Download presentation

Presentation is loading. Please wait.

Published byRoberto Petitt Modified over 2 years ago

1
Scan Conversion Algorithms www.smartphonetech.org

2
Line Drawing Description : Given the specification of a straight line, find the collection of addressable pixels which closely approximate this line. Goals : Straight lines should appear straight Lines should start and end accurately Lines should be drawn as quickly as possible. Line drawing algorithms: DDA (Digital Differential Analyzer) Algorithm Bresenham’s Line Algorithm www.smartphonetech.org

3
DDA Algorithm Consider the slope-intercept equation of a line y = m. x + b where m slope & b y-intercept Given two end-points (x 1,y 1 ) & (x 2,y 2 ) m = (y 2 -y 1 ) / (x 2 -x 1 ) b = y1 – m. x 1 Now, given any x-interval or y-interval we can calculate the corresponding y-interval or x- interval through the slope ∆y = m. ∆x ∆x = ∆y / m Thus considering an interval of 1 pixel, we have y k+1 = y k + m x k+1 = x k + 1/m If m > 1, we should calculate x otherwise we should calculate y www.smartphonetech.org

4
void lineDDA(int x1, int y1, int x2, int y2){ int dx = x2 –x1, dy = y2 –y1, steps, k; float x = x1, y = y1, xIncr, yIncr; if(abs(dx) > abs(dy)) setps = abs(dx); else steps = abs(dy); xIncr = dx / steps; yIncr = dy / steps; setPixel(Round(x), Round(y)); for(k = 0; k

5
Also called Midpoint Line Algorithm. Considering slope and end point ordering, a line can belong to one of the eight octants. Given the current pixel, which one do we chose next : E or NE (for octant 1) is based on the Midpoint. If the line goes above the midpoint, M then NE is chosen otherwise E is chosen as the next pixel. Let us consider two forms of a Line equation y = (dy / dx). x + B F(x,y) = a. x + b. y + c = 0 Thus we get a = dy, b = - dx & c = B. dx ------- (1) Again the midpoint M is below the line if F(M) > 0 otherwise M is above the line. Bresenham’s Line Algorithm E NE M 1 23 4 5 6 7 8 www.smartphonetech.org

6
The value of F(M) can be calculated in an incremental way as follows: Consider a decision variable d = F(x p +1, y p + ½) The decision variable helps to chose E or NE. If d <= 0 chose E otherwise chose NE. The next decision variable is calculated as follows: Set d old = d Case E is chosen: (current point is x p + 1, y p ) d new = F(x p + 2, y p + ½) = a. x p +2a + b. y p + b/2 +c (∆d) E = d new – d old = a = dy Case NE is chosen: (current point is x p + 1, y p + 1) d new = F(x p + 2, y p + 3/2) = a. x p +2a + b. y p + 3b/2 +c (∆d) NE = d new – d old = a + b = dy – dx Bresenham’s Line Algorithm (cont.) x p +1,y p E NE M x p, y p x p +1,y p +1 x p +1,y p +1/2 www.smartphonetech.org

7
The initial decision value of the decision variable is calculated as d start = F(x 0 +1, y 0 +1/2) = a. x 0 + a + b. y 0 + b/2 + c = a + b/2 = dy – dx/2 = ½ (2dy - dx) To avoid fractional computation we can assume decision variable values for 2.F(x,y). Thus d start = 2dy – dx (∆d) E = 2dy (∆d) NE = 2(dy –dx) Thus we can plot the successive points as follows while(x < x1){ if(d<=0) /*chose E */ d = d + (∆d) E ; else{ /* chose NE */ d = d + (∆d) NE ; y = y + 1; } x = x + 1; setPixel(x, y); } Bresenham’s Line Algorithm (cont.) www.smartphonetech.org

8
Bresenham’s Line Algorithm (cont.) What’s about lines belong to other octants? 1 23 4 5 6 7 8 OctantChange 1None 2Swap x & y 3Plot from P 1 to P 0 ; Swap x & y; Use y = y - 1 4Plot from P 1 to P 0 ; Use y = y - 1 5Plot from P 1 to P 0 6Plot from P 1 to P 0 ; Swap x & y 7Swap x & y; Use y = y - 1 8Use y = y - 1 www.smartphonetech.org

9
Due to symmetry in circle, it needs to plot only octant 2 Others can be plotted though mirror effect. Thus setCirclePixel(x c, y c, x, y){ setPixel(x c + x, y c + y); 2 setPixel(x c - x, y c + y); 3 setPixel(x c - y, y c + x); 4 setPixel(x c - y, y c - x); 5 setPixel(x c - x, y c - y); 6 setPixel(x c + x, y c - y); 7 setPixel(x c + y, y c - x); 8 setPixel(x c + y, y c + x); 1 } Now for octant 2 choice is between E and SE and the decision function is F(x, y) = x 2 +y 2 –R 2 = 0; Midpoint Circle Algorithm 1 23 4 5 6 7 8 E SE M www.smartphonetech.org

10
The decision variable now becomes d = F(x p + 1, y p – ½) (∆d) E = F(x p + 2, y p – ½) - F(x p + 1, y p – ½) = 2 x p + 3 (∆d) SE = F(x p + 2, y p – 3/2) - F(x p + 1, y p – ½) = 2 x p - 2y p + 5 d start = F(x0 + 1, y0 – ½) = F(1, R – ½) = 5/4 – R To get rid of the fraction, let h = d – ¼ => h start = 1 – R Since h is initialized and incremented by integers we can use h instead of d for the decision. Midpoint Circle Algorithm (cont.) E SE M www.smartphonetech.org

11
The algorithm x = 0; y = R; h = 1 – R; setCirclePixel(x c, y c, x, y); while(y > x){ if(h <= 0) /* chose E */ h = h + 2x + 3; else{ /* chose SE */ h = h + 2(x – y) + 5; y = y – 1; } x = x + 1; setCirclePixel(x c, y c, x, y); } Midpoint Circle Algorithm (cont.) www.smartphonetech.org

12
Midpoint Ellipse Algorithm a b -b -a Slope = -1 R2 R1 E SE M S M setEllipsePixel(x c, y c, x, y){ setPixel(x c + x, y c + y); 1 setPixel(x c - x, y c + y); 2 setPixel(x c - x, y c - y); 3 setPixel(x c +x, y c - y); 4 } www.smartphonetech.org

13
Let us consider the following ellipse equation F(x, y) = b 2 x 2 + a 2 y 2 –a 2 b 2 = 0 Slope of the equation at a point (x, y) is dy/dx = - b 2 x/a 2 y The magnitude of the slope is 0 in (0, b) and gradually increments and finally becomes infinity at (a, 0) Thus the condition for R1 is b 2 x = a 2 y Analysis for R1: The decision variable becomes d = F(x p + 1, y p – ½) (∆d) E = F(x p + 2, y p – ½) - F(x p + 1, y p – ½) = b 2 (2x p + 3) (∆d) SE1 = F(x p + 2, y p – 3/2) - F(x p + 1, y p – ½) = b 2 (2x p + 3) -2a 2 (y p - 1) d start = F(1, b – ½ ) = b 2 + a 2 (1/4 – b) Analysis for R2: The decision variable becomes d = F(x p + ½, y p – 1) (∆d) SE2 = F(x p + 3/2, y p – 2) - F(x p + ½, y p – 1) = 2b 2 (x p + 1) – a 2 (2y p - 3) (∆d) S = F(x p + ½, y p – 2) - F(x p + ½, y p – 1) = - a 2 (2y p - 3) Midpoint Ellipse Algorithm (cont.) a b -b -a E SE M S M www.smartphonetech.org

14
The Algorithm: sa = sqr(a); sb = sqr(b); d = b 2 + a 2 (1/4 – b); setEllipsePixel(x c, y c, 0, b); while(b 2.(x+1) < a2.(y – ½ )){ /* For R1 */ if(d<0) /* chose E */ d += sb. ((x << 1)+3); else{ /* chose SE */ d += sb. ((x << 1)+3) – sa. ((y<<1) - 2) ; y--; } x++; setEllipsePixel(x c, y c, x, y); } Midpoint Ellipse Algorithm (cont.) d = sb. sqr(x+1/2) + sa. sqr(y-1) – sa. sb; while(y>0){ /* For R2 */ if(d<0){ /* chose SE */ d += sb. ((x++)+2) – sa. ((y<<1)-3); x++; }else{ d -= sa. ((y<<1)-3); } y--; setEllipsePixel(x c, y c, x, y); } www.smartphonetech.org

15
Thank You www.smartphonetech.org www.smartphonetech.org

Similar presentations

OK

CS 4731: Computer Graphics Lecture 20: Raster Graphics Part 1 Emmanuel Agu.

CS 4731: Computer Graphics Lecture 20: Raster Graphics Part 1 Emmanuel Agu.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on infosys company history Ppt on online banking security Ppt on ovarian cycle Download ppt on conservation of water resources Ppt on ashoka the great emperor Ppt on media research institute Ppt on virtual laser keyboard Ppt on photosynthesis and respiration Ppt on 5 elements of earth Ppt on db2 architecture