To start, this is cutter type Euclidean 4 dimensional or more, there are other ways that fit in the entire dimensional space into 3d like our normal tesseracts. That said, cutting is a complex subject and one that is at the heart of 4 dimensions or more having any known visualization. To start, there is a view point and a space within one three dimensional space within a bigger 4 or more dimensional space. All seen will be the parts of the 4d+ that have exactly the same coordinates as the view space, except 3 dimensions it was in. Now in that case we can say excluding 3 and only three dimensions, where all else is equal it is viewed with no other parts in view. We have effectively made our any dimensional object 3d or even 2d for our viewers.
Now comes a fun part. First, there may be a matrix where multiplying by it from a center, say the view point, many views may be as simple as the one above. To do the matrix, first subtract the center e.g. view point from all, then multiply all by the matrix of same dimension as the points. What if they don't match up dimensions no matter how you spin them though? For this, examine another case. The matrix and relativity may be applied but 2 or more directions of travel other than the x and y planes are. In sum, there is a space of 2+(number of dimensions moving in from view point to view plane) being cut to and it is like a 3d. Objects now may have bounding volumes with bounding volumes and eventually, planes. This like 3 dimensional thing is indeed always able to be moved in with only three directions if there is a 2d viewing screen, but it views more than 3 dimensions.
As long as any object is convex, the result should be convex and a single volume. It should be that all vertices of our result cut shape are along the outer edges or volumes or whatever they are. In very many, in fact possibly all cases of computer graphics, but excluding curves that are not to be broken down, this applies to some set or subset of our objects and thus we can search the bounding volumes which eventually have planar parts for where on them the cut shape will have its boundaries and vertices. Thus any shape which has a possibility to be represented in computer graphics may be cut using only planar finding of linear feasibility. The search may be limited to only where it exactly equals the coordinates that do not move, not x, y, z, and whatever others. This effectively makes our ND space lose a set number of dimensions for our viewing.
In a blocky world, this can be simple enough. This may also work for spherical or hyperboloid graphics simulators with some oddity such as 2 of one object with opposite viewing sides in spherical. For the blocky world there may even be just linear cutting in many cases. For example in axis aligned everything including view, just do like find excluding xyz where if anywhere from a minimum to a maximum in each dimension where it may be equal. This may involve cutting up shapes but it will yield simpler correctness. As long as we stay within one 3d with our view, it is 3d, and it is similar for 4d, 5d, 6d, 7d, 10d, 1000d, 1 million dimensions, and any more that can be made or imagined.
4d objects will operate in 4d. If there were 3d screens or eyes then it may be possible to point it correctly to see a 4d if we angle it correctly. We have 2d screens so can see like 3d. It would need to all sense at once. We can do number minus one projection but that is it except multiple views which is an interesting thought. We truly can see more than 3d with multiple view points. This may weird out virtual reality enthusaists but it is real. Every object can be effectively broken down into planes and then connection of those planes forms the whole. I have yet to do this because I do not know linear programming the math yet, but the preceding should be accurate. Whatever way you slice it, most views break down to (view point)+sum of three(scalar times vector) and that with two not three for a plane, setting them equal there is any point in view+vectorX*scalarX+vectorY*scalarY+vectorZ*scalarZ=any point in plane+vectorA*scalarA+vectorB*scalarB. This is an oversimplification because objects are finite, but add your constraints to this and you effectively have cut your shape. Graphics force triangles and pixels so this effectively applies everywhere with precision.
Step 1 is define the shape mathematically. Step 2 is set the small or large representations to the view space equation. Step 3 may be beyond me now but effectively it is find the solution set to your problem and connect all of it. Step 4 makes people enjoy the fun. Some advice for this includes try to break things down and run things threaded or on a GPU because 1 million is a lot of dimensions, cutting it to view is a lot more. Also, utilize the max possible space dimension rule for viewing. There is also 2d and like 3d cutting for cut to only in front of view and only on the finite screen. What would you make with this?