Smooth Path joining any two points in $U=\{(x,y)\in \mathbb{R^2}|1<x^2+y^2<4\}$.

Question

Please help me to solve the problem:

Let $U=\{(x,y)\in \mathbb{R^2}|1<x^2+y^2<4\}$. Let $p,q \in U$. Show that there is a continuous map $\gamma:[0,1] \to U$ such that $\gamma(0)=p$ and $\gamma(1)=q$ and such that $\gamma$ is differentiable on $(0,1)$.

Now, $U$ is pathconnected because any $p,q \in U$ can be join by means of polygonal paths inside $U$. But such a path is not differentiable on $(0,1)$.

Again, the straight line joining any $p,q \in U$ may not lie inside $U$.

So How to construct a desired map? Please help.Thank you.

Answer 1

The question only asks us to show that such a map exists. So fix apoint $a \in U$ and consider the set $E$ of all points $b \in U$ such that there is a differentiable path in $U$ from $a$ to $b$. It is enough to show that this set is open and closed. Suppose $x \in E$ and choose an open disk $D$ around $x$ contained in $U$. We claim that this disk is contained in $E$, proving that $E$ is open; a similar argument shows that $U\setminus E$ is also open in $U$. Now working within the disk we only have to shaow that any two points can be connectes inside the disk by a differentiable path, say $\gamma :[0,1] \to D$ with a specified value for the right hand derivative at 0. This is easy: there exists a path of the type $a(t-a)+(t-b)(\alpha t+\beta)$ with these properties.

Answer 2

Hint 1: $f(x)=(1-x)p + x\cdot q$ and $0\le x\le 1$

Hint 2: What function makes arcs for a fixed distance from the origin?

Hint 3: It is possible to change the angle from $p$ to $q$'s angle then apply Hint 1 to get the right distance.

Hint 4: Make Hint 3 smooth where Hint 1 and Hint 2 "connect".

Answer 3

Consider $p=r_1e^{i\varphi_1}$ and $q=r_2e^{i\varphi_2}$ as complex numbers. Then the “spiral” path $$\gamma\colon[0,1]\to \mathbb C,\quad t\mapsto\bigl(r_1+t(r_2-r_1)\bigr)e^{i(\varphi_1+t(\varphi_2-\varphi_1))}$$ satisfies all needs.