In higher dimensions, the 3d people cannot see it all well. If a point was unshielded to the higher dimensions, light could come in to the eye from any unshielded angle. We cannot simulate such a thing very well but such is life, here is one interpretation. There are other ways of defining this such as projecting all to a lower dimension repeatedly too, so keep in mind this is only one of many ways. For instance, the classic tesseract image is not this. Using this per view may be good.
4d, 5d, 6d, 7d, 25d, 50d, 100d, 100000d, 1 million dimensional and beyond, how to define it? Please note that this applies to euclidean, spherical, hyperbolic, and possibly some yet to be imagined spaces. To start, define the view plane, its shape, and define that from a view point, there may be more than one as in VR, if it passes the view plane on the way to an object exactly then it is viewed, otherwise it is not viewed and may be discarded. Tetrahedra are an industry standard here, they can be used in all of these spaces, higher dimensional or not. It is from these that I draw my inspiration for this slicing technique, but it is better than them for higher dimensional convex shapes. It should be stressed that this is only for convex shapes, but for example a hammer could be 2 shapes or 3 or 4 or more depending how you shape it. Once we have that, we know for any convex shape, the result cut shape will have its furthest points at the bounds, this is known. A 5d box has 4d bounding volumes and a 4d bounding box has 3d bounding volumes and all know that a 3d box has 2d faces. This applies up to any dimensional, the big shape has a 1 lesser dimension bounding volume until it is 2d. Tetrahedra have all of them in, or from points on their 1 dimensional line edges, a shape like a plane, a line, or a point.
From the theory set forth so far, we may conclude that we may use the triangle like shapes as would be with tetrahedra on convex shapes much like tetrahedra would be and then connect the resultant points. How is that better? See an example of an ND box. Define it as extruded all possible ways within a minimum and a maximum. It has (N!/((N-2)!2!))2^(N-2) 2d faces and N! tetrahedra needed about, where ! is factorial. The results are 2N points maximum and N(2N-1) connections could result from those 2N points. At 20 or even 50 dimensional, you can square and multiply by 1000 the number of 2d faces and get close on the 20 dimensional and not even near on the 50 dimensional box. For connections of the triangle like things, we take the result points, connect them, and if the midpoint is in view, the whole line is in view. After this we can triangulate the resultant planar shapes. The numbers are reasonable for this 2d plane slicer to do 50 dimensional, the numbers for about 20 dimensional are where it is not reasonable to use tetrahedra for slicing. At 4 dimensions this approach may have negligible benefits, but it can also be done such that no inner parts are shown, less like tetrahedra, although the excess lines can be eliminated regardless.
This can be used for unbounded dimensional geometry, it is not infinite dimensional because our computers are finite, but it is significantly more efficient. At these dimensions, space efficiency is critical, you cannot store many trillions of shapes in small amounts of space and storage. With this is left to choice the best way to slice a triangle or a tetrahedron. I suggest using the fact that at one rotation or another, things get simpler and we can say in the rotation, a corresponding coordinate part may be used to find one value and check all the others for being in view with that value. Similar applies for spherical and hyperbolic geometries of spaces. Matrices and linear algebra are quite good here. This may also apply to curves so far as they can be tessellated and used like tetrahedra would be. Note that only as long as the shape would be a valid tetrahedralization is the way of slicing here certain to work. The methods here may be valid for many other geometries of space and for high dimensions, tetrahedra may well be nearly obsolete, but useful.