Computer Graphics Line drawing Algorithms soham jisitgmail com

Computer Graphics: Line drawing Algorithms soham. jisit@gmail. com 1/23/2022 1

Draw a Straight Line in MS-Paint Now increase Line Width

Is It a Stair-step or what! Pixels are discrete, they belong to Computer’s Display Unit Why did these happen? • It is not a continuous Line Coordinates are Continuous

Into The Straight Line c intercept on Y-axis

q m=0 the Line is parallel to X-axis q m=∞ the Line is parallel to Y-axis q m=1 the line is inclined at 45 degree Note: See-Saw Steps are less in 3 cases above

Because, Pixels are Discrete ∴ Let is Plot now! x y 0 1 2 3 4 Decision? 2 Exactly 2 3. 732 5. 464 7. 196 8. 928 3. 8 (Impossible) or 4? 5 or 5. 5 (Impossible) 7. 2 (Impossible) or 7 9 Or 9. 3 (Impossible) Thus, plotting on Graph is not accurate For, pixels are discrete while points are not!

Line drawing Algorithm-1: DDA

Case -1 : |m| ≤ 1 Switch roles of x and y for |m| >1

DDA: Easier and Generic Approach Plotting a Pixel Differs across Programming languages

Do Yourself: Deduction from DDA Algorithm

Implementing DDA: A sample Turbo C-Code This is need for Graphics in Turbo C This is need for mathematical functions Common to standard Graphics programs Ø initgraph(int* graphics. Driver , int* graphics. Mode, char* driver. Path. As. String): starts graphics system Ø DETECT: is a macro whose value is 0: indicates to detect graphics system automatically soham. jisit@gmail. com 1/23/2022 12

Implementing DDA: A sample Turbo C-Code This is need for Graphics in Turbo C This is need for mathematical functions putpixel(int x. Coord, int y. Coord, int color. As. Int) soham. jisit@gmail. com 1/23/2022 13

Draw back of DDA Algorithm 1. Stair –case Output is not optimized considerably 2. Ceil() or Floor() : Rounding-off is not dynamically decided 3. Floating point Arithmetic is an overhead 4. Rounding-off is another computational overhead, too! It also yields low accuracy soham. jisit@gmail. com 1/23/2022 14

Bresenham’s Line Drawing Algorithm Given two End Points (X 1, Y 1) and (X 2, Y 2) Assumption: X 1 < X 2 ∴ (X 1, Y 1) is to the left y= m*x + c y. K= m*x K + c Increment of X happens from left to right ∴ x. K+1 = x K +1 But YK+1 =? Should be: y. K+1= m*(x. K+1) + c But it is often a fraction!!! So, YK+1 is either YK or (YK +1) Original value of YK+1 But it is neither y. K nor (y. K +1) D 1 = y. True – y. K D 2 = (YK +1) – y. True IF (D 1 < D 2 ) YK+1 = YK Else YK+1 = YK + 1 Bresenhams’ dynamically optimizes it 1/23/2022 15

Bresenhams Line Drawing Algorithm D 1 = y True (K+1) – y. K D 2 = (YK +1) – y True(K +1) y= m*x + c y True-(K+1)= m* x K+1 + c 1/23/2022 16

Bresenham’s Line Drawing Algorithm p 0 =? y= m*x + c y True-(K+1)= m* x K+1 + c 1/23/2022 This is the foundation of Bresenham’s Algorithm 17

Bresenhams Line Drawing Algorithm This is Bresenhem’s Algorithm 1/23/2022 18

Implementation in Turbo C When do you get p. K = 0 ? Browse any study material (maybe, on Internet ) and tell this old teacher “How Bresenham’s Algorithm is different from DDA Algorithm” soham. jisit@gmail. com 1/23/2022 19

Everything is not Straight: Circles Ahead x 2 + y 2= r 2 Example: (3, 4) is a point on a circle whose center is the origin (0, 0) and radius r= 5 unit x 2 + y 2= r 2 ∵ 3 2 + 4 2= 5 2 Again, (4, 3) is another point on the same circle! ∵ 42 + 32= 52 ∴ Interestingly, the same Quadrant, there exists to isometric points, (a, b) and (b, a) on the same Circle! ∴ There is 8 way Isometry in all 4 Quadrants This is very useful to ease drawing a Circle Generate points for only one Quadrant And use 8 -way Isometry for rest 3 quadrants soham. jisit@gmail. com 8 way Isometry in all 4 Quadrants 1/23/2022 20

Ready-On your mark-… Mid-Point Circle Algorithm Next Point XK +1 , YK XK , YK 6 Which point should I plot next? This one ? (XK +1 , YK) IT lies outside the circle Then, go for (XK , YK -1) ? 5 XK , YK-1 4 3 1 2 3 4 It’s a fraction value!

Mid-Point Circle Algorithm f( x, y)= x 2 +y 2 –r 2 A point ( a, b) lies outside, on or inside the circle according as f( a, b)( x, y)= ⋛ 0 XK +1 , YK XK , YK 6 Mid point ü q Ø • v v 5 Kth (XK , YK) is the point We are in the 1 st Quadrant We move from left to right ∴ XK+1 = X K + 1 But YK = ? Is it YK or YK+1 X K , Y K- 1 4 3 1 2 3 4

Mid-Point Circle Algorithm f( x, y)= x 2 +y 2 –r 2 XK +1 , YK XK , YK 6 Mid point ü q Ø • v v 5 Kth (XK , YK) is the point We are in the 1 st Quadrant We move from left to right ∴ XK+1 = X K + 1 But YK = ? Is it YK or YK+1 X K , Y K- 1 4 3 1 2 3 4

Mid-Point Circle Algorithm The same can be But pas 0 =? written

Circle Drawing Algorithm: For one region K=0 XK =0 , YK = r p. K <0? XK+1 = XK + 1 YK+1 = YK K=K+1 XK <YK ? soham. jisit@gmail. com XK+1 = XK + 1 YK+1 = YK -1 1/23/2022 25

Circle Drawing Algorithm: For one region Shift origin from (0, 0) to (xc, yc) 1/23/2022 soham. jisit@gmail. com 26