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Algorithm to compute the remaining polygon after subtraction

I have a big polygon (Pa). Inside the polygon there are a lot of small "holes", as shown:
Here are a few condition for the holes:
The holes cannot overlap one another
The holes cannot go outside the outer polygon
However, the holes can touch the outer polygon edge
How to obtain the remaining polygon ( or the polygon list) in an efficient manner? The easiest way ( brute force way) is to take the Pa, and gradually computing the remaining polygon by subtracting out the holes. Although this idea is feasible, but I suspect that there is a more efficient algorithm.
Edit: I'm not asking about how to perform polygon clipping ( or subtraction) algorithm! In fact that's something I would do by brute force. I'm asking in addition to the polygon clipping method ( take the main polygon and then gradually clip the holes out), is there other more efficient way?

This is very hard to do in a general manner. You can find source code for a solution here:
General Polygon Clipper (GPC)

Well, if you use the right representation for your polygon you would not need to do anything. Just append the list of edges of the holes to the list of edges of Pa.
The only consideration you should have is that if some hole vertex or edge can touch Pa edge, you will have to perform some simplification there.
A different problem is rendering that polygon into a bitmap!

You can do like this.
Draw the main polygon with a color in a bitmap.
Draw the holes with another color in the same bitmap.
Then extract the polygon by running marching square algorithm with the main polygons color as threshold.
The output will contain all the points that belong to that polygon.
You can sort the points if you want it as a continous closed polygon.

I agree with salva, but my post is going to address the drawing part. Basically, you can add up all lines of the main and the hole polygons together and thereby get a single complex polygon.
The algorithm itself is not very complicted and it is nicely explained in the Polygon Fill Teaching Tool.

Draw 2D Curve in XNA

Is there any way to generate a Curve class and then draw that curve in 2D on the screen in XNA?
I want to basically randomly generate some terrain using the Curve and then draw it. Hoping that I can then use that curve to detect collision with the ground.

It sounds like what you want is the 2D equivalent of a height-map. I'd avoid making a true "curve" and simply approximate one with line segments.
So basically you'll have an array or list of numbers that represent the height of your terrain at a series evenly spaced (horizontally) points. When you need a height between two points, you simply linearly interpolate between the two.
To generate it - you could set a few points randomly, and then do some form of smooth interpolation to set the rest. (It really depends on what kind of curve you want.)
To render it you could then just use a triangle strip. Each point in your height-map will have two vertices associated with it - one at the bottom of the screen, the other at the height of that point in the height-map.
To do collision detection - the easiest way is to have your objects be a single point (it sounds like you're making a artillery game like Scorched Earth) - simply take the X position of your object, get the Y position of your terrain at that X position, if the Y position of your object is below the terrain, set it so that it is on the terrain's surface.
That's the rough guide, anyway :)

Subdividing 3D mesh into arbitrarily sized pieces

I have a mesh defined by 4 points in 3D space. I need an algorithm which will subdivide that mesh into subdivisions of an arbitrary horizontal and vertical size. If the subdivision size isn't an exact divisor of the mesh size, the edge pieces will be smaller.
All of the subdivision algorithms I've found only subdivide meshes into exact powers of 2. Does anyone know of one that can do what I want?
Failing that, my thoughts about a possible implementation is to rotate the mesh so that it is flat on the Z axis, subdivide in 2D and then translate back into 3D. That's because my mind finds 3D hard ;) Any better suggestions?
Using C# if that makes any difference.

If you only have to work with a rectangle in 3D, then you simply need to obtain the two edge vectors and then you can generate all the interior points of the subdivided rectangle. For example, say your quad is defined by (x0,y0),...,(x3,y3), in order going around the quad. The edge vectors relative to point (x0,y0) are u = (x1-x0,y1-y0) and v = (x3-x0,y3-y0).
Now, you can generate all the interior points. Suppose you want M edges along the first edge, and N along the second, then the interior points are just
(x0,y0) + i/(M -1)* u + j/(N-1) * v
where i and j go from 0 .. M-1 and 0 .. N-1, respectively. You can figure out which vertices need to be connected together by just working it out on paper.
This kind of uniform subdivision works fine for triangular meshes as well, but each edge must have the same number of subdivided edges.
If you want to subdivide a general mesh, you can just do this to each individual triangle/quad. This kind of uniform subdivision results in poor quality meshes since all the original flat facets remain flat. If you want something more sophisticated, you can look at Loop subidivision, Catmull-Clark, etc. Those are typically constrained to power-of-two levels, but if you research the original formulations, I think you can derive subdivision stencils for non-power-of-two divisions. The theory behind that is a bit more involved than I can reasonably describe here.

Now that you've explained things a bit more clearly, I don't see your problem: you have a rectangle and you want to divide it up into rectangular tiles. So the mesh points you want are regularly spaced in both orthogonal directions. In 2D this is trivial, surely ? In 3D it's also trivial though the maths is a little trickier.
Off the top of my head I would guess that transforming from 3D to 2D (and aligning the rectangle with the coordinate axes at the same time) then calculating the mesh points, then transforming back to 3D is probably about as simple (and CPU-time consuming) as working it all out in 3D in the first place.
Yes, using C# means that I'm not able to propose a code to help you.
Comment or edit you question if I've missed the point.

How to color a mesh with values at the vertices in WPF 3D?

We've got a sphere which we want to display in 3D and color given a function that depends on spherical coordinates.
The sphere was triangulated using a regular grid in (theta, phi), but this produced a lot of small triangles near the poles. In an attempt to reduce the number triangles at the poles, we've changed out mesh generation to produce more evenly sized triangles over the surface.
The first triangulation method had the advantage that we could easily create a texture and drape it over the surface. It seems that in WPF it isn't possible to assign colors to vertices the way one would go about in OpenGL or Direct3D.
With the second triangulation method it isn't apparent how to go about generating the texture and setting the texture coordinates, since the vertices aren't aligned to a grid any more.
Maybe it would be possible to create a linear texture containing a color for each vertex, but then how will that effect the coloring? Will it still render smoothly over the triangle surfaces as one would expect by applying per vertex coloring?

I've converted the algorithm to use a linear texture which is really just a lookup into the colormap. This seems to work great and is a much better solution than the previous one. Instead of creating a texture of size ThetaSamples * PhiSamples, I'm now only creating a fixed texture of 256 x 1.

Transforming co-ordinates from a rectangle to a parallelogram

I am stuck on a simple yet vexing problem with basic geometry. Too bad I don;t remember my high-school co-ordinate geometry and looking for some help.
My problem is illustrated in this diagram: A rectangle rotated, scaled, and warped into a parallelogram http://img248.imageshack.us/img248/8011/transform.png
I am struggling with transforming a co-ordinate from the rectangle to a resized parallelogram. Any tips, pointers and/or code-examples would be wonderful!
Thanks,
M.

There are several steps in this transformation.
Scale about (x,y) to adjust to the final size W', H'. (Possibly unequal
scaling on X and Y axes).
Apply a shear transform to convert
the rectangle to a parallelogram
(keeping x,y invariant).
Rotate about (x,y) to align to the
final coordinate orientation.
Translate to the new location.
Create the coordinate matrices for each of these and composite (multiply) them together to create the overall transform. Wikipedia could be your starting point to know about these transformation matrices.
Tip: Might be simplest to apply a translation to move (x,y) to the origin first. Then, the shear, scaling and rotation are a lot simpler to do. Then move it to the new location.