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Complete HOW TO of Polygons

3D Graphics Programming

by Voltaire/OTM (Zach Mortensen)

Table of Contents

Introduction

After receiving numerous requests to do so, I have compiled a HOW TO doc on polygon filling. It seems that a lot of people out there are a lot like myself, they really dislike using other people's code because it is extremely difficult to figure out, especially if it is highly optimized. Sometimes text files are the answer, often times however they do more harm than good. When I was writing my 3d engine a few erroneous text files caused me to waste several days debugging, and in the end I wound up deriving everything on my own. Hopefully I have learned from my painful experiences and will make my explanations clear and concise, yet still offer ideas for optimization. My polygon routines are among the fastest I have seen on the PC, but that is only because I am a perfectionist obsessed with being the best ;)

I have attempted to arrange the sections of this document in a somewhat logical fashion; that is, you should feel confident in one area before attempting to move on to the next. This is the order in which I did things, so I figured what worked for me will most likely work for everyone else as well. For my code examples I will use C++, and I will give ideas for assembler optimizations of these routines.

Everything here applies to triangles only, although it is possible to adapt the techniques used here to four sided polygons (quadrangles?) as well. Why use triangles? The simple and straightforward answer: triangles are the most mathematically correct. Take any three points in space, ANY three, and you can make a two dimensional plane out of them. If you were to take four points on the other hand, it is very likely that you will end up with a 3 dimensional shape instead. Triangles also make vector operations (dot and cross product) much simpler.

Scan Conversion

The process of determining the coordinates along the edges of a polygon is known as SCAN CONVERSION. The name comes from the fact that most people use the horizontal scanlines on the screen as the basis for doing this. You will need to scan convert each edge of every polygon you draw, and to do this you need two points (the endpoints of the edge) P1 (X1, Y1) and P2 (X2, Y2)

              P1 .  (X1, Y1)
                  \
                    \
                      \
                        \
                          \
                            \
                              \
                            P2  \.  (X2, Y2)

Now it's time to take a little trip back in time to the Algebra you suffered through in Junior High (if you ever thought you could be a coder without using any math you may as well turn off your computer right now and throw it out the window). The equation

y = mx + b

should be indelibly etched in your mind. To refresh the memory those who were asleep in class, m is the slope (change in Y/change in X or dy/dx if you are a calculus type), and b is the y intercept (value of y when x = 0). A variation of this formula you learned in your Algebra II class is

y - Y1 = m(x - X1)

which is much more useful in our case, because we really don't care about the value of y at (x = 0).

What I am about to say will confuse a lot of you, but I'm going to say it anyway. When filling polygons, we need to switch all the Xs and Ys in the above equation. WHY??? The reason is simple: the above equation is solved for y in terms of x, which means "you feed me an x value, and I'll tell you the y value at that point." When filling polygons, we would much rather draw horizontal lines than vertical lines, because horizontal line drawing is MUCH easier and much faster. Therefore, we want an equation that will give us the x value at a given y:

x - X1 = m(y - Y1)

That's simple enough, but there is a catch. The m in the above equation is no longer dy/dx, instead it is dx/dy. This is purely logical, dy/dx can be read as "change in y with respect to change in x." Since we are changing y instead of x in the above equation, it is logical to have m = "change in x with respect to change in y."

When you begin scan converting an edge, you will only know X1, Y1, X2, and Y2. Therefore, you need to calculate m. The formula for this is

                    (X2 - X1)
                m = ---------
                    (Y2 - Y1)

Once we have a value for m, we can determine the x value of any given y. We will find x values for all integer y values between Y1 and Y2 in order to determine the edges of our polygon, so we know where to start and stop drawing horizontal lines when we fill it. One equation which will give us these values is

x = m(y - Y1) + X1

Now, you could solve this equation for every integer y value between Y1 and Y2, but this is extrememly slow. If you remember that we are starting at y = Y1 and we only interested in the value of x at each scanline (we are using integers for y), you can do a little trick to simplify this equation:

point 0 -> x = m((Y1 + 0) - Y1) + X1
point 1 -> x = m((Y1 + 1) - Y1) + X1
point 2 -> x = m((Y1 + 2) - Y1) + X1

                ...

point n -> x = m((Y1 + n) - Y1) + X1

It is obvious that ((Y1 + n) - Y1) is simply n, so we arrive at the general equation

x = m(n) + X1

Now, if we take the difference between two consecutive x values (x values at scanlines n + 1 and n), we find that

(m(n + 1) + X1) - (m(n) + X1)
= mn + m + X1 - mn - X1
= m

WOW, that makes life easy. The difference between the x values of consecutive y values is simply m. That means that we now have a recursive definition for x (recursive means that the value of each term is based on the value of the previous one)

Xn = X(n-1) + m

So by adding m to the x value of the previous scanline, we can determine the x value at the current scanline. Of course we also know the starting point of this sequence, X1. Some C++ code that implements this might look like

float x, m, edge[200];          // use edge to store the x
int count;                      // values at each scanline,
                                // there are a maximum of 200
                                // lines on the screen, so we
                                // need room for 200 x values


m = (X2 - X1)/(Y2 - Y1);
x = X1;

for (count = Y1; count < Y2; count++)
{
	edge[count] = x;
        x += m;
}

There is another problem here, we are dealing with floating point numbers, which are incredibly slow in calculations if you don't have a coprocessor. The solution to this problem is to use fixed point integer math. In fixed point, you multiply each number by a scaling factor. When you perform any calculations, divides in particular, the precision of your answer is accurate to a fixed number of decimal places, hence the name fixed point.

The most common implementations of fixed point scale by factors of either 256 or 65536 because multiplying and dividing by these numbers can be accomplished by shifting bits, and each of these numbers is equivalent to half a register in assembler. In order to use 16 bit or 8.8 fixed point, scale by 256 (you now have 8 bits for the integer part and 8 bits for the decimal part), and scale by 65536 in order to use 32 bit or 16.16 fixed point (16 integer bits, 16 decimal bits). When you are done with all your calculations and need an integer answer, you simply use the top 8 or 16 bits of your fixed point integer, depending on what your scale factor was.

The above example converted to fixed point would look something like this

int x, m, count, edge[200];     // if you are using 16 bit
                                // code, these should be of
                                // type long rather than int

m = (X2 - X1) << 16;            // dx scaled to 16.16 fixed
                                // point, I'm assuming you are
                                // using 32 bit code


m /= (Y2 - Y1)                  // notice that you do NOT scale
                                // the denominator, if you did
                                // you would lose all precision
                                // in your answer

x = X1 << 16;                   // scale the starting x value

for (count = Y1; count < Y2; count++)
{
    edge[count] = x >> 16;      // store only the integer part
    x += m;
}

Now that you have an incredibly fast way to scan convert a single edge, you must do this for all the edges in your polygon. Here's another place where triangles are better than quadrangles: on any triangle, there is a top, middle, and bottom point. The side connecting the top and bottom points is ALWAYS the longest. Therefore, the best strategy for scan converting an entire triangle is as follows:

  1. Set up two edge lists, one for the right and one for the left side of each scanline.
  2. Scan convert the longest edge and store it in the left edge list.
  3. Scan convert the remaining two edges and store them in the right edge list.
  4. Well...FILL IT OF COURSE!

Flat Filling

Flat filling is the simplest and least impressive form of polygon filling. The entire polygon is filled with one color. In order to flat fill a polygon, you need to write two routines

  1. A routine to scan convert an entire polygon
  2. A routine to draw a horizontal line

THAT'S IT!

Horizontal line drawing is very simple in chained (packed pixel) video modes such as VGA mode 13h. For the sake of simplicity, I will use mode 13h as the example here.

Your horizontal line routine should definitely be in assembler, because it is going to get called a LOT. Preferably, it will be integrated into your poly filler so you don't have to waste time pushing arguments and calling another procedure.

To draw a horizontal line you need to know 4 things: the starting and ending x values (X1, X2), the y value (Y), and the color (C).

The first thing to do is make sure that X1 < X2. If not, switch them before you continue.

I'm not going to cover mode 13h graphics basics here because they are too basic. If you don't know mode 13h yet, you have no business writing anything until you learn it! The easiest way to draw a horizontal line in mode 13h is to

  1. Determine the starting memory address of the line (A000h + (320 * Y) + X1)
  2. Determine the length of the line in pixels
  3. Store (length) bytes of value (color) starting at the starting memory address.

In order to make use of the 32 bit processor in your machine, you will want to store doublewords instead of bytes. This makes flat filling in mode 13h almost as fast as flat filling in mode x. Too store doublewords, you need to

  1. Perform steps 1 and 2 listed above
  2. Convert your byte color value into a dword (just make it 4 bytes in a row of your original color)
  3. Store (length / 4) doublewords
  4. Store (length % 4) bytes (a quick way to do (length % 4) is (length & 3))

Once you have your horizontal line routine up and running, you need to integrate it into your scan converter. The easiest way to do this is to make a loop from Y1 to Y2 where you draw horizontal lines between the edges of the polygon. Here's some pseudo code

scanLongSide();
scanMidSide();
scanShortSide();

for (count = Y1; count < Y2; count++)
	hline(lEdge[count], rEdge[count], count, color);

Easy as pi...

Clipping

We quicly run into a problem in mode 13h, images drawn off the screen wrap around to the next scanline. This is not very aesthetically pleasing to say the least. In protected mode, this poses an even nastier problem. Any memory writes outside the 64k window reserved for the VGA produce a general protection fault, very nasty. The answer to these problems is to "stay inside the lines" when we are drawing, to "clip" our drawings so we only draw what is physically on screen.

When incorporated in a poly filler, the most rudimentary form of clipping checks y values at scan conversion time and x values when drawing horizontal lines.

Scan conversion C++ code that implements y clipping would look somewhat like this

for (count = Y1; count < Y2; count++)
{
	if ((count >= 0) && (count < 200))
		edge[count] = x;
	x += m;
}

Notice that you change the x value every time through the loop, but you only store the edges that are on the screen. This assures that the x values you get later in the loop are accurate.

Clipping in the x direction is best if done in the horizontal line routine, and is even easier to implement than y clipping in the scan conversion. Once you make sure that X1 < X2, all you need to do is substitute 0 for X1 if X1 is smaller than 0, and 319 for X2 if X2 is greater than 0. Also, if X1 > 319 or X2 < 0 you don't need to draw anything, the line is entirely off screen.

Some more complicated clipping that will improve your performance significantly involves checking to see if the polygon is on screen before you draw. Check your vertices to see if the maximum x value is less than 0, the minimum x value is greater than 319, the maximum y value is less than 0, or the minimum y value is greater than 199. If any of these are true, your polygon is completely off screen and you have no need to scan convert or fill.

Goraud Shading

If you have never heard of or seen gouraud shading, I pity you. It is the easiest way to make your boring flat shaded polygons come to life. In order to explain gouraud shading, I ask you to bear with my while I digress and explain a bit about 3d math.

According to Lambert's law, the intensity of light falling on a plane is directly related to the angle made when the light vector intersects the normal vector of a plane. Lambert shading a polygon involves taking the dot product of the normal vector and the vector of the light intersecting it, which gives the cosine of the angle made by the intersection. Based on this value, the color of a plane can be calculated. I'm not going to give any further explanation than this because I don't want to be up all night typing.

Gouraud shading is a simple extension of Lambert shading. Instead of finding the intensity of light falling on a plane, you determine the average intensity of light at a vertex based on a normalized average of the normal vectors of all the planes that share that point. Once again, I'm not going to give any further explanation than this of the math behind Gouraud shading. Suffice to say that Gouraud shading uses a separate color value for each vertex of each polygon rather than a single color for the entire plane.

After mathematically determining the color of each vertex, the color of each point in the plane can be approximated using linear interpolation. First, the color values along the edges are interpolated between the color values at each vertex. Then color values along each scanline are interpolated between the color values at the edges of the line

Start with      /16 interpolate  16/16  interpolate  16/16
color values  / /   edge        E/ /E   scanline    E/E/E 
at vertices /  /    colors    C/  /D    colors    C/CD/D 
          /   /             A/   /B             A/AAB/B 
        /    /            8/    /A            8/899A/A 
      /     /           6/     /8           6/67778/8 
    /      /          4/      /7          4/455667/7 
  /       /         2/       /5         2/2333445/5 
/--------/        0/--------/4        0/--------/4
0       4          001122334           001122334

  step 1             step 2              step 3

Scan conversion is a form of linear interpolation in which x values are interpolated between two known points. Scan conversion for Gouraud shading very similar, but instead of interpolating only x values, we interpolate x values and colors. A Gouraud polygon filler must be given more information than a flat filler. For each vertex, we need to know (X, Y, C) instead of just (X, Y). As I said before, we also need to trace color values while scan converting. This makes the scan conversion more complicated, however, color is traced (interpolated) in exactly the same way as x

mx  = (X2 - X1) << 16;
mx /= (Y2 - Y1);

mc  = (C2 - C1) << 16;
mc /= (Y2 - Y1);

x = X1 << 16;                   // scale the starting x value
c = C1 << 16;                   // scale the starting c value

for (count = Y1; count < Y2; count++)
{
	edge[count] = x >> 16;      // store only the integer part
	color[count] = c >> 16;     // store only the integer part
	x += mx;
	c += mc;
}

The resulting color values are stored in color lists corresponding with the existing edge lists. Now we have two x values and two color values for each scanline. The color values for each scanline are not equal, so we need to interpolate colors along each scanline as we draw. This is done almost exactly as scan conversion is done, with the sole exception that we use dc/dx instead of dx/dy.

The Gouraud horizontal line routine needs to be passed x values for the start and end of each scanline (X1, X2) as well as color values (C1, C2) and a y value (Y). Once again, you need to make sure that X1 < X2. If you switch X1 and X2, be sure to switch C1 and C2 as well! Here is some C++ code for a gouraud hline routine

mc  = (C2 - C1) << 8;
mc /= (X2 - X1);

c = C1 << 8;

for (count = X1; count < X2; count++)
{
	setPixel(count, y, c >> 8);  // set pixel at (count, y) to
                                     // color c
	c += mc;
}

Notice that I used 8.8 fixed point here. This is because you are GUARANTEED that you will not have any color greater than 255 when you are using mode 13h, so you can shift the value left 8 bits without any word overflow. The reason you want to use 8.8 fixed point is so you can do a little trick when you convert to assembler that will allow you to eliminate the shift right

; first, get the starting memory address of the line in edi
; and the number of pixels to draw in ecx

mov edx, mc     ; (dc/dx * 256)
mov ebx, C1     ; starting color
shl ebx, 8      ; C1 * 256

ghlLoop:

mov [edi], bh   ; draw the upper 8 bits (integer part)
add ebx, edx    ; c += mc
inc edi         ; go to next screen location

dec ecx         ; pixels to draw --
jnz ghlLoop

That inner loop is VERY FAST, it should almost run in one memory wait state, which means that you get all the cpu cycles for free while you are waiting to be able to write to memory again.

Clipping here is implemented exactly the same as for flat polygons.

Z-Bufferring

Ah yes, the crux of the biscuit indeed. Z-buffering is a technique used by more advanced 3d systems, it allows you to do several things. First, z-buffering speeds up 3d code by eliminating plane sorting. You can draw planes in any order, and they will come out looking right. This is a big advantage when drawing objects with many faces, because sorting routines take exponentially longer to sort larger data sets. Second, z-buffering correctly draws intersecting polygons WITHOUT having to calculate where they intersect, which produces some impressive effects and doesn't require any extra calculation.

Z-buffering accomplishes these feats by interpolating z values between vertices and scanline edges much in the same way Gouraud shading interpolates color. The resulting z values for each pixel on the screen are stored in a 'z-buffer' containing (screenWidth * screenHeight) elements (64000 in mode 13h). Before a new pixel is drawn, the z-buffer value for that pixel is examined. If the new pixel's z-value is less than (closer to the viewer than) the z-buffer value, the new z value is stored in the z-buffer and the pixel is drawn to the screen. Otherwise, the z-buffer and the screen remain unchanged. Through this process, the pixel at each screen location which is closest to the viewer (and therefore not obscured by anything else) is always displayed.

Because we don't want to have a 3d world which is only 128 pixels deep, the z-buffer elements cannot be bytes; we need to store at least 16 bits of z data in each array element. This allows us to have a world which is 32768 pixels deep, adequate for most applications. Right away, we see a big problem with z-buffering, MEMORY! It takes a LOT of memory to store all those z values, 128k for a 16 bit z-buffer in mode 13h. In my opinion, the advantages of z-buffering disappear when you are in real mode due to the incredible amount of memory required, so switch to protected mode before attempting to implement z-buffering.

As I said before, z-buffering is almost identical to Gouraud shading, with the exception that z values are interpolated instead of color values. Of course, you need to pass some more information to your z-buffered poly routine, namely the z values of each vertex. When scan converting, store z values in z lists the same way you store color values in color lists. Here's some C++ code

mx  = (X2 - X1) << 16;
mx /= (Y2 - Y1);

mz  = (Z2 - Z1) << 16;
mc /= (Y2 - Y1);

x = X1 << 16;                   // scale the starting x value
z = Z1 << 16;                   // scale the starting z value

for (count = Y1; count < Y2; count++)
{
	edge[count] = x >> 16;      // store only the integer part
	zval[count] = z >> 16;      // store only the integer part
	x += mx;
	z += mz;
}

When you are tracing horizontal lines, the z values are interpolated in exactly the same way as they are with Gouraud shading. After you determine a z value, check it against the z value stored in the z-buffer for the current screen location to see if the current pixel is visible. If it is, draw it; otherwise skip it and go on. Here's some C++ again

mz  = (Z2- Z1) << 16;
mz /= (X2 - X1);

z = Z1 << 16;

for (count = X1; count < X2; count++)
{
	if ((z >> 16) < zBuffer[y * 320 + x])
	{
		setPixel(count, y, c);
		zBuffer[y * 320 + x] = z >> 16;
	}

	z += mz;
}

Of course, you don't perform the slow z-buffer address calculation every time through the loop. Just find the starting offset into the screen, use that to find the starting offset into the z-buffer, and then increment the z-buffer pointer by 2 (for words) every time through the loop. Also make sure you use 16.16 fixed point here.

Don't forget to clear the z-buffer every time you clear the screen. This is accomplished by filling it with 32767 (7fffh), or whatever your maximum depth is.

By now you should know how to clip, it's exactly the same for z-buffered polygons as it is for all others.

Notice that all the above examples deal with flat shaded polygons. This is because once you have written a Gouraud poly filler, it will take you about a half hour to convert it to a flat z-buffered filler if you were smart about how you wrote your code. If you are feeling REALLY brave, you should try writing a Gouraud shaded z-buffered poly routine; but let me warn you - YOU _WILL_ RUN OUT OF REGISTERS! Besides that, the code will be huge. My gouraud shaded z-buffered poly routine was well over 1200 lines of assembler, about twice as long as this file! But it's by no means impossible, and it's a lot of fun just sitting and watching two gouraud shaded objects intersecting each other when you are done.

Closing Comments

WOW, I didn't intend to write this entire text in one night. Oh well, at least it's done. Hopefully I have been of some assistance to just about everyone who reads this file. I apologize if some of the basics were too simple or if some of the more complex points were not covered in enough detail, but that is what you risk by catering to such a wide group of people.

If you need any tips on optimization feel free to mail me (mortens1@nersc.gov). The source to my flat and Gouraud poly fillers has already been released as part of my 3d engine (V3DT090.ZIP availible via ftp at hornet.eng.ufl.edu /demos/code/library/graph/), and my z-buffering code will be released shortly (as soon as I clean it up and document it so that it is READABLE ;))

Greets

	Siri                    - I LOVE YOU I LOVE YOU I LOVE YOU
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