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

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

7. Which point is closer? y = mx + b A yi -1+1 yi -1 B ideal line xi xi -1 error(A) = (yi -1 + 1) – y* error(B) = y* - (yi -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(xi – x0) + 2(y0 – yi -1) -1 • Remember: m = Δy/Δx (slope of line) • Multiply through by Δx (to avoid fractions) • Let di = Δx( error(B) – error(A) ) • Rule is: choose B if and only if di < 0

9. Computing di+1 from di di+1 = 2(Δy)(xi+1 – x0) +2(Δx)(y0 – yi) – Δx di = 2(Δy)(xi – x0) + 2(Δx)(y0 – yi-1) – Δx The difference can be expressed as: di+1 = di + 2(Δy)(xi+1 – xi) – 2(Δy)(yi – yi-1) Recognize that xi+1 – 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(Δy) - Δx • And, at each step thereafter: if ( d i < 0 ) { di+1 = di + 2(Δy); yi+1 = yi; } else { di+1 = di + 2(Δy-Δx); yi+1 = yi + 1; } xi+1 = 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 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 Y 0 XLO XHI 1 2 8 8 3 7 9 4 6 10 5 5 11 6 7 12 7 9 13 8 11 14 10 13 15 11 15 16 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