Projections (1)
Projections are a method of representing an N-dimensional object using only (N-1) dimensions. There are several different methods of projection, but they all have the same underlying idea: imaginary rays called projectors are emanated from the object towards an (N-1)-dimensional projection plane. The intersections of these rays with the projection plane produce an image of the object. This is like taking a picture with a camera: light from the object travels in straight lines (rays) and strike the film (the projection plane), producing an image of the object.
Projections are an easier method of exploring higher dimensions, because it gives us an integrated view of the object, rather than isolated bits of information such as in the cross-section method. In fact, our own eyes work this way: the retina captures a projection of a 3D object outside of us, producing a 2D image which we then reconstruct into a 3D model in our mind. The image captured this way retains valuable information about the 3D object: such features as corners, edges, and the shape and number of faces, are represented as an integrated whole. Consider the following image:
We can immediately see where the corners of the cube are, and we can see that its faces are tetragonal. It is more immediately obvious that this is the image of a cube, compared with the sequence of cross-sections we saw in the previous chapter. Although the cross-section method does give us valuable information, the projection method is easier to understand.
Types of Projections
There are two main categories of projections: parallel projection and perspective projection. In parallel projection, the projection rays are parallel to each other. In perspective projection, the rays converge on a single point behind the projection plane. Both types of projection are useful in examining higher-dimensional objects.
Parallel Projection
Parallel projection may be further classified into orthographic projection, where the rays are always perpendicular to the projection plane, and oblique projection, where the rays intersect the projection plane at an angle.
For example, here is a 3D cube projected into 2D face-on using orthographic projection:
Here is the same cube projected using oblique projection:
For comparison, here's a 4D hypercube in orthographic projection:
And here's the 4D hypercube in oblique projection:
Two of the hypercube's cells have been highlighted to make them easier to see. Note the similarity between these images and the images of the 3D cube. The image of the cube under orthographic projection a 2D square (because we are looking at it face-on). In the same way, the image of the hypercube in orthographic projection is a 3D cube. The oblique projection of the cube looks like two squares connected by 4 lines. In the same way, the oblique projection of the hypercube looks like two cubes connected by 8 lines. You can see dimensional analogy at work here.
Here are some of the properties of images produced by parallel projection:
The size of the image does not depend on the distance of the object. No matter how far away the object is, the parallel rays will always produce the same image size.
Parallel lines in the object remain parallel in the image. For example, in the oblique projection of the cube, the vertical edges of the cube all appear vertical in the image. The front-to-back edges appear slanted, but they are still parallel to each other, just as in the real cube.
