This is more of a conceptual question rather than a pure calculation problem, but what is the difference between a projection and a cross section. For instance in the diagram below, in the regular tetrahedon with a plane passing through a vertex and the midpoints of two opposite sides, what would the respective projections and cross sections look like?
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There are two more orthogonal projections also to be considered in the given example even if we cut the regular tetrahedron parallel to xy planes, namely the planes represented by straight lines parallel to xy plane.
The cross sections in this simple rough sketch are quite different like the elliptical blue, yellow, red of different cross sections formed by different inclination of cutting planes leaving behind truncated segments but all having the same base projection of three sections at the bottom shown.
So much depends ons on the inclination of the cutting plane.
I don't think there is a good way to compare sections and projections. They are essentially different constructions.
You might enjoy playing with the sections and projections of a cube.
Consider a plane perpendicular to the long diagonal. If you project onto that plane you get a regular hexagon. If you imagine the plane moving parallel to itself to make a sequence of sections you will see triangles and hexagons. At the center you have a regular hexagon, but one that's smaller than the projection.
There are sections of the cube that are trapezoids and pentagons, but the only projections are hexagons and rectangles.
In general the projection of a figure onto a plane is the union of all the sections by slices parallel to that plane.