UNIT 4-geometry of which and line drawing.pdf
- 1. 1 UNIT 4 Geometry and Line Generation Line Generation Algorithm A line connects two points. It is a basic element in graphics. To draw a line, you need two points between which you can draw a line. In the following three algorithms, we refer the one point of line as X0, Y0 and the second point of line as X1, Y1. DDAAlgorithm Digital Differential Analyzer (DDA) algorithm is the simple line generation algorithm which is explained step by step here. Step 1: Get the input of two end points (X0, Y0) and (X1, Y1). Step 2: Calculate ΔX, ΔY and M from the given input. These parameters are calculated as- • ΔX = Xn – X0 • ΔY =Yn – Y0 • M = ΔY / ΔX Step 3: Find the number of steps or points in between the starting and ending coordinates. if (absolute (ΔX) > absolute (ΔY)) Steps = absolute (ΔX); else Steps = absolute (ΔY); Step 4: Suppose the current point is (Xp, Yp) and the next point is (Xp+1, Yp+1). Find the next point by following the below three cases-
- 2. 2 Step 5: Keep repeating Step-04 until the end point is reached or the number of generated new points (including the starting and ending points) equals to the steps count. Example 1 : Calculate the points between the starting point (5, 6) and ending point (8, 12). Solution- Given- • Starting coordinates = (X0, Y0) = (5, 6) • Ending coordinates = (Xn, Yn) = (8, 12) Step-01: Calculate ΔX, ΔY and M from the given input. • ΔX = Xn – X0 = 8 – 5 = 3 • ΔY =Yn – Y0 = 12 – 6 = 6 • M = ΔY / ΔX = 6 / 3 = 2 Step-02: Calculate the number of steps. As |ΔX| < |ΔY| = 3 < 6, so number of steps = ΔY = 6
- 3. 3 Step-03: As M > 1, so case-03 is satisfied. Now, Step-03 is executed until Step-04 is satisfied. Xp Yp Xp+ 1 Yp+ 1 Round off (Xp+1, Yp+1) 5 6 5.5 7 (6, 7) 6 8 (6, 8) 6.5 9 (7, 9) 7 10 (7, 10) 7.5 11 (8, 11) 8 12 (8, 12) Example 2: Calculate the points between the starting point (1, 7) and ending point (11, 17). Given- • Starting coordinates = (X0, Y0) = (1, 7) • Ending coordinates = (Xn, Yn) = (11, 17) Step-01: Calculate ΔX, ΔY and M from the given input. • ΔX = Xn – X0 = 11 – 1 = 10
- 4. 4 • ΔY =Yn – Y0 = 17 – 7 = 10 • M = ΔY / ΔX = 10 / 10 = 1 Step-02: Calculate the number of steps. As |ΔX| = |ΔY| = 10 = 10, so number of steps = ΔX = ΔY = 10 Step-03: As M = 1, so case-02 is satisfied. Now, Step-03 is executed until Step-04 is satisfied. Xp Yp Xp+ 1 Yp+ 1 Round off (Xp+1, Yp+1) 1 7 2 8 (2, 8) 3 9 (3, 9) 4 10 (4, 10) 5 11 (5, 11) 6 12 (6, 12) 7 13 (7, 13) 8 14 (8, 14) 9 15 (9, 15) 10 16 (10, 16) 11 17 (11, 17)
- 5. 5 Advantages of DDAAlgorithm- • It is a simple algorithm. • It is easy to implement. • It avoids using the multiplication operation which is costly in terms of time complexity. Disadvantages of DDAAlgorithm- • There is an extra overhead of using round off( ) function. • Using round off( ) function increases time complexity of the algorithm. • Resulted lines are not smooth because of round off( ) function. • The points generated by this algorithm are not accurate. Bresenham’s Line Generation Algorithm The Bresenham algorithm is another incremental scan conversion algorithm. The big advantage of this algorithm is that, it uses only integer calculations. Moving across the x axis in unit intervals and at each step choose between two different y coordinates. For example, as shown in the following illustration, from position (2, 3) you need to choose between (3, 3) and (3, 4). You would like the point that is closer to the original line. At sample position xk+1, the vertical separations from the mathematical line are labelled as dupper and dlower. convert the output primitives into frame buffer
- 6. 6 Step-01: Calculate ΔX and ΔY from the given input. These parameters are calculated as- • ΔX = Xn – X0 • ΔY =Yn – Y0 Step-02: Calculate the decision parameter Pk. It is calculated as- Pk = 2ΔY – ΔX Step-03: Suppose the current point is (Xk, Yk) and the next point is (Xk+1, Yk+1). Find the next point depending on the value of decision parameter Pk. Follow the below two cases-
- 7. 7 Step-04: Keep repeating Step-03 until the end point is reached or number of iterations equals to (ΔX-1) times. Example : Calculate the points between the starting coordinates (9, 18) and ending coordinates (14, 22). Given- • Starting coordinates = (X0, Y0) = (9, 18) • Ending coordinates = (Xn, Yn) = (14, 22) Step-01: Calculate ΔX and ΔY from the given input. • ΔX = Xn – X0 = 14 – 9 = 5 • ΔY =Yn – Y0 = 22 – 18 = 4 Step-02: Calculate the decision parameter. Pk= 2ΔY – ΔX = 2 x 4 – 5 = 3
- 8. 8 So, decision parameter Pk = 3 Step-03: As Pk >= 0, so case-02 is satisfied. Thus, • Pk+1 = Pk + 2ΔY – 2ΔX = 3 + (2 x 4) – (2 x 5) = 1 • Xk+1 = Xk + 1 = 9 + 1 = 10 • Yk+1 = Yk + 1 = 18 + 1 = 19 Similarly, Step-03 is executed until the end point is reached or number of iterations equals to 4 times. (Number of iterations = ΔX – 1 = 5 – 1 = 4) Pk Pk+ 1 Xk+ 1 Yk+1 9 18 3 1 10 19 1 -1 11 20 -1 7 12 20 7 5 13 21 5 3 14 22
- 9. 9 Advantages of Bresenham Line Drawing Algorithm- The advantages of Bresenham Line Drawing Algorithm are- • It is easy to implement. • It is fast and incremental. • It executes fast but less faster than DDAAlgorithm. • The points generated by this algorithm are more accurate than DDAAlgorithm. • It uses fixed points only. Disadvantages of Bresenham Line Drawing Algorithm- The disadvantages of Bresenham Line Drawing Algorithm are- • Though it improves the accuracy of generated points but still the resulted line is not smooth. • This algorithm is for the basic line drawing. • It can not handle diminishing jaggies. Mid-Point line drawing Algorithm Mid-point algorithm is due to Bresenham which was modified by Pitteway and Van Aken. Assume that you have already put the point P at (x, y) coordinate and the slope of the line is 0 ≤ k ≤ 1 as shown in the following illustration. Now you need to decide whether to put the next point at E or N. This can be chosen by identifying the intersection point Q closest to the point N or E. If the intersection point Q is closest to the point N then N is considered as the next point; otherwise E.
- 10. 10 The points generation using Mid Point Line Drawing Algorithm involves the following steps- Step-01: Calculate ΔX and ΔY from the given input. These parameters are calculated as- • ΔX = Xn – X0 • ΔY =Yn – Y0 Step-02: Calculate the value of initial decision parameter and ΔD. These parameters are calculated as- Dinitial = 2ΔY – ΔX • ΔD = 2(ΔY – ΔX) Step-03: The decision whether to increment X or Y coordinate depends upon the flowing values of Dinitial. Follow the below two cases-
- 11. 11 Step-04: Keep repeating Step-03 until the end point is reached. For each Dnew value, follow the above cases to find the next coordinates. Example: Calculate the points between the starting coordinates (20, 10) and ending coordinates (30, 18). Given- • Starting coordinates = (X0, Y0) = (20, 10) • Ending coordinates = (Xn, Yn) = (30, 18) Step-01: Calculate ΔX and ΔY from the given input. • ΔX = Xn – X0 = 30 – 20 = 10 • ΔY =Yn – Y0 = 18 – 10 = 8 Step-02: Calculate Dinitial and ΔD as- • Dinitial = 2ΔY – ΔX = 2 x 8 – 10 = 6 • ΔD = 2(ΔY – ΔX) = 2 x (8 – 10) = -4 Step-03: As Dinitial >= 0, so case-02 is satisfied., • Xk+1 = Xk + 1 = 20 + 1 = 21 • Yk+1 = Yk + 1 = 10 + 1 = 11 • Dnew = Dinitial + ΔD = 6 + (-4) = 2 Similarly, Step-03 is executed until the end point is reached. Dinitia l Dne w Xk+ 1 Yk+1 20 10 6 2 21 11 2 -2 22 12 -2 14 23 12 14 10 24 13 10 6 25 14 6 2 26 15 2 -2 27 16 -2 14 28 16 14 10 29 17
- 12. 12 10 30 18 Advantages of Mid Point Line Drawing Algorithm- The advantages of Mid Point Line Drawing Algorithm are- • Accuracy of finding points is a key feature of this algorithm. • It is simple to implement. • It uses basic arithmetic operations. • It takes less time for computation. • The resulted line is smooth as compared to other line drawing algorithms. Circle Generation Algorithm Drawing a circle on the screen is a little complex than drawing a line. There are two popular algorithms for generating a circle: Bresenham’s Algorithm and Midpoint Circle Algorithm. These algorithms are based on the idea of determining the subsequent points required to draw the circle. Let us discuss the algorithms in detail: The equation of circle is X2 + Y2 = r2, where r is radius. Bresenham’s Circle Drawing Algorithm Given-
- 13. 13 • Centre point of Circle = (X0, Y0) • Radius of Circle = R The points generation using Bresenham Circle Drawing Algorithm involves the following steps- Step-01: Assign the starting point coordinates (X0, Y0) as- • X0 = 0 • Y0 = R Step-02: Calculate the value of initial decision parameter P0 as- P0 = 3 – 2 x R Step-03: Suppose the current point is (Xk, Yk) and the next point is (Xk+1, Yk+1). Find the next point of the first octant depending on the value of decision parameter Pk. Follow the below two cases-
- 14. 14 Step-04: If the given centre point (X0, Y0) is not (0, 0), then do the following and plot the point- • Xplot = Xc + X0 • Yplot = Yc + Y0 Here, (Xc, Yc) denotes the current value of X and Y coordinates. Step-05: Keep repeating Step-03 and Step-04 until Xplot => Yplot. Step-06: Step-05 generates all the points for one octant. To find the points for other seven octants, follow the eight symmetry property of circle. This is depicted by the following figure- Example: Given the center point coordinates (0, 0) and radius as 8, generate all the points to form a circle. Given- • Centre Coordinates of Circle (X0, Y0) = (0, 0) • Radius of Circle = 8 Step-01: Assign the starting point coordinates (X0, Y0) as- • X0 = 0
- 15. 15 • Y0 = R = 8 Step-02: Calculate the value of initial decision parameter P0 as- P0 = 3 – 2 x R P0 = 3 – 2 x 8 P0 = -13 Step-03: As Pinitial < 0, so case-01 is satisfied. Thus, • Xk+1 = Xk + 1 = 0 + 1 = 1 • Yk+1 = Yk = 8 • Pk+1 = Pk + 4 x Xk+1 + 6 = -13 + (4 x 1) + 6 = -3 Step-04: This step is not applicable here as the given centre point coordinates is (0, 0). Step-05: Step-03 is executed similarly until Xk+1 >= Yk+1 as follows- Pk Pk+1 (Xk+1, Yk+1) (0, 8) -13 -3 (1, 8) -3 11 (2, 8) 11 5 (3, 7) 5 7 (4, 6) 7 (5, 5) Algorithm Terminates These are all points for Octant-1. Algorithm calculates all the points of octant-1 and terminates. Now, the points of octant-2 are obtained using the mirror effect by swapping X and Y coordinates. Octant-1 Points Octant-2 Points (0, 8) (5, 5) (1, 8) (6, 4) (2, 8) (7, 3)
- 16. 16 (3, 7) (8, 2) (4, 6) (8, 1) (5, 5) (8, 0) These are all points for Quadrant-1. Now, the points for rest of the part are generated by following the signs of other quadrants. The other points can also be generated by calculating each octant separately. Here, all the points have been generated with respect to quadrant-1- Quadrant-1 (X,Y) Quadrant-2 (-X,Y) Quadrant-3 (-X,-Y) Quadrant-4 (X,-Y) (0, 8) (0, 8) (0, -8) (0, -8) (1, 8) (-1, 8) (-1, -8) (1, -8) (2, 8) (-2, 8) (-2, -8) (2, -8) (3, 7) (-3, 7) (-3, -7) (3, -7) (4, 6) (-4, 6) (-4, -6) (4, -6) (5, 5) (-5, 5) (-5, -5) (5, -5) (6, 4) (-6, 4) (-6, -4) (6, -4) (7, 3) (-7, 3) (-7, -3) (7, -3) (8, 2) (-8, 2) (-8, -2) (8, -2) (8, 1) (-8, 1) (-8, -1) (8, -1) (8, 0) (-8, 0) (-8, 0) (8, 0) These are all points of the Circle. Advantages of Bresenham Circle Drawing Algorithm- The advantages of Bresenham Circle Drawing Algorithm are- • The entire algorithm is based on the simple equation of circle X2 + Y2 = R2. • It is easy to implement. Disadvantages of Bresenham Circle Drawing Algorithm • Like Mid Point Algorithm, accuracy of the generating points is an issue in this algorithm. • This algorithm suffers when used to generate complex and high graphical images. • There is no significant enhancement with respect to performance. Mid Point Algorithm
- 17. 17 The points generation using Mid Point Circle Drawing Algorithm involves the following steps- Step-01: Assign the starting point coordinates (X0, Y0) as- • X0 = 0 • Y0 = R Step-02: Calculate the value of initial decision parameter P0 as- P0 = 1 – R Step-03: Suppose the current point is (Xk, Yk) and the next point is (Xk+1, Yk+1). Find the next point of the first octant depending on the value of decision parameter Pk. Follow the below two cases- Step-04: If the given centre point (X0, Y0) is not (0, 0), then do the following and plot the point- • Xplot = Xc + X0 • Yplot = Yc + Y0 Here, (Xc, Yc) denotes the current value of X and Y coordinates.
- 18. 18 Step-05: Keep repeating Step-03 and Step-04 until Xplot >= Yplot. Step-06: Step-05 generates all the points for one octant. To find the points for other seven octants, follow the eight symmetry property of circle. Example: Given the center point coordinates (0, 0) and radius as 10, generate all the points to form a circle. Given- • Centre Coordinates of Circle (X0, Y0) = (0, 0) • Radius of Circle = 10 Step-01: Assign the starting point coordinates (X0, Y0) as- • X0 = 0 • Y0 = R = 10 Step-02: Calculate the value of initial decision parameter P0 as- P0 = 1 – R P0 = 1 – 10 P0 = -9 Step-03: As Pinitial < 0, so case-01 is satisfied. Thus, • Xk+1 = Xk + 1 = 0 + 1 = 1 • Yk+1 = Yk = 10 • Pk+1 = Pk + 2 x Xk+1 + 1 = -9 + (2 x 1) + 1 = -6 Step-04: This step is not applicable here as the given centre point coordinates is (0, 0). Step-05: Step-03 is executed similarly until Xk+1 >= Yk+1 as follows- Pk Pk+1 (Xk+1, Yk+1) (0, 10) -9 -6 (1, 10) -6 -1 (2, 10) -1 6 (3, 10)
- 19. 19 6 -3 (4, 9) -3 8 (5, 9) 8 5 (6, 8) Algorithm Terminates These are all points for Octant-1. Algorithm calculates all the points of octant-1 and terminates. Now, the points of octant-2 are obtained using the mirror effect by swapping X and Y coordinates. Octant-1 Points Octant-2 Points (0, 10) (8, 6) (1, 10) (9, 5) (2, 10) (9, 4) (3, 10) (10, 3) (4, 9) (10, 2) (5, 9) (10, 1) (6, 8) (10, 0) These are all points for Quadrant-1. Now, the points for rest of the part are generated by following the signs of other quadrants. The other points can also be generated by calculating each octant separately. Here, all the points have been generated with respect to quadrant-1- Quadrant-1 (X,Y) Quadrant-2 (-X,Y) Quadrant-3 (-X,-Y) Quadrant-4 (X,-Y) (0, 10) (0, 10) (0, -10) (0, -10) (1, 10) (-1, 10) (-1, -10) (1, -10) (2, 10) (-2, 10) (-2, -10) (2, -10) (3, 10) (-3, 10) (-3, -10) (3, -10) (4, 9) (-4, 9) (-4, -9) (4, -9) (5, 9) (-5, 9) (-5, -9) (5, -9) (6, 8) (-6, 8) (-6, -8) (6, -8) (8, 6) (-8, 6) (-8, -6) (8, -6)
- 20. 20 (9, 5) (-9, 5) (-9, -5) (9, -5) (9, 4) (-9, 4) (-9, -4) (9, -4) (10, 3) (-10, 3) (-10, -3) (10, -3) (10, 2) (-10, 2) (-10, -2) (10, -2) (10, 1) (-10, 1) (-10, -1) (10, -1) (10, 0) (-10, 0) (-10, 0) (10, 0) These are all points of the Circle. Advantages of Mid Point Circle Drawing Algorithm- The advantages of Mid Point Circle Drawing Algorithm are- • It is a powerful and efficient algorithm. • The entire algorithm is based on the simple equation of circle X2 + Y2 = R2. • It is easy to implement from the programmer’s perspective. • This algorithm is used to generate curves on raster displays.