Bresenhams Line Generation Algorithm 23-03-2020

Instruction: First of all carefully watch video lecture then read theory.
Video Lecture: Bresenhams Line Algorithm 

This algorithm is used for scan converting a line. It was developed by Bresenham. It is an efficient method because it involves only integer addition, subtractions, and multiplication operations. These operations can be performed very rapidly so lines can be generated quickly.

In this method, next pixel selected is that one who has the least distance from true line.

The method works as follows:

Assume a pixel P₁'(x₁',y₁'),then select subsequent pixels as we work our may to the night, one pixel position at a time in the horizontal direction toward P₂'(x₂',y₂').

Once a pixel in choose at any step

The next pixel is

1. Either the one to its right (lower-bound for the line)
2. One top its right and up (upper-bound for the line)

The line is best approximated by those pixels that fall the least distance from the path between P₁',P₂'.

To chooses the next one between the bottom pixel S and top pixel T.

If S is chosen            

                               We have x_i+1=x_i+1   and       y_i+1=y_i

If T is chosen                       

                                We have x_i+1=x_i+1    and      y_i+1=y_i+1

The actual y coordinates of the line at  x = x_i+1is           

y=mx_i+1+b

The distance from S to the actual line in y direction                       

                                                      s = y-y_i

The distance from T to the actual line in y direction                       

t = (y_i+1)-y.

Now consider the difference between these 2 distance values s - t.

When (s-t) <0 ⟹ s < t. The closest pixel is S. 

When (s-t) ≥0 ⟹ s < t. The closest pixel is T.

This difference is           

 s-t = (y-yi)-[(yi+1)-y] 

= 2y - 2yi -1

                            

Substituting m by  and introducing decision variable                

d_i=△x (s-t)                                   

d_i=△x (2  (x_i+1)+2b-2y_i-1)                                     

=2△xy_i-2△y-1△x.2b-2y_i△x-△x                                                      

d_i=2△y.x_i-2△x.y_i+c

Where c= 2△y+△x (2b-1).

We can write the decision variable d_i+1 for the next slip on                    

d_i+1=2△y.x_i+1-2△x.y_i+1+c                                   

d_i+1-d_i=2△y.(x_i+1-x_i)- 2△x(y_i+1-y_i)

Since x_(i+1)=x_i+1,we have                     

d_i+1+d_i=2△y.(x_i+1-x_i)- 2△x(y_i+1-y_i).

If chosen pixel is at the top pixel T        

(i.e., d_i≥0)⟹ y_i+1=y_i+1                                   

d_i+1=d_i+2△y-2△x.

If chosen pixel is at the bottom pixel T      

(i.e., d_i<0)⟹ y_i+1=y_i                                                              

d_i+1=d_i+2△y.

Finally, we calculate d₁                       

d₁=△x[2m(x₁+1)+2b-2y₁-1]                       

d₁=△x[2(mx₁+b-y₁)+2m-1].

Since 
mx₁+b-y_i=0     and                      m = , we have                                     

d₁=2△y-△x

Advantage:

1. It involves only integer arithmetic, so it is simple.

2. It avoids the generation of duplicate points.

3. It can be implemented using hardware because it does not use multiplication and division.

4. It is faster as compared to DDA (Digital Differential Analyzer) because it does not involve floating point calculations like DDA Algorithm.

Disadvantage:

1. This algorithm is meant for basic line drawing only Initializing is not a part of Bresenham's line algorithm. So to draw smooth lines, you should want to look into a different algorithm.