Minimum distance between two curves What is the condition that a straight line connecting two non-intersecting curves should satisfy in order to be of minimum length?
For example how to find minimum distance between
$$y=1+ (x-3)^2,\; x^2+y^2= 1\;? $$
Does this turn out that there should be parallel tangents at minimum distance endpoints of each curve?

Appreciate all help.
 A: Let $\gamma , \varphi: D \to \mathbb{R}^2$ be 2 curves in the plane. Then you can define the following function: $g: D \times D \to \mathbb R$ such that $g(t,u)= |\gamma(t) - \varphi(u)|$.
You then need to find the local minimum of this function, lets call it $(t',u') = \min \{g(t,u), (t,u) \in D \times D\}$.
Then you know that the straight line from $\gamma(t')$ to $\varphi(u')$ whould be the smallest straight line that starts in $\gamma$ and ends in $\varphi$.
This processes helps us to find that straight line. Now we go the opposite way to find out if a line is the smallest stright line:
Let $r$ be a line that starts in $(a,b) \in \gamma$ and ends in $(c,d) \in \varphi$. Then let $t_0,u_0 \in D$ such that $\gamma(t_0)=(a,b)$ and $\varphi (u_0) = (c,d)$.
This line is the smallest line if $(t_0,u_0)$ is a local minimum of $g$. So we have to have that:
$$\to \nabla g(t_0,u_0) = 0$$
$$\to \det \left(
\begin{matrix}
\partial_{xx}g & \partial_{yx}g\\
\partial_{xy}g & \partial_{yy}g
\end{matrix}
\right) > 0  \ \ \ \ \ \wedge \ \ \ \partial_{xx}g > 0, \ \ \text{for } (t,u)= (t_0,u_0)$$
If a straight line satisfies there two criteria than it is the smallest straight line between those two curves
