1. computer graphics
2D graphics

2. contents Lines Circles DDA algorithm Bresenham’s algorithm
The midpoint algorithm
Circles
Symmetry algorithm

3. line drawing

4. a line can be expressed as

5. Digital Differential Analyzer Algorithm (|m| ≤ 1)
scan conversion

6. 要点:增量计算,变乘法为加法

7. Example 2.1
Using the DDA line drawing algorithm, determine the pixels along a line segment
that goes from [3, 4]T to [8, 6]T.

8. current x
current y
floor(y+0.5)
next y 3 4
4.4
4.8 5
5.2 6
5.6 7 8
6.4

9. when |m| > 1

10. another solution current x current y floor(y+0.5) next y 3 4 4.4 4.8 5
5.2 6
5.6 7 8
6.4

11. special case

12. generalization
general case

14. Bresenham’s algorithm
replace floating point numbers with integer numbers

16. Example 2.3
You are asked to draw a line segment between the points [1, 1] and [4, 3]. Use Bresenham’s line drawing algorithm to specify the locations of pixels that should approximate the line.

17. The Midpoint Algorithm

18. implicit equation of curve
implicit equation of line

19. derivation of incremental expression
test midpoint (x+1,y+1/2) with f(x,y)
f(x+1,y+1/2) = a(x+1)+b(y+1/2)+c
if f(x+1,y+1/2)>0, P2 is closer
if f(x+1,y+1/2)<0, P1 is closer
if f(x+1,y+1/2)=0, equal closer, chooe any

20. if P1 is selected, next midpoint to be tested is (x+2,y+1/2)
f(x+2,y+1/2) = a(x+2)+b(y+1/2)+c = a(x+1)+b(y+1/2)+c+a = f(x+1,y+1/2)+a
if P2 is selected, next midpoint to be tested is (x+2,y+3/2)
f(x+2,y+3/2) = a(x+2)+b(y+3/2)+c = a(x+1)+b(y+1/2)+c+a+b = f(x+1,y+1/2)+a+b

21. initial value
f(x0+1,y0+1/2) = a(x0+1)+b(y0+1/2)+c = ax0+by0+c+a+b/2 = f(x0,y0)+a+b/2=a+b/2

24. Example 2.4
You are asked to draw a line segment between the points [1, 1] and [4, 3]. Use the midpoint line drawing algorithm to specify the locations of pixels that should approximate the line.

25. circle drawing
x=x0+rcos(θ)
y=y0+rsin(θ)

26. Two-Way Symmetry Algorithm

29. Four-Way Symmetry Algorithm

31. Eight-Way Symmetry Algorithm

33. The Midpoint Algorithm

34. derivation of the incremental expression (ref. p.28)
implicit equation of the circle centered at origin (0,0)
f(x,y)=x2+y2-r2=0