I am looking for a method to calculate the nearest distance from a point (x-y-z) to a volume in 3-dimensional space. Assumption: The point is not inside the volume. Assumption: The volume is a closed coherent shape.
Is there a conventional name for this problem?
Bonus points if you can link to or include a function in a common programming language for doing this.
If I interpreted the problem correctly, let your surface be $z=f(x,y)$ and your point $P(x_0,y_0,z_0)$, which is fixed, using the distance formula $$D(x,y)=\sqrt{(x-x_0)^2+(y-y_0)^2+(f(x,y)-z_0)^2}$$ you want to solve the system $$\begin{cases}\partial_xD(x,y)=0\\ \partial_yD(x,y)=0 \end{cases}$$ and check using the $Hessian $ that the points found provide a minimum. Add the additional constraint that $f(x^*,y^*)>0$, where $x^*$ and $y^*$ are the possible mins.
You can also use Lagrange Multipliers, which exploits the fact that the shortest distance from a point to a surface is perpendicular to the surface.