mg.metric geometry – tetrahedral interpolation and integration along a segment


Let’s say we have a several tetrahedrons $T_i$ whose faces touch so that each face belong to two tetrahedrons. Each tetrahedron contain a value $V_{i}$.

Given a position $P$ inside the tetrahedron $T_0$, and neighboring tetrahedron are labeled $T_1, T_2, T_3, T_4$.

How to compute the value $V(P)$ such that its value is a linear interpolation between all $V_i$?

Following this, given a direction $vec{d}$ and the origin $O$ and a scalar $t$ such that $P(t)=O+d*t$, what is the equation giving the interpolated value along this segment $V(t)$, considering only the part where the segment is inside $T_0$?

I tried to use barycentric coordinates, and I think it confused me more than it helped.

What would be a simple explanation for solving such a problem?