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 < slope < 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 < slope < 1 y increases by 1y does not change X-axis Y-axis

6. Integer endpoints ΔXΔX ΔYΔY slope = ΔY/ΔX (X0,Y0) (X1,Y1) ΔY = Y1 – Y0 ΔX = X1 – X0 0 < ΔY < ΔX

7. Which point is closer? A B x i -1 xixi y = mx + b error(A) = (y i -1 + 1) – y* error(B) = y* - (y i -1 ) ideal line y i -1 y i -1 +1

8. The Decision Variable Choose B if and only if error(B)<error(A) Or equivalently: error(B) – error(A) < 0 Formula: error(B) – error(A) = 2m(x i – x 0 ) + 2(y 0 – y i -1 ) -1 Remember: m = Δy/Δx (slope of line) Multiply through by Δx (to avoid fractions) Let d i = Δx( error(B) – error(A) ) Rule is: choose B if and only if d i < 0

9. Computing d i+1 from d i d i+1 = 2(Δy)(x i+1 – x 0 ) +2(Δx)(y 0 – y i ) – Δx d i = 2(Δy)(x i – x 0 ) + 2(Δx)(y 0 – y i-1 ) – Δx The difference can be expressed as: d i+1 = d i + 2(Δy)(x i+1 – x i ) – 2(Δy)(y i – y i-1 ) Recognize that x i+1 – x i = 1 at every step And also: y i – y i-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 (x 0,y 0 ) Thus, at step i =1, our formula for d i is: d 1 = 2(Δy) - Δx And, at each step thereafter: if ( d i < 0 ) { d i+1 = d i + 2(Δy); y i+1 = y i ; } else { d i+1 = d i + 2(Δy-Δx); y i+1 = y i + 1; } x i+1 = x i + 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 Bresenham’s 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 0 1 2 3 4 5 6 7 8 10 11 12 88 79 610 11 12 13 14 15 16 5 7 9 11 13 15 Y XLOXHI 13 17

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