,0)$.

As for internal and external tangencies, I am not able to find any reference to those notions on general curves. However, since we do have such notions for circles, we can intuitively extend it to general curves using osculating circle. If the two osculating circles of the curves at some point $(x,y)$ are internally tangent, then we say that the two curves are internally tangent to each other at $(x,y)$, and similarly for external tangent.

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Edit:

I found an answer here which inspires a perhaps more general definition for internal and external tangencies. Let $f,g$ be $\mathbb{R}^2$-curves having a common tangent line $L$ at point $(a,b)$. Let $C_f$, $C_g$, $C_L$ each be the respective range of $f$, $g$, $L$.

Suppose that there exists an open disk $B$ centered at $(a,b)$ such that the following is true:

1. $C_L$ divides $B$ into two disconnected regions $B_1$ and $B_2$.
2. $B_1\cap C_f=\emptyset$ and $B_1\cap C_g=\emptyset$.
3. $B_2\cap C_f\neq\emptyset$ and $B_2\cap C_g\neq\emptyset$.

Then we say that $f$ and $g$ are internally tangent to each other at $(a,b)$.

On the other hand, if there exists an open disk $B$ centered at $(a,b)$ such that the following is true instead:

1. $C_L$ divides $B$ into two disconnected regions $B_1$ and $B_2$.
2. $B_1\cap C_f=\emptyset$ and $B_1\cap C_g\neq\emptyset$.
3. $B_2\cap C_f\neq\emptyset$ and $B_2\cap C_g=\emptyset$.

Then we say that $f$ and $g$ are externally tangent to each other at $(a,b)$.

Examples and non-examples:

Take $g(t)=(t,t^2)$ and consider the tangent line $L$ at point $(0,0)$ with the following $f$:

1. $f(t)=(t,0)$. In this case, $B\setminus C_L$ does not intersect $C_f$ at all, but our definitions require that there must be some nonempty intersection. Thus, we say that $f$ and $g$ are neither internally nor externally tangent to each other (which makes intuitive sense).
2. $f(t)=(t,t^3)$. In this case, both $B_1$ and $B_2$ have some nonempty intersection with $C_f$, but our definitions require that either $B_1$ or $B_2$ does not have intersection with $C_f$. Thus, we say that $f$ and $g$ are neither internally nor externally tangent to each other (which again makes sense).
3. $f(t)=(t, t^2\sin(1/t))$. Just as the previous case, both $B_1$ and $B_2$ have nonempty intersection with $C_f$ (because no matter how small $B$ is, there are always some $(x, y)\in C_f$ and $(-x,-y)\in C_f$ inside $B$, where $y>0$). It also makes sense to say that $f$ and $g$ are neither internally nor externally tangent to each other in this case.
4. $f(t)=(t, |t|^{3/2})$. Note that $f$ does not take any value in the lower half-plane, so (say) $B_1\cap C_f=\emptyset$. Also it is clear that $B_2$ must contain some $(x,y)\in C_f$ where $x,y>0$. The same is true for $C_g$. Thus, we say that $f$ and $g$ are internally tangent to each other. This demonstrates that twice-differentiability is not required for internal and external tangencies (because $f$ is not twice-differentiable).