How to calculate distance between point and object in 3d space I have object in 3d space created from points $P_i(x, y, z)$ from which I can create triangles, and I need to calulate distance from point X to this object.
I try to take 3 points from smallest distance and calulate height of tetrahedron created from this 3 points and X, but this will be not the distance from the object.
So my question is how to calculate this distance.
 A: I don't know whether this is a correct solution. But I will make a try.
Let $P_1$, $P_2$, $P_3$ be three closest points to $X$. Consider plane $\pi$ containing $P_1$, $P_2$, $P_3$.
Let $P$ be the orthogonal projection of $X$ on $\pi$. If $P$ is inside triangle $P_1P_2P_3$ then the distance is $PX$, otherwise $\min(\mathrm{dist}(P_1,X),\mathrm{dist}(P_2,X),\mathrm{dist}(P_3,X))$. 
Here is a picture, for clarification

Well, how to determine that $P$ is inside $P_1P_2P_3$? Just check the equality 
$$
\mathrm{area}(P_1P_2P_3)=
\mathrm{area}(PP_2P_3)+\mathrm{area}(P_1PP_3)+\mathrm{area}(P_1P_2P)
$$
In order to determine projectoin $P$ you can use the following formula
$$
P=X+tN
$$
where
$$
N=[P_3P_1,P_2P_1],\qquad
t=\frac{\langle P_1-X,N\rangle}{\langle N,N\rangle}
$$
A: Use the GKJ algorithm.
A good online tutorial including pictures and code snippets is here:
http://entropyinteractive.com/2011/04/gjk-algorithm/
A good video explanation of the concept is here:
http://mollyrocket.com/849
The basic principle is that the spatial relationship between 2 convex objects can be studied by considering the properties of their minkowski difference.
