When drawing objects that exist in 3D space to a 2D plane (like your monitor or an image), there are a number of spaces that are useful to work in:
- Global Coordinates - Somewhere in your 3D world there is an origin, and objects will be placed relative to that origin. The camera will also be placed somewhere relative to that origin and will point in some direction in the world.
- Object Coordinates - If you have 3D objects in your scene, they are generally created outside of the world and then transformed in some way to be placed in the world. You might create, say, a chair, with the origin of the chair where the front right leg touches the ground. In Object space, that's (0,0,0). But you'll transform that chair to where it needs to go in the scene and put it at some location in world (global) coordinates.
- View Coordinates - Your virtual camera, as mentioned above, is placed somewhere and is pointing at something. Whatever your camera is pointing at will generate a space relative to the camera. You can think of the camera as always being at the origin of camera space.
Clip Space - eventually during rendering, you will use a projection matrix of some sort (perspective, orthographic, etc.) to render the final image. When running your geometry through this matrix, any coordinates that end up outside of [-1..1] will be clipped out. (In some cases [0..1] is used instead, but this detail doesn't change the point.)
Screen Space - finally, after you've clipped everything and gotten it into normalized coordinates, you then have to flatten it to 2D and display it to the user. This is screen space, and is often the same as the pixels (or some portion of them) on the user's physical display (or in an image if your rendering to an image).
So in the original problem, you've been given the requirement of having a field of view for your camera of 135° (3/4π radians), and a window that's 15x15 pixels large. So you need to construct a projection matrix with a 135° field of view and an aspect ratio of 1:1.
The view parameters aren't specified in your problem, so I guess the camera is at the world's origin, so camera/view space and world space are the same for now.
Normally, you'd take the world coordinates, (
p2) and multiply them by the view matrix (identity here, so no work to do), then multiply that result by the projection matrix to get to clip space. Once you have coordinates in the normalized range, you can multiply them by the window's size (15x15 in this case) to get them into screen space and determine which pixels to set.