1. Raster conversion algorithms for line and circle
Introduction
- Pixel addressing
- Primitives and attributes
Line drawing algorithms
- DDA
- Bresenham
Circle generating algorithms
- Direct method
- Bresenham algorithm

2. Pixel addressing in raster graphics

3. Raster conversion algorithms: requirements
visual accuracy
spatial accuracy
speed

4. Line drawing algorithms

5. Line – raster representation

6. DDA ( Digital Differential Algorithm )

7. DDA ( Digital Differential Algorithm )

8. DDA ( Digital Differential Algorithm )

9. Digital Differential Algorithm
input line endpoints, (x0,y0) and (xn, yn)
set pixel at position (x0,y0)
calculate slope m
Case |m|≤1: repeat the following steps until (xn, yn) is reached:
yi+1 = yi + y/ x
xi+1 = xi + 1
set pixel at position (xi+1,Round(yi+1))
Case |m|>1: repeat the following steps until (xn, yn) is reached:
xi+1 = xi + x/ y
yi+1 = yi + 1
set pixel at position (Round(xi+1), yi+1)

10. Bresenham's line algorithm
y = mx + b
Bresenham's line algorithm
y = mx + b
y = m(x+1) + b d2 d1 y x
x+1

11. Bresenham's line algorithm (slope ≤ 1)
input line endpoints, (x0,y0) and (xn, yn)
calculate x = xn - x0 and y = yn - y0
calculate parameter p0 = 2 y - x
set pixel at position (x0,y0)
repeat the following steps until (xn, yn) is reached:
if pi < 0
set the next pixel at position (xi +1, yi )
calculate new pi+1 = pi + 2 y
if pi ≥ 0
set the next pixel at position (xi +1, yi + 1 )
calculate new pi+1 = pi + 2(y - x)

12. DDA versus Bresenham’s Algorithm
DDA works with floating point arithmetic
Rounding to integers necessary
Bresenham’s algorithm uses integer arithmetic
Constants need to be computed only once
Bresenham’s algorithm generally faster than DDA

13. Circle generating algorithms
Direct
Polar coordinate based
Bresenham’s

14. Direct circle algorithm
Cartesian coordinates
Circle equation:
( x - xc )2 + ( y - yc )2 = r2
Step along x axis from xc - r to xc + r and calculate
y = yc ±  r2 - ( x - xc )2

15. Polar coordinates Polar coordinate equation x = xc + r cos
y = yc + r sin
step through values of  from 0 to 2π

16. Optimisation and speed-up
Symmetry of a circle can be used
Calculations of point coordinates only for a first one-eighth of a circle
(-x,y)
(x,y)
(-y,x)
(y,x)
(-y,-x)
(y,-x)
(-x,-y)
(x,-y)

17. Bresenham’s circle algorithm
1. Input radius r
2. Plot a point at (0, r)
3. Calculate the initial value of the decision parameter as p0 = 5/4 – r ≈ 1 – r

18. 4. At each position xk, starting at k = 0, perform the following test:
if pk < 0
plot point at (xk +1, yk)
compute new pk+1 = pk + 2xk+1 + 1
else
plot point at (xk + 1, yk – 1)
compute new pk+1 = pk + 2xk – 2yk+1
where xk+1 = xk + 1 and yk+1 = yk - 1

19. 5. Determine symmetry points in the other seven octants and plot points
6. Repeat steps 4 and 5 until x  y