This can be done using a number of strategies.
The goal is 2, with a stretch goal of an additional one.
Probe on vertices of the mesh
Probe on face centers of the mesh
[Stretch goal] Probe on random locations on the surface.
The random probing could be done through randomly selecting a face, then randomly generate a point on the face based on interpolation between the corners.
This can be done using a number of strategies. The goal is 2, with a stretch goal of an additional one.
The random probing could be done through randomly selecting a face, then randomly generate a point on the face based on interpolation between the corners.