I'm interested in a problem and have no idea how to approach solving it. Could you please point me in the right direction.
Given 3 smooth paths $\varphi_{1}:[0,1]\to \mathbb{R}^{2}$, $\varphi_{2}:[0,1]\to \mathbb{R}^{2}$ and $\gamma:[0,1]\to \mathbb{R}^{2}$ such that:
1) $\varphi_{1}(1)=\varphi_{2}(1)=\gamma(1)=(x_{0},y_{0})$
2) $\varphi_{1}(0) = (0,y_{1})$, $\varphi_{2}(0) = (0,y_{2})$ and $\gamma(0) = (0,y_{3})$, where $y_{1} < y_{2} < y_{3}$.
3) All three intersect the $x = 0$ line perpendicularly.
4) (Edit) The curvature of each decreases monotonically from $0$ to $1$.
What conditions on the curvature of these three paths ensure that they do not intersect other than at $(x_{0},y_{0})$?
The more specific problem I am interested in is when $(x_{0},y_{0})$ is a point at infinity and all three paths tend to being parallel.
What subject/books concern questions like this? Which theorems would be useful for proving such properties.
Badly drawn picture to illustrate:
P.S. Parameterization of the curves is not important.
Thanks!