Subliminal Extacy
#03
01 апреля 2001 

Draw Routine with Bresenham Algorithms
Draw Routine with Bresenham Algorithms Written by Mikezt This article is dedicated to baze/Зsc. One of the most basic graphic routines is a line draw. When you try to write a draw routine you may find it a bit complicated. Don't worry, here's one of way of doing it. Bresenham algorithms have an advantage over methods because they work with integer numbers. So they can be easily written in machine code. Before we start, get ready for some mathematic expressions. I'll write lower indexes in []. Abscissa is defined with two points (x1, y1) and (x2, y2) and can be written in the expression: y = kx + q where k is the direction and q is the shift on the y axis. Direction k indicates inclination of the line and divide coordination system to eight parts (see pic. 1). We get a situation where k is positive and less than 1. Let's presume that point (x[i], y[i]) was drawn. For each pixel we have to compute the differences from the ideal course of abscissa (pic. 2). Difference of both possible points are: D1 = y  y[i] = k * (x[i] + 1) + q  y[i] D2 = y[i] + 1  y = y[i] + 1  k * (x[i] + 1)  q Difference of these two values is equal to: D = D1  D2 = 2 * k * (x[i] + 1)  2 * y[i] + 2 * q  1 Along this difference we can choose the right point. It's clear that the factor of decision is it's sign. A negative value will select the point (x[i] + 1, y[i] + 1), and a positive one will select (x[i] + 1, y[i]). The disadvantage is real character of values. We can remove this disadvantage by multiply difference expression by (x2  x1) and we get the expression which is called decision: d[i]=D*(x2x1)=2*(y2y1)*(x[i]+1)2*y[i]*(x2x1)+(2*q1)*(x2x1) and after a slight adjustment: d[i]=2*(y2y1)*x[i]2*y[i]*(x2x1)+2*(y2y1)+(2*q1)*(x2x1) Expression 2*(y2y1)+(2*q1)*(x2x1) is constant and we indicate it with c. so: d[i] = 2 * (y2  y1) * x[i]  2 * y[i] * (x2  x1) + c Now this is an integer value, but it's quite complicated. If we express decision for the next point we can write a recursive relation. Here is decision for the next point: d[i+1] = 2*(y2  y1)*(x[i] + 1)  2*y[i+1]*(x2  x1) + c After substracting these two decisions we can express the decision value in a recursive form: d[i+1] = d[i] + 2 * (y2  y1)  2 * (x2  x1) * (y[i+1]  y[i]) Expression (y[i+1]  y[i]) can only store values 0 and 1 and we can write modified expressions to compute decision: d[i] < 0 => d[i+1] = d[i] + 2 * (y2  y1) d[ ] >= 0 => d[i+1] = d[i] + 2 * (y2  y1)  2 * (x2  x1) Last thing you need to know is the decision for the first step and here is: d[1] = 2 * (y2  y1)  (x2  x1) So all you must do is check the decision for it's sign, if's negative, move the y coordinate, compute decision for the next step and test if you've drawn the last point. If not, do it all again. For the other direction that 0x1 (or any other one and choose variants with this comparison) and if it's true then there's no change at the start and end points.  decision equations should be divided by 4 (two rotations) so you can store it in an 8bit register, and no precision's lost. Uf, that's all. My routine is available (I hope) on the HighTech project (http://hightech.da.ru). I hope you understand what I've tryed to say here. If not feel free to write to me. Bye! mike/zt/ctl jurica@sknet.sk
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