1. 1 Computer Graphics Chapter 4 2D Viewing Algorithms

2. [4]-2RM World Coordinates: User-Defined Limits Floating point values Device Coordinates: Device dependent Limits Positive Integer values Coordinate Systems x y 0 v 0 u (x, y) (u,v)

3. [4]-3RM (0,300) (360,0) (0,0) Window: A rectangular region in World Coordinate System. Viewport: A rectangular region in Device Coordinate System.

4. [4]-4RM Window to Viewport Transformation We denote the boundaries of the world window by four real values xmin, ymin, xmax, ymax denoting its left, bottom, right, top margins respectively. Similary the boundaries of the viewport on the screen are defined by four integer values umin, vmin, umax, vmax. When a graphics display is generated using arbitrary coordinates on the world window, the important problem encountered in viewing this display on the actual viewport on the screen is to have a function which maps every point (x,y) on the window to a corresponding point (u,v) on the viewport. The following window to viewport transformation achieves this relationship.

5. [4]-5RM (xmin,ymin) (xmax, ymax) (umin, vmin) (umax, vmax) (x, y) (u, v) (x, y)(u, v) Window to Viewport Transformation

6. [4]-6RM Window to Viewport Transformation u = c 1 x + c 2 v = d 1 y + d 2

7. [4]-7RM Window to Viewport Transformation -0.05+0.05 0.1 0.2 100 400 250 550 u = 3000 x + 400 v = 3000 y  200

8. [4]-8RM Aspect Ratio = Width/Height. Aspect ratio of a world window = (xmax-xmin) / (ymax-ymin). Aspect ratio of the viewport = (umax-umin) / (vmax-vmin). The transformation from the window to viewport is said to preserve the aspect ratio, if both the above quantities are same. In such a case, we have c 1 = d 1. When this condition is satisfied, the mapping from the window to the viewport is distortion-free. Aspect Ratio

9. [4]-9RM Window to Viewport Transformation -0.05+0.05 0.1 0.2 100 400 250 350 u = 1000 x + 300 v = 3000 y  200 For distortion-free mapping from the window to the viewport, we must have c 1 = d 1 (in magnitude)

10. [4]-10RM W-V Transform (OpenGL)

11. [4]-11RM A line is required to be clipped against a rectangular clipping window such that the portion of the line that falls outside the window is not displayed. There are many possible arrangements of a segment with respect to the window. We therefore need an organized approach to identify the correct situation and to compute the new end points of each clipped segment. Efficiency is important because a typical picture contains thousands of line segments, each of which must be clipped against a window. Line Clipping

12. [4]-12RM Window Clipping Window Line Clipping

13. [4]-13RM Cohen-Sutherland Algorithm Clipping window A rapid divide-and-conquer approach to the line clipping Trivial Accept: When both end points are inside the window, and therefore the line is completely visible. Trivial Reject: When both end points are outside the window and on the same side of the window. Then the line is completely outside, and hence can be rejected.

14. [4]-14RM A point is assigned a unique region code depending on the location of the point with respect to the window. 100110001010 0001 0000 0010 010101000110 Clipping window BRL Region Codes A

15. [4]-15RM Trivial Accept 0000 r 1 == 0000 and r 2 == 0000 (r 1 OR r 2 ) == 0000 (OR: Bitwise) Region Codes

16. [4]-16RM Trivial Reject 0010 1010 r 1 and r 2 have a common bit set to 1 (r 1 AND r 2 )  0000 (AND: Bitwise) Region Codes

17. [4]-17RM “Other Cases” 0000 1010 Region Codes 1001 0100

18. [4]-18RM Region Codes i. The point that lies outside the window is considered. (This is the point whose region code is non-zero). ii. For the above point, an edge outside which the point lies is identified. (If a particular bit is non-zero, the corresponding edge of the window is considered). iii. The intersection of the line with the edge is computed. The initial point in (i) is now replaced by the new intersection point. Its region code is computed. iv. The conditions for the “trivial accept” or “trivial reject” or “other case” is again checked for the new line segment. “Other Cases”

19. [4]-19RM Region Codes 0000 1010 0000 Trivial Accept

20. [4]-20RM Computing Intersection Point (Eg.) The equation of the line PQ is The intersection point A lies on the left edge and therefore its x-coordinate is x = xmin. To get the y-coordinate of A, we substitute for x in the above equation of PQ: y = P Q A x=xmin

21. [4]-21RM Computing Intersection Point (Eg.) Now P is replaced by the point A (effectively discarding the segment PA), and the whole process of checking is repeated for the segment AQ. Now both A, Q will have region codes 0000, and hence will satisfy the condition “trivial accept”. P Q A x=xmin

22. [4]-22RM Line Clipping (OpenGL)