1. CS 376 Introduction to Computer Graphics 02 / 12 / 2007 Instructor: Michael Eckmann

2. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Today’s Topics Questions? World & Screen coordinates, Windows & Viewports Clipping –points –lines –polygons

3. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports Recall that we define our objects in World coordinates and our screen shows (some part of) the world in Screen coordinates. Now that we know about transformations like translation and scaling, we can talk about how the world coordinates are transformed into screen coordinates. In the world, we define a clipping window which contains that part of the world we wish to be displayed. On the screen we create and position a display window and within that display window we create and position some viewport which can be all or part of the display window. Diagram on the board.

4. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports

5. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports

6. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports clipping window in world –specified in openGL in the call to glu.gluOrtho2D(MIN_X, MAX_X, MIN_Y, MAX_Y); display window on the screen –specified in openGL in the calls testFrame = new Frame("TestFrame"); testFrame.setLocation(10, 10); // positions it on screen at 10,10 testFrame.setSize( 410, 452 ); // size in pixels by default the viewport is the entire display window to make a smaller viewport (part of that display window) use gl.glViewport(lowLeftX, lowLeftY, width, height); openGL still automatically performs the viewing transform from the clipping window in the world to the viewport in the display window.

7. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports The transformation that converts the clipping window to the viewport is called the 2d viewing transformation, (aka window-to-viewport transformation, aka windowing transformation). So a way to do this is: –define a window in your world coordinates –clip the scene to that window (that is, “clip” or ignore everything outside the window) –translate this clipping window to the origin –scale it to the size of the viewport –translate from the origin to correct position within the display window. Diagram on board.

8. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports Different effects can be generated when you change viewport / clipping window. –1) if we just change the viewport's position same exact objects will be shown but in a different position within the display window. –2) if we change size of viewport changes the size of the objects shown within that viewport –3) if we keep the viewport constant but change the clipping window larger clipping window will cause more of the scene to be shown but these objects will be smaller smaller clipping window will cause less of the scene to be shown but the objects will be larger if you continually make the clipping window smaller this has the effect of zooming in making the clipping window larger --- zooms out.

9. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports Different effects can be generated when you change viewport / clipping window. –4) if we keep the viewport constant, keep the clipping window the same size but change it's position in the world what kind of effect will be displayed?

10. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports Different effects can be generated when you change viewport / clipping window. –4) if we keep the viewport constant, keep the clipping window the same size but change it's position in the world what kind of effect will be displayed? –panning Aspect ratio: if the clipping window and viewport have the same ratio of x to y, then proportions will be correct (no stretching etc.) –but if they have different aspect ratios, for example a square in the world may not be a square in the viewport

11. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Windows and Viewports Example on the board of a viewing transformation –define a clipping window in our world coordinates –define a viewport in screen coordinates translate clipping window to origin scale translate to correct screen coords

12. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping To clip our world into a window we need to determine what in the world is inside our clipping window and what is outside it. we'll talk about clipping points, lines and polygons let's say our clipping window has the following vertices in the world –(xL, yB), (xR, yB), (xR, yT), (xL, yT) –L = left, R = right, T = top, B = bottom A point (x,y) is visible if xL <= x <= xR yB <= y <= yT

13. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping let's say our clipping window has the following vertices in the world –(xL, yB), (xR, yB), (xR, yT), (xL, yT) –L = left, R = right, T = top, B = bottom –draw clipping window on board A line segment is trickier –a line is visible if both its endpoints are in the clipping window –could solve simultaneous equations for the intersections of the window edge with the lines, but this is expensive The Cohen-Sutherland algorithm for line clipping is a more efficient way. –each vertex (endpoint) is assigned a region code (aka outcode) that defines that vertex to be in one of 9 regions

14. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping this region code is 4 bits 1 st bit is the sign of yT-y 2 nd bit is the sign of y-yB 3 rd bit is the sign of xR-x 4 th bit is the sign of x-XL 0 = positive 1 = negative Clipping window is the center area in the right diagram with the 0000 | | 1001 | 1000 | 1010 --------------|-----------------------|-------------- | | 0001 | 0000 | 0010 --------------|-----------------------|-------------- | | 0101 | 0100 | 0110 | |

15. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping Trivially accept if both endpoints have 0000 region codes Trivially reject if any corresponding bits in the two region codes are both 1 (and them together and if result != 0000, then reject) Example on board of trivial accepts and rejects. | | 1001 | 1000 | 1010 --------------|-----------------------|-------------- | | 0001 | 0000 | 0010 --------------|-----------------------|-------------- | | 0101 | 0100 | 0110 | |

16. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping Example on board of several non-trivial rejects / accepts.

17. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping If can't trivially accept or reject a line then we compute some intersections with the line boundaries Text book figure 6-14 has a mistake that shows a line that would have been trivially rejected (P3 to P4) but they consider it as not rejected until intersections are computed... | | 1001 | 1000 | 1010 --------------|-----------------------|-------------- | | 0001 | 0000 | 0010 --------------|-----------------------|-------------- | | 0101 | 0100 | 0110 | |

18. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping Besides Cohen-Sutherland line clipping, there are several faster line clipping algorithms. Our text talks about two more: Liang-Barsky line clipping and Nicholl- Lee-Nicholl line clipping. Let's discuss how Liang-Barsky Line Clipping works.

19. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Clipping Liang-Barsky line clipping –based on parametric equation of a line given end points (x 0, y 0 ) and (x end, y end ) the parametric equations of a line are: x = x 0 + u (x end – x 0 ) y = y 0 + u (y end – y 0 ) 0 <= u <= 1 –recall that our clipping window has the following vertices in the world (xL, yB), (xR, yB), (xR, yT), (xL, yT) –and a point (x,y) is visible if xL <= x <= xR yB <= y <= yT –replace the parametric equations for x and y in those inequalites xL <= x 0 + u (x end – x 0 ) <= xR yB <= y 0 + u (y end – y 0 ) <= yT

20. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Liang-Barsky Line Clipping xL <= x 0 + u (x end – x 0 ) <= xR yB <= y 0 + u (y end – y 0 ) <= yT –which can be expressed as 4 inequalities of the form u p k <= q k, where k = 1, 2, 3, or 4 and –p 1 = (x 0 – x end ) p 2 = (x end – x 0 ) p 3 = (y 0 – y end ) p 4 = (y end – y 0 ) –q 1 = (x 0 – xL) q 2 = (xR – x 0 ) q 3 = (y 0 – yB) q 4 = (yT – y 0 ) –a line parallel to a clipping edge has p k = 0, where k represents the edge to which it is parallel: 1=Left, 2=Right, 3=Bottom, 4=Top –if (p k == 0 and q k < 0) then line completely outside boundary, done –if (p k == 0 and q k >= 0) then the line is inside the boundary –if (p k < 0) then the line goes from outside to inside of the infinite extension of a clipping window edge –if (p k > 0) then the line goes from inside to outside of the infinite extension of a clipping window edge

21. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Liang-Barsky Line Clipping –Recall what k's mean: 1=Left, 2=Right, 3=Bottom, 4=Top –when p k is non-zero, we find the intersection of the line with the infinite extension of the clipping window edge k recall that u p k <= q k so this intersection is simply when u = q k /p k –initialize u 1 to be 0 and u 2 to be 1 –For each of the four infinite extensions of a clipping window edge we calculate either r 1 or r 2 r 1 is the u parameter for the intersection of the line with the edge when coming from the outside of the clipping window to the inside (p < 0) r 2 is the u parameter for the intersection of the line with the edge when coming from the inside of the clipping window to the outside (p > 0) if p u1) if p>0, update u2 to be r2 if it results in a shorter line (i.e. if r2 < u2) if u1 > u2 we reject the line, and we're done. –if it makes it through all four boundary checks without rejection, then the line to be drawn is between u1 and u2.

22. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Liang-Barsky Line Clipping Example of Liang-Barsky line clipping algorithm in action –(x 0,y 0 ) = (10,5) –(x end,y end ) = (30,30) –Clipping window: xL = 20, xR = 35, yB = 10, yT = 20. –p 1 = (x 0 – x end ) = -20 –p 2 = (x end – x 0 ) = 20 –p 3 = (y 0 – y end ) = -25 –p 4 = (y end – y 0 ) = 25 –q 1 = (x 0 – xL) = -10 –q 2 = (xR – x 0 ) = 25 –q 3 = (y 0 – yB) = -5 –q 4 = (yT – y 0 ) = 15 –u 1 = 0 and u 2 = 1 to start, let's run the algorithm on the board

23. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Liang-Barsky Line Clipping Another example –(x 0,y 0 ) = (10,18) –(x end,y end ) = (25,25) –Clipping window: xL = 20, xR = 35, yB = 10, yT = 20. –p 1 = (x 0 – x end ) = -15 –p 2 = (x end – x 0 ) = 15 –p 3 = (y 0 – y end ) = -17 –p 4 = (y end – y 0 ) = 17 –q 1 = (x 0 – xL) = -10 –q 2 = (xR – x 0 ) = 25 –q 3 = (y 0 – yB) = 8 –q 4 = (yT – y 0 ) = 2 –u 1 = 0 and u 2 = 1 to start, let's run the algorithm on the board

24. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Line Clipping Liang-Barsky (L-B) vs. Cohen-Sutherland (C-S) –L-B in general more efficient updates of u1 & u2 require only one divide window edge intersections are computed only once when u1 & u2 are final –C-S repeatedly calcs intersections along a line path even though line may be completely outside clipping window each intersection calc requires a divide and a multiply Both line clipping algorithms can be extended to 3-d A third line clipping algorithm, Nicholl-Lee-Nicholl (N-L-N), which we will not go into is faster than both L-B & C-S but it cannot be extended to 3-d.

25. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Polygon Clipping Polygon clipping –Polygons can be clipped against successive infinite extensions of the clipping window edges –Keep track of new vertices (intersections with edge of clipping window) at each stage. –Notice the number of vertices increased in the example below.

26. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Polygon Clipping

27. Michael Eckmann - Skidmore College - CS 376 - Spring 2007 Polygon Clipping The text describes two polygon clipping algorithms. The Sutherland-Hodgman algorithm and the Weiler-Atherton algorithm. A major difference between the two is that the Sutherland-Hodgman algorithm produces as output one polygon. Think about a concave polygon that when clipped, really should result in two polygons. Example on board.