Editing Line drawing algorithm (section)

=== Simple Methods ===
The simplest method of drawing a line involves directly calculating pixel positions from a line equation. Given a starting point <math>(x_1, y_1)</math> and an end point <math>(x_2, y_2)</math>, points on the line fulfill the equation <math>y = m(x-x_1) + y_1</math>, with <math>\textstyle m = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1}</math> being the [[slope]] of the line. The line can then be drawn by evaluating this
equation via a simple loop, as shown in the following pseudocode:

 dx = x2 − x1
 dy = y2 − y1
 m = dy/dx
 '''for''' x '''from''' x1 '''to''' x2 '''do'''
     y = m × (x − x1) + y1
     plot(x, y)

Here, the points have already been ordered so that <math>x_2 > x_1</math>.

This algorithm is unnecessarily slow because the loop involves a multiplication, which is significantly slower than addition or subtraction on most devices. A faster method can be achieved by viewing the Difference between two consecutive steps:

:{|
|-
| <math>y_{i+1} - y_i \!</math>
| <math>= (m(x_{i+1}-x_1) + y_1) - (m(x_i-x_1) + y_1) \!</math>
|-
|  || <math>= m(x_{i+1}-x_i) \!</math>
|-
|  || <math>= m \!</math>.
|}

Therefore, it is enough to simply start at the point <math>(x_1, y_1)</math> and then increase <math> y </math> by <math> m </math> once on every iteration of the loop. This algorithm is known as a [[Digital differential analyzer (graphics algorithm) | Digital differential analyzer]].

Because rounding <math> y </math> to the nearest whole number is equivalent to rounding <math> y+0.5 </math> down, rounding can be avoided by using an additional control variable that is initialized with the value 0.5. <math> m </math> is added to this variable on every iteration. Then, if this variable exceeds 1.0, <math> y </math> is incremented by 1 and the control variable is decremented by 1. This allows the algorithm to avoid rounding and only use integer operations. However, for short lines, this faster loop does not make up for the expensive division <math>\textstyle m = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1}</math>, which is still necessary at the beginning.

These algorithm works just fine when <math>dx \geq dy</math> (i.e., slope is less than or equal to 1), but if <math>dx < dy</math> (i.e., slope greater than 1), the line becomes quite sparse with many gaps, and in the limiting case of <math>dx = 0</math>, a division by zero exception will occur.