Clipping Polygons
Line clipping is all you need if you are doing transparent wire-frame modelling, but most 3D applications will be modelling 3D polygons. Polygon clipping is somewhat of a natural extension from line clipping, because polygons are defined as a set of line segments that make up the edges of the polygons, and clipping a polygon against the Z=1, X=Z, X=-Z, Y=Z, and Y=-Z clipping planes will involve clipping the polygon edges to the clipping planes.
I assume here that for a polygon P I can get P.numVerts = number of vertices in P, and for i = 0..P.NumVerts-1 I can get P.Vert[i] = the i'th vertex of P. The edges (line segments) of P are from P.Vert[0]..P.Vert[1], P.Vert[1]..P.Vert[2], ..., P.Vert[P.NumVerts-2]..P.Vert[P.NumVerts-1], and P.Vert[P.NumVerts-1] to P.Vert[0]. IOW the vertices are listed either clock-wise or counter-clockwise around the poly edges, and the poly is implied to be closed.
Also note the following on how we will treat a polygon:
- If P.NumVerts == 0 then the poly has no edges, no area, etc (ignore it).
- If P.NumVerts == 1 then the poly has 1 edge of 0 length (ie a point - probably ignore it).
- If P.NumVerts == 2 then the poly has 2 edges that lie on top of each other (poly has no area - probably ignore it)
- If P.NumVerts >= 3 then the poly probably has an area (unless the 3+ vertices are colinear), and should be handled.
Usually in 3D graphics you consider a polygon to be invisible if it has no visible area. This is the equivalent of viewing the poly "edgewise", and the poly is considered to have zero width.
As you might expect, drawing the polygon will, at a minimum, involve transforming it from world-space to clip-space, clipping it, and then (if any of it is visible) drawing whatever is left over. Transforming the polygon from world-space to clip-space just involves transforming each of its vertices.
The next step is to clip the poly to the Z=1, X=Z, X=-Z, Y=Z, and Y=-Z clipping planes. How is that done? Let's take it one plane at a time.
Consider the Z=1 clipping plane. As the line-clipping page showed this can result in, at most, one new vertex (actually, in line clipping we replaced one of the vertices with the point where the line intersects the plane). But unlike line-clipping, we cannot just replace vertices as we find a line segment that crosses a clipping plane, as the following diagram shows:
[Image] diagram 1.1
Here, we started with the line segment from p1 to p2. It crossed the Z=1 plane, so the line-clip algorithm replaced p2 with p2'. But now we end up checking a line segment from p2' to p3, when we wanted to check from p2 to p3! We have changed the characteristics of the polygon!
What we would instead like to do is:
- Check p1. It is visible, so add it to the output list of vertices.
- Check p1 to p2. It crosses the Z=1 plane, so find the intersect point (p2') and add that to the output list (p2 was not visible, so do not add it to the output list).
- Check p2 to p3. It crosses the Z=1 plane, so find the intersect point and add that to the output list. p3 was visible, so also add it to the output list.
- Continue for the rest of the vertices in the polygon.
This will yield a diagram like this instead:
[Image] diagram 1.2
Now our output polygon has the edges:
- p1..p2'
- p2'..p2''
- p2''..p3
- p3..
We have shortened two edges (p1..p2 and p2..p3), and introduced a new edge (p2'..p2''). From a vertex point of view we have added one new vertex; we originally had a 5-sided (and 5-vertex) poly, and now we have a 6-sided (and 6-vertex) poly. You should also note that, if the polygon is convex ("inside" angle between any two adjacent edges is <= 180 degrees) then clipping the polygon against a plane will yield at most one additional vertex. Conceptually, clipping the polygon will entail:
- Let P = poly to clip, P1 = an output poly, P2 = an output poly
- Clip P to the Z=1 plane (ie "near" plane), storing the output vertices in P1. Note that P1.NumVerts can be as large as P.NumVerts+1. If P1.NumVerts < 3 then consider the whole polygon clipped.
- Clip P1 to the X=-Z (ie "left" plane), storing the output vertices in P2. P2.NumVerts can be as large as P.NumVerts+2. If P2.NumVerts < 3, consider the polygon clipped.
- Clip P2 to the X=Z (ie "right" plane), storing the output vertices in P1. P1.NumVerts can be as large as P.NumVerts+3. If P1.NumVerts < 3, consider the poly clipped.
- Clip P1 to the Y=-Z (ie "top") plane and store in P2. P2.NumVerts can be up to P.NumVerts+4. If P2.NumVerts < 3, consider the poly clipped. * Clip P2 to the Y=-Z (ie "bottom") plane and store in P1. P1.NumVerts can be up to P.NumVerts+5. If P1.NumVerts < 3, consider the poly clipped.
- Transform P1.Vert[...] from clip-space (3D) to pixel-space (2D).
- Paint the polygon to the screen.
Notice that this algorithm clips a complete polygon against each of the five clipping planes, stopping early if it is completely clipped against one of the planes. Alternatively you might come up with an algorithm that clips each line segment against the 5 clipping planes - but this would mean you would never short-circuit (stop early because the poly is completely clipped).
Notice also that there is a LOT of operations required here - and this is just to render a poly from 3D to 2D! We haven't even examined shading, texture mapping, bump mapping, etc! And this is going on for each poly you want to render, indicating two important concepts:
- Optimize the poly rendering section. See comments below.
- Minimize the number of polys drawn. Go to 3D Basics page and follow the Polygon Culling (and other) links.
At this point (even after reading about line-clipping) you may ask: why all this overhead? I have to clip against Z=1 to prevent the "mirror effect" in the perspective transform (3D to 2D transform), but won't it be cheaper to clip in the 2D functions instead of the 3D functions?
The answer is: it depends on what you want to do. For wire-frame models and solid-color polygons, it would be cheaper to clip in 2D instead of 3D. But shading, texture mapping, bump mapping, etc are expensive on a per-pixel basis (vs 3D clipping, which is expensive on a per-vertex basis - and a poly usually has a LOT more pixels than vertices!) For example: you might have a texture-mapped poly which is only 75% visible - 25% of the area is invisible and would be clipped if doing 3D clipping. If you don't clip in 3D however, you will have to texture map that 25% - and this will undoubtedly cost more (in time) than if you had first done clipping in clip-space.
Keep in mind however that I do not have empirical evidence for this statement <g>, and provide no mathematical proofs. Not only that, but many factors in 3D graphics are app-dependent, and what might be most efficient in general may turn out to be less efficient under a certain set of game design specs. If you have the development time available, you might want to code in both techniques, have a switch (pre-processor macros, etc), and test both methods.
I do not include sample code for polygon clipping for the simple fact that optimizations will be critical in this section, and optimization techniques will depend greatly on the data structures you use. For example: in the above code snippet I just blithely transformed all the poly vertices from world-space to clip-space (by multiplying by the camera->ParentToClip() matrix, presented in the line-clipping article). However a real-world 3D application will probably have "objects" that are made up of multiple polys, and these polys will "share" a vertex. The above algorithm would then "waste" rendering time by transforming shared vertices more than once. For example: instead of a poly containing a list of Point3Ds, you may have it containing a Point3D (the world-space coordinate), another Point3D (the clip-space coordinate), flags (Sutherland-Cohen clipping outcodes, boolean flags to indicate if this point has been transformed to clip-space yet, etc). In that case any code I wrote would not remotely work, because you are using different data structures.
