Given $a,b \in R^2$, how to find a piecewise $C^{1} \beta$ joining $a,b$ such that $\int \beta^{*}(ydx)=0$

Question

Given $a,b \in R^2$, how to find a piecewise $C^{1} \beta$ joining $a,b$ such that $\int \beta^{*}(ydx)=0$. The most straightforward way to do is the straight line, but it does not work.

Answer 1

It's easy if $a$ and $b$ are vertically aligned; then the line segment works. If not, you need the graph of a function, say, with $f(a_1)=a_2$ and $f(b_1)=b_2$ so that $\int_{a_1}^{b_1} f(x)\,dx = 0$. Obviously, $f$ will have to change sign in the case the $b_i$ have the same sign. There are infinitely many solutions. You can give a concrete formula for one of them by constructing a simple polygonal path with its edges parallel to the coordinate axes. Then make it $C^1$ by rounding the corners with quarter-circle arcs.

Answer 2

Project $a$ and $b$ vertically on the horizontal axis $y=0$ on to point $a',b'$. On $[a,a'],[a',b'],[b',b]$ the form $ydx$ is $0$. Parametrize smoothly this union of 3 segments.