I assume the grid you are using is always going to be flat. You can try this. Gather all the edge info for the grid and form an array of line segment per edge. You should then use the inner line curve to form an extruded polygon strip (extended in both direction perpendicular to the grid). Now perform line segment/polygon face intersection test to using each line segment against all the polygon faces. You should then be able to find the red points by using barycentric coordinates then. Hope this help. :)
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