On https://stackoverflow.com/a/8862483/4918319, an algorithm is put forward with which to determine whether a ray in 3-space goes through a given rectangle, if we specify the ray by two vectors (one specifying the origin of the ray and the other its direction) and the rectangle by three vectors (one specifying the location of one corner and the other two giving the relative positions of the two corners adjacent to that one).
The author then asserts that this algorithm is not limited to squares & rectangles, but also works for any parallelogram. However, while that initially appears to be true, and works to a limited extent, the fact is that, because projection is used, which involves moving a point onto a line by a course perpendicular to the line, it does not work, as it is, for just any parallelogram.
However, it appears that every parallelogram, once you have selected a corner, has one complementary parallelogram. Each parallelogram covers the area which that algorithm would attribute to its complement, and vice versa. For example, if you draw a rhombus and select an obtuse corner of it, and then shade the area of the paper that projects to each of the two sides which that corner joins, the shaded areas will overlap to form another rhombus, with one acute corner at the point which you selected. Then, if you draw the acute rhombus and then shade the areas of the paper which project to the sides adjacent to the originally selected corner, the shaded area will form the same shape, size, and orientation as your original rhombus.
The question remains, how to derive the complement of a parallelogram once you know the three vectors which define the first?