David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory

1. CS 430/585Computer Graphics I3D ClippingBackface Culling,Z-buffering,Ray Casting, Ray TracingWeek 7, Lecture 14 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University http://gicl.cs.drexel.edu

2. Overview • 3D Clipping • Advanced topics • Z-buffering • Back-Face Culling • Ray Tracing (Ray Casting) 1994 Foley/VanDam/Finer/Huges/Phillips ICG

3. Simple Mesh Format (SMF) • Michael Garland http://graphics.cs.uiuc.edu/~garland/ • Triangle data • Vertex indices begin at 1

4. 3D Clipping • Cohen-Sutherland and Cyrus-Beck can be trivially extended to 3D • We will cover: • Cohen-Sutherland for 3D, (parallel projection) • Cohen-Sutherland for 3D, (perspective projection)

5. Line is completely visible iff both code values of endpoints are 0, i.e. If line segments are completely outside the window, then Recall: Cohen-Sutherland Pics/Math courtesy of Dave Mount @ UMD-CP

6. Cohen-Sutherland for 3D,Parallel Projection • Use 6 bits • Trivially accept if both end-codes are 0 • Trivially reject if bit-by-bit AND of end-codes is not 0 • Up to 6 intersections may have to be computed

7. Cohen-Sutherland for 3D computing intersection points. • Use parametric representation of the line to compute intersections • So for y=1 replace y with 1 and solve for t • If 1  t  0use it to find x and z • Test if x and z are in valid range • Repeat for planes y=-1, x=1, x=-1, z=-1, z=0

8. Cohen-Sutherland for 3D, Perspective Projection • Use 6 bits identical to parallel view volume clipping • Conditions on the codes are different • Trivially accept/reject lines using same roles • Intersection points computed differently

9. Cohen-Sutherland for 3D computing intersection points. • Intersections with planes z=-1, z=zminis the same. • Calculating intersections with a sloping plane … • For plane y=z these two equations are equal • Repeat for planes y=-z, x=z, x=-z

10. More Efficient Alternative? • Use Cohen-Sutherland to do trivial accept/reject • Project remaining edges onto view plane • Clip lines in 2D

11. end clipping

12. Mesh/Faceted Model

13. Back-Face Culling • Assumptions: • Object approximated as closed polyhedron • Polyhedron interior is not exposed by the front cutting plane • Eye-point not inside object • Polygons not facing the viewer called Back-Facing • Right-hand vertex ordering defines outward normal • Back-Face Culling is a technique for eliminating these polygons • On average eliminates half of the polygons

14. Back-Face Culling • After projection, examine normal Nk(xk , yk, zk)to the face. • If zk < 0, face is a Back-Face - don’t draw it • More general test looks at Nk•V • V - View vector • The only test necessary for a single convex polyhedron 1994 Foley/VanDam/Finer/Huges/Phillips ICG

15. Back-Face Culled Wire-Frame

16. Z-buffering • Z-buffering (depth-buffering) is a visible surface detection algorithm • Implementable in hardware and software • Requires data structure (z-buffer) in addition to frame buffer. • Z-buffer stores values [0 .. ZMAX] corresponding to depth of each point. • If the point is closer than one in the buffers, it will replace the buffered values

17. Z-buffering + = + = 1994 Foley/VanDam/Finer/Huges/Phillips ICG

18. Z-buffering for (y = 0; y < YMAX; y++) for (x = 0; x < XMAX; x++) { F[x][y] = BACKGROUND_VALUE; Z[x][y] = ZMIN; } for (each polygon) for (each pixel in polygon’s projection) { pz = polygon’s z-value at pixel coordinates (x,y) if (pz > Z[x][y]) {/* New point is closer */ Z[x][y] = pz; F[x][y] = polygon’s color at pixel coordinates (x,y) } } 1994 Foley/VanDam/Finer/Huges/Phillips ICG

19. Z-buffering • We can simplify the calculation of z by exploiting the fact that triangle is planar. • Interpolate z values along the edges • Interpolate z values along scan line • Special cases: vertices & horizontal edge 1994 Foley/VanDam/Finer/Huges/Phillips ICG

20. Back-Face Culled & Z-Buffered Wire-Frame

21. See the Difference From HW of Andrew Mroczkowski

22. Depth Cueing • Objects that are closer are brighter • Objects farther away are darker • Color = BaseColor(z - far)/(near - far) www.siggraph.org www.dm.unibo.it/~casciola

23. Important Reminder • Recall: near and far planes are transformed by Nparand Nper • Parallel projection • Near: z = 0 • Far: z = -1 • Perspective projection • Near: z = • Far: z = -1

24. Ray Casting (Ray Tracing ) • Determines visible surfaces by tracing rays of light from the viewers eye to the objects in the world.

25. Ray Casting • Determines visible surfaces by tracing rays of light from the viewers eye to the objects • View plane is divided on the pixel grid • The eye ray is fired from the center of projection through each pixel 1994 Foley/VanDam/Finer/Huges/Phillips ICG

26. Ray Casting Select the center of projection and viewplane window for (each scan line) { for (each pixel on the scan line) { Find ray from the center of projections through the pixel for (each object in the scene) { if (object is intersected and intersection is closest considered so far) { Record the intersection and the object name } } Set pixel color to that of the closest intersected object } } 1994 Foley/VanDam/Finer/Huges/Phillips ICG

27. Computing Intersections • Using parametric equation of the line: • Simplifying for speed: • Resulting ray: (x1,y1,z1) (x0,y0,z0) 1994 Foley/VanDam/Finer/Huges/Phillips ICG

28. Computing Intersections with Sphere • Sphere with radius r and center (a,b,c) is: • Intersection is found by substituting values for (x,y,z) • With some straightforward algebraic transformations: • Equation is quadratic in terms of t • No real roots, No intersection • One real root, Ray grazes the sphere • Two roots, There are two points of intersection 1994 Foley/VanDam/Finer/Huges/Phillips ICG

29. Computing Intersections with Polygon • First intersect ray with plane • The substitution results in: • If denominator is 0, ray is parallel to the plane • Project polygon and point orthographically on the coordinate plane • Polygon containment test can be performed in 2D 1994 Foley/VanDam/Finer/Huges/Phillips ICG

30. Polygon Containment Test • Jordan Curve Theorem: • Point is inside if, for any ray, there is an odd number of crossings • Otherwise it is outside • Be careful with all the special cases • Wide variety of other techniques exist

31. Why Trace Rays? • More elegant • Testbed for techniques: • modeling (reflectance, transport) • rendering (e.g. Monte Carlo) • texturing (e.g. hypertexture) • Easiest photorealistic renderer to implement Boat reflected in wavy water rendered in openGLusing an environment map Compiled from: Lecture notes of Dr. John C. Hart @ University of Illinois

32. Ray Tracing • Extension of ray casting • Idea: Continue to bounce the ray in the scene • Shoot rays to light sources • Simple and powerful • Reflections, shadows, transparency and multiple light sources • Can be used to produce highly realistic images

33. Ray Traced Image Blessed State, by Ken Musgrave

34. cos in - i i i m t t = ? -n To Ray Trace, We Need Refraction • Snell’s Law: i sin i = t sin t • Let  =i/t = sin t / sin i • Let m = (cos in - i) / sin i • Then… t = sintm - cos tn = (sint / sini) (cos in -i) - cos tn = ( cos i- cos t )n -i Can be negative for grazing angles when  >1, say when going from glass to air, resulting in total internal reflection (no refraction). Compiled from: Lecture notes of Dr. John C. Hart @ University of Illinois

35. Refraction in Action

36. More Refractions Photon Mapping! Justin Hansen University of Utah Henrik Wann Jensen UC, San Diego

37. Efficiency Considerations • Partition the bounding box on a regular grid with equal sized extents • Associate each partition with the list of objects contained in it 1994 Foley/VanDam/Finer/Huges/Phillips ICG

38. Efficiency Considerations • Ray only intersected with the objects contained in the partitions it passes through • If partitions are examined in the order ray travels, we can stop after first intersection is found • Need to check if intersection itself is in the partition 1994 Foley/VanDam/Finer/Huges/Phillips ICG

39. Computer Graphics II Image produced by Horace Ip