Today we will do some things vital to most programs: lines and circles.

```DENTHOR, coder for ...
_____   _____   ____   __   __  ___  ___ ___  ___  __   _____
/  _  \ /  ___> |  _ \ |  |_|  | \  \/  / \  \/  / |  | /  _  \
|  _  | \___  \ |  __/ |   _   |  \    /   >    <  |  | |  _  |
\_/ \_/ <_____/ |__|   |__| |__|   |__|   /__/\__\ |__| \_/ \_/
smith9@batis.bis.und.ac.za
```

Grant Smith, alias Denthor of Asphyxia, wrote up several articles on the creation of demo effects in the 90s. I reproduce them here, as they offer so much insight into the demo scene of the time.

These articles apply some formatting to Denthor's original ASCII files, plus a few typo fixes.

## Circle Algorithim

You all know what a circle looks like. But how do you draw one on the computer?

You probably know circles drawn with the degrees at these points:

Anyway, Pascal doesn’t work that way… it works with radians instead of degrees (you can convert radians to degrees, but I’m not going to go into that now). Note though that in Pascal, the circle goes like this:

Even so, we can still use the famous equations to draw our circle. (You derive the following by using the theorem of our good friend Pythagoras.)

(This is standard 8(?) maths … if you haven’t reached that level yet, take this to your dad, or if you get stuck leave me a message and I’ll do a bit of basic Trig with you. I aim to please ;-))

To simplify matters, we rewrite the equation to get our X and Y values:

This obviously is perfect for us, because it gives us our X and Y coordinates to put into our `putpixel` routine (see part 1). Because the `Sin` and `Cos` functions return a `Real` value, we use the `round` function to transform it into an `Integer`.

In the above example, the smaller the amount that `deg` is increased by, the closer the pixels in the circle will be, but the slower the procedure. 0.005 seem to be best for the 320x200 screen. Note: ASPHYXIA does not use this particular circle algorithm, ours is in assembly language, but this one should be fast enough for most.

## Line algorithms

There are many ways to draw a line on the computer. I will describe one and give you two. (The second one you can figure out for yourselves; it is based on the first one but is faster)

The first thing you need to do is pass what you want the line to look like to your line procedure. What I have done is said that `x1`, `y1` is the first point on the screen, and `x2`, `y2` is the second point. We also pass the color to the procedure. (Remember the screen’s top left-hand corner is (0,0); see part 1)

To find the length of the line, we say the following:

The `Abs` function means that whatever the result, it will give you an absolute, or positive, answer. At this stage I set a variable stating whether the difference between the two x’s are negative, zero or positive. (I do the same for the y’s) If the difference is zero, I just use a loop keeping the two with the zero difference positive, then exit.

If neither the x’s or y’s have a zero difference, I calculate the X and Y slopes, using the following two equations:

As you can see, the slopes are real numbers. NOTE: XSlope = 1 / YSlope

Now, there are two ways of drawing the lines:

The question is, which one to use? If you use the wrong one, your line will look like this: