159.235 Graphics & Graphical Programming

1. 159.235 Graphics & Graphical Programming Lecture - Projections - Part 2 Graphics

2. Projections Part 2 - Outline • Mathematics of Projections • Perspective Transforms • Stereo • Orthographic • 3D Clipping • Homogeneous Coordinates (again) Graphics

3. Mathematics of Viewing • Need to generate the transformation matrices for perspective and parallel projections • Should be 4x4 matrices to allow general concatenation Graphics

4. Perspective Projection – Simplest Case Centre of projection at the origin, Projection plane at z=d. Projection Plane. y P(x,y,z) x Pp(xp,yp,d) z d Graphics

5. x P(x,y,z) xp z d y d P(x,y,z) z x Pp(xp,yp,d) yp P(x,y,z) z d y Perspective Projection – Simplest Case Graphics

6. Perspective Projection Graphics

7. Perspective Projection Graphics

8. Perspective Projection Trouble with this formulation : Centre of projection fixed at the origin. Graphics

9. Finding Vanishing Points • Recall : An axis vanishing point is the point where the axis intercepts the projection plane  point at infinity. Graphics

10. d x z P(x,y,z) xp yp P(x,y,z) z y d Projection plane at z = 0 Centre of projection at z = -d Alternative Formulation Graphics

11. d x z P(x,y,z) xp yp P(x,y,z) z y d Projection plane at z = 0, Centre of projection at z = -d Now we can allow d Alternative Formulation Graphics

12. l Left eye x r E z object l E Right eye r Max effect at 50cm display screen Stereo Projection • Stereo projection is just two perspective projections : Graphics

13. Stereo Projection E is the interocular separation, typically 2.5cm to 3cm. d is the distance of viewer from display, typically 50cm. Let 2E = 5cm: Graphics

14. View as an Anaglyph (Red/Green) Graphics

15. Orthographic Projection Orthographic Projection onto a plane at z = 0. xp = x , yp = y , z = 0. Graphics

16. Implementation of Viewing • Transform into world coordinates. • Perform projection into view volume or eye coordinates. • Clip geometry outside the view volume. • Perform projection into screen coordinates. • Remove hidden lines. Graphics

17. left Far c lipping plane. Near clipping plane Need to calculate intersection With 6 planes. right 3D Clipping • For orthographic projection, view volume is a box. • For perspective projection, view volume is a frustrum. Graphics

18. Speeding up Projection • Can always place camera at origin ! • OpenGL , Renderman. • Much easier to clip lines to unit cube • Define canonical view volumes. • Define normalising transformations that project view volume into canonical view volume. • Clip, transform to screen coordinates. Graphics

19. 3D Clipping • Can use Cohen-Sutherland algorithm. • Now 6-bit outcode. • Trivial acceptance where both endpoint outcodes are all zero. • Perform logical AND, reject if non-zero. • Find intersect with a bounding plane and add the two new lines to the line queue. • Line-primitive algorithm. Graphics

20. 3D Polygon Clipping • Sutherland-Hodgman extends easily to 3D. • Call ‘CLIP’ procedure 6 times rather than 4 • Polygon-primitive algorithm. Graphics

21. Four cases of polygon clipping : Inside Outside Inside Outside Inside Outside Inside Outside First Output Output Vertex Output Intersection Second Output Case 3 No output. Case 1 Case 2. Case 4 Sutherland-Hodgman Algorithm Graphics

22. Clipping and Homogeneous Coordinates • Efficient to transform frustrum into perspective canonical view volume – unit slope planes. • Even better to transform to parallel canonical view volume • Clipping must be done in homogeneous coordinates. • Points can appear with –ve W and cannot be clipped properly in 3D. Graphics

23. Transforming between canonical view-volumes • The perspective canonical view-volume can be transformed to the parallel canonical view-volume with the following matrix: • Derivation left as an exercise :-) Graphics

24. Clipping in 3D. • 3D parallel projection volume is defined by: -1x 1, -1 y 1 , -1 z 0 • Replace by X/W,Y/W,Z/W: -1X/W 1, -1 Y/W 1 , -1 Z/W 0 • Corresponding plane equations are : X= -W, X=W, Y=-W, Y=W, Z=-W, Z=0 Graphics

25. Clipping in W • If W>0 , multiplication by W doesn’t change sign. W>0: -W X W, -W Y W, -W Z 0 • However if W<0, need to change sign : W<0: -WX W,-W Y W, -W Z 0 Graphics

26. W P1=[1 2 3 4]T W=1 Projection of P1 and P2 onto W=1 plane X or Y P2=[-1 –2 –3 –4]T Points in Homogeneous Coordinates Graphics

27. W P1=[1 2 3 4]T W=1 Region A Projection of P1 and P2 onto W=1 plane X or Y Region B Need to consider both regions when Performing clipping. P2=[-1 –2 –3 –4]T Points in Homogeneous Coordinates Graphics

28. Clipping in Homogeneous Coordinates • Could clip twice – once for region B, once for region A. • Expensive. • Check for negative W values and negate points before clipping. • What about lines with endpoints in each region ? Graphics

29. W P1 Projection of points. W=1 X P2 Lines in Homogeneous Coordinates Graphics

30. Lines in Homogeneous Coordinates W P1 Clipped lines W=X W=1 X Solution : Clip line segments, negate both endpoints, clip again. W=-X P2 Graphics

31. Final Step • Divide by W to get back to 3-D • Divide by z. • Where the perspective projection actually gets done – perspective foreshortening. • Now we have a ‘view volume’. • Don’t flatten z due to hidden line calculations. Graphics

32. Projections in Java • Use Java 3D package • Or can DIY! Graphics

33. Projections - Summary • Orthographic matrix - replace (z) axis with point. • Perspective matrix – multiply w by z. • Clip in homogeneous coordinates. • Preserve z for hidden surface calculations. • Can find number of vanishing points • Acknowledgments - thanks to Eric McKenzie, Edinburgh, from whose Graphics Course some of these slides were adapted. Graphics