Midpoint Line Algorithm

Intro
Algorithm
Algorithm Derivation
Hand Trace

Intro

I had a read of "Computer Graphics: Principles and Practice" 2nd edition around pages 78, which gives the Midpoint Line Algorithm, and I can throughly say I didn't understand it.

The use of Implicit functions to derive the algorithm - well where did the implicit form come from? Its at best a proof without meaning!

Long ago I came to the conclusion that such maths was bullshit and so I derive my own. I have pointed the superficialness of the proof so that you may recognize fluff when you see it - underlining their fluff is a really good idea and as far as implicit functions go they are general but how was the equation formed in the first place?

Algorithm

Line translated to origin for simplicity
Line from $(0, 0)$ to $(x1, y1)$

Initial Conditions
$dx = x1$
$dy = y1$
$d = 2 dy - dx$
$x = 0$
$y = 0$

while $x \leq dx$
{
write point

if $d > 0$
$++ y$
$d -= 2 dx$

$++ x$
$d += 2 dy$
}

Algorithm Derivation

$y_{i} = ⌊ \frac{dy}{dx} i ⌋$

$y_{i} = \frac{dy}{dx} i - \frac{1}{2} t$

Lets discuss this. The negative sign is rounding down. 1/2 is the point at which the rounding function balances and hence makes a decision at.

The equation itself is in integer arithmetic with rounding down. In the next step the equation will be brought fully into the integer domain by removing division. At present its in a gray area but I like it here.

Increment the equation

$y_{i + 1} = \frac{dy}{dx} (i + 1) - \frac{1}{2} t$
$y_{i} + w = \frac{dy}{dx} i + \frac{dy}{dx} - \frac{1}{2} t$

Now as discussed in Computing a Straight Line in Integers the change of one incremented equation relative to another in $t$ is one way, so redefine $y_{i}$ .

$y_{i} = \frac{dy}{dx} i$
relative to $y_{i + 1}$

Subtract the equations for y.

$w = \frac{dy}{dx} - \frac{1}{2} t$
$2 w dx = 2 dy - t dx$
$t dx = 2 dy - 2 w dx$

Note that $t dx$ corresponds with $d$ in the algorithm. After some reflection this is to be expected as a consequence of considering the change about the floor operator. Relabel.

$w = (d_{i} \leq 0)$
$d_{i + 1} = 2 dy - 2 w dx$

There is still a bit of work to be done in the algorithm details, eg the correct w condition needs to be solved for in detail. Similarly I have not derived the initial algorithm condition which in itself is interesting, if someone can do it shoot me an email.

To conclude this method is generalizable. I have constructed two of the integer arithmetic algorithms and they share similar assumptions which seem to work. The best thing about these constructions is that they are direct eg from the line equation identify the change and perform analysis to get the algorithm, something which the authors of the algorithm could not do.