Drawing Lines

1
Drawing Lines

• The Bresenham Algorithm for drawing lines and
  filling polygons

2
Plotting a line-segment

• Bresenham published algorithm in 1965
• It was originally to be used with a plotter
• It adapts well to raster scan conversion
• It uses only integer arithmetic operations
• It is an iterative algorithm each step is
  based on results from the previous step
• The sign of an error term governs the choice
  among two alternative actions

3
Scan conversion
The actual line is comprised of points drawn from
a continuum, but it must be approximated using
pixels from a discrete grid.
4
The various cases

• Horizontal or vertical lines are easy cases
• Lines that have slope 1 or -1 are easy, too
• Symmetries leave us one remaining case 0 lt
  slope lt 1
• As x-coodinate is incremented, there are just two
  possibilities for the y-coordinate (1)
  y-coordinate is increased by one or (2)
  y-coordinate remains unchanged

5
0 lt slope lt 1
Y-axis
X-axis
y increases by 1
y does not change
6
Integer endpoints
?Y Y1 Y0 ?X X1 X0
(X1,Y1)
0 lt ?Y lt ?X
?Y
(X0,Y0)
?X
slope ?Y/?X
7
Which point is closer?
y mx b
A
yi -11
yi -1
B
ideal line
xi -1
xi
error(A) (yi -1 1) y error(B) y - (yi
-1)
8
The Decision Variable

• Choose B if and only if error(B)lterror(A)
• Or equivalently error(B) error(A) lt 0
• Formula error(B) error(A) 2m(xi x0)
  2(y0 yi -1) -1
• Remember m ?y/?x (slope of line)
• Multiply through by ?x (to avoid fractions)
• Let di ?x( error(B) error(A) )
• Rule is choose B if and only if di lt 0

9
Computing di1 from di

• di1 2(?y)(xi1 x0) 2(?x)(y0 yi) ?x
• di 2(?y)(xi x0) 2(?x)(y0 yi-1) ?x
• The difference can be expressed as
• di1 di 2(?y)(xi1 xi) 2(?y)(yi yi-1)
• Recognize that xi1 xi 1 at every step
• And also yi yi-1 will be either 0 or 1
• (depending on the sign of the previous d)

10
How does algorithm start?

• At the outset we start from point (x0,y0)
• Thus, at step i 1, our formula for di is
  d1 2(?y) - ?x
• And, at each step thereafter
• if ( d i lt 0 ) di1 di 2(?y) yi1 yi
• else di1 di 2(?y-?x) yi1 yi 1
• xi1 xi 1

11
bresdemo.cpp

• The example-program is on class website
• http//nexus.cs.usfca.edu/cruse/cs686/
• It draws line-segments with various slopes
• The Michener algorithm (for a circle-fill) is
  also included, for comparative purposes
• Extreme slopes (close to zero or infinity) are
  not displayed in this demo program
• They can be added by you as an exercise

12
Filling a triangle or polygon

• The Bresenhams method can be adapted
• But an efficient data-structure is needed
• All the sides need to be handled together
• We let the y-coordinate steadily increment
• For sides which are nearly horizontal the
  x-coordinates can change by more than 1

13
Triangle Illustration
14
Non-Convex Polygons
15
Bucket-Sort
Y
0
XLO
XHI
1
2
8
8
3
7
9
4
6
10
5
11
5
6
12
7
7
13
9
8
14
11
10
15
13
11
16
15
12
17
17
13
16
Handling Corners
17
In-class exercises

• For the bresdemo.cpp program
• Supply a function that tests the capability of
  the Breshenham line-drawing algorithm to draw
  lines having the full range of slopes
• For the fillpoly.cpp program
• Modify the program code so that it will work with
  polygons having more than three sides