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plotting - Mathematica function intersection points with 3D grid



I need to produce a 3-dimensional equispaced grid over a given function in a way, that I can calculate intersection points of the function with the grids edges. So my first question is how to produce this grid in Mathematica. I don't really get if this is possible by modeling the grid by Polygons? For this I might need some help then, too. Or can it be done by using more simple graphics objects like done here?




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