--- title: "Bresenham's algorithm for all slopes" subject: "Computer Graphics" chapter: 2 key: "semester_5/computer_graphics:bresenham-all-slopes" source: https://www.bctengineer.com/q/semester_5/computer_graphics/bresenham-all-slopes site: BCT Engineering --- # Bresenham's algorithm for all slopes Computer Graphics · Chapter 2: Scan-Conversion - Asked in: 2067 Mangsir (old course), 10 marks - Video: [Bresenham's Line Drawing Algorithm With Example](https://www.youtube.com/watch?v=GKnzwTMmO1Y) — QuickCS - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/computer_graphics/bresenham-all-slopes - Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-2 ## Question Write down Bresenham's line drawing algorithm for drawing a straight line with consideration of all the slope categories. ## Short answer All-slope Bresenham uses step signs sx and sy. If |Δy| ≤ |Δx|, x is driving axis, p0 = 2Δy − Δx: x += sx; if p<0 p+=2Δy, else y += sy, p+=2Δy−2Δx. If |Δy| > |Δx|, y drives, p0 = 2Δx − Δy: y += sy; if p<0 p+=2Δx, else x += sx, p+=2Δx−2Δy. ## Answer Bresenham's algorithm works with one coordinate as the unit-step axis (the "driving" axis) and the decision parameter chooses the other coordinate. For a line (x1,y1) to (x2,y2), let Δx = |x2 − x1|, Δy = |y2 − y1|, sx = sign(x2 − x1), sy = sign(y2 − y1). #### Slope categories | Slope | Driving axis | Other coordinate | p0 | |---|---|---|---| | 0 < m < 1 | x (+1) | y increases by 1 or stays | 2Δy − Δx | | m > 1 | y (+1) | x increases by 1 or stays | 2Δx − Δy | | −1 < m < 0 | x (+1) | y decreases by 1 or stays | 2Δy − Δx | | m < −1 | y (−1) | x increases by 1 or stays | 2Δx − Δy | | m = 0 | x | y stays | horizontal line | | m = ∞ | y | x stays | vertical line | #### Algorithm ```text 1. Input end points (x1,y1), (x2,y2). 2. Δx = |x2 − x1|, Δy = |y2 − y1|, sx = sign(x2−x1), sy = sign(y2−y1). 3. If Δy ≤ Δx (|m| ≤ 1): case A, else case B. Case A (x is the driving axis): p = 2Δy − Δx; x = x1; y = y1; plot (x, y) repeat Δx times: x = x + sx if p < 0: p = p + 2Δy else: y = y + sy; p = p + 2Δy − 2Δx plot (x, y) Case B (y is the driving axis): p = 2Δx − Δy; x = x1; y = y1; plot (x, y) repeat Δy times: y = y + sy if p < 0: p = p + 2Δx else: x = x + sx; p = p + 2Δx − 2Δy plot (x, y) ``` #### C code ```c void bres(int x1,int y1,int x2,int y2){ int dx=abs(x2-x1), dy=abs(y2-y1); int sx=(x2>x1)?1:-1, sy=(y2>y1)?1:-1; int x=x1, y=y1, i, p; putpixel(x,y,WHITE); if(dy<=dx){ /* |m| <= 1 */ p=2*dy-dx; for(i=0;i 1 */ p=2*dx-dy; for(i=0;i 1 in Bresenham's algorithm, which coordinate is the unit-step (driving) axis? A. y, stepping by 1 each time while the decision parameter chooses x B. x, stepping by 1 each time C. Both x and y step by 1 D. The axis with the smaller coordinate value Answer: A. For steep lines y changes every step, and x either stays or changes by 1. (Key point: Driving axis) 2. For |m| <= 1 (case A) the initial decision parameter is: A. p0 = 2Δx − Δy B. p0 = Δy − 2Δx C. p0 = 2Δy − Δx D. p0 = 2(Δy − Δx) Answer: C. Case A uses x as the driving axis, giving p0 = 2Δy − Δx; steep lines swap the roles. (Key point: Initial parameter) 3. In case A, what happens when p >= 0? A. y stays and p = p + 2Δy B. y = y + sy and p = p + 2Δy − 2Δx C. y = y + sy and p = p + 2Δy D. x stays and p = p + 2Δx − 2Δy Answer: B. A non-negative p means the next pixel is the diagonal one, so y moves and p is increased by 2Δy − 2Δx. (Key point: Update when p >= 0) 4. What is the purpose of sx = sign(x2−x1) and sy = sign(y2−y1)? A. They swap the end points so the line goes left to right B. They scale the decision parameter C. They decide whether the line is horizontal D. They make the same code draw lines in any direction, covering all slope categories Answer: D. The signs give the direction of each step, so positive/negative slopes and any quadrant are handled. 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