Drawing Lines The Bresenham Algorithm for drawing lines

Drawing Lines The Bresenham Algorithm for drawing lines and filling polygons

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

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

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

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

Integer endpoints ΔY = Y 1 – Y 0 ΔX = X 1 – X 0 (X 1, Y 1) 0 < ΔY < ΔX ΔY (X 0, Y 0) ΔX slope = ΔY/ΔX

Which point is closer? A yi -1+1 yi -1 B xi -1 y = mx + b ideal line xi error(A) = (yi -1 + 1) – y* error(B) = y* - (yi -1)

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) = 2 m(xi – x 0) + 2(y 0 – 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

Computing di+1 from di • • • di+1 = 2(Δy)(xi+1 – x 0) +2(Δx)(y 0 – yi) – Δx di = 2(Δy)(xi – x 0) + 2(Δx)(y 0 – 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)

How does algorithm start? • At the outset we start from point (x 0, y 0) • Thus, at step i =1, our formula for di is: d 1 = 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;

‘bresdemo. cpp’ • The example-program is on class website: http: //nexus. cs. usfca. edu/~cruse/cs 686/ • 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

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

Triangle Illustration

Non-Convex Polygons

Bucket-Sort Y 0 1 2 3 4 5 6 7 8 10 11 12 13 6 5 XLO 8 7 7 XHI 8 9 10 11 12 9 13 11 14 13 15 15 16 17 17

Handling Corners

Legacy Software • • • We have a polygon-fill demo: ‘polyfill. cpp’ You can find it now on our class website But it was written in 1997 for MS-DOS We’d like to run it on our Linux system But it will need a number of modifications This is our first programming assignment: to “adapt” this obsolete application so it can execute on a contemporary platform

What are some issues? • • GNU compiler enforces stricter standards Sizes of some data-types are now larger Physical VRAM now must be “mapped” I/O instructions now require permissions Real-Mode addresses must be converted Not all the header-files are still supported Interface to CPU registers is a bit different

Converting addresses • • TURBO-C++ allowed use of ‘far’ pointers Use a macro to create a pointer to VRAM: uchar *vram = MK_FP( 0 x. A 000, 0 x 0000 ); Real-mode address has two components: (16 -bit segment and 16 -bit offset) • For Linux we convert to a 32 -bit address: uchar *vram = (uchar*)0 x 000 A 0000; • Formula: address = 16*segment + offset

Hardware Issues? • • • Pentium CPUs are “backward compatible” BIOS firmware can use Virtual-8086 mode SVGA still supports older graphics modes The keyboard’s mechanism is unchanged Feasibility test: if we will “boot” MS-DOS, we can easily run the ‘polyfill’ application • So it’s just a software problem to give our MS-DOS program a new life under Linux!