CSG and Raytracing

1. CSG and Raytracing CPSC 407

2. What is CSG? • Computational Solid Geometry • Volume Representation • Very powerful modeling paradigm • Difficult to render CSG models

3. Volumes, Not Surfaces • CSG modeling tools allow a user to apply Boolean operations to volumes • Union, Intersection, Difference • Only works with volumes • Closed surfaces can be interpreted as volumes… • In 3D, a volume is bounded by a surface • In 2D, the boundary is a curve

4. Volumes are Half-Spaces • We can think of volumes as half-spaces • Each primitive divides space into a set of points inside the volume, and a set of points outside • The sets of points are infinitely dense • Now our Boolean operations are set operations on these sets of points

5. Simple 3D Half-Spaces • Sphere • Cylinder • Cone • Torus • Box • Plane • Plane is tricky - it splits space into two infinite half-spaces • Note that the cylinder and cone are capped. This is not necessary • In fact, you can use an infinite cylinder and two planes to make a capped cylinder • Infinite cylinders are easier to implement • You can also get a box from 6 planes…

6. Boolean CSG Operations • Union • Addition, A Ú B • Intersection • A Ù B • Difference • Subtraction, A – B, A Ùnot B • Difference is not commutative

7. A more complicated example • Difference of: • Intersection of Sphere and Cube • Union of 3 Cylinders - =

8. Raytracing CSG Objects • Need a data structure to describe primitives and operations • Binary CSG • Binary tree • Leaf nodes are primitives • Interior nodes are Boolean operations

9. Binary Tree Example

10. Ray Intervals • Define a ray as a set of points (r+td) • r is eye point • d is the direction of the ray • t is the scalar distance along the ray • For now, assume that a primitive is convex • A ray (r+td) intersects a convex primitive at most 2 times • Let’s say the ray enters the primitive at distance t1 and leaves at distance t2 • Now we can define an interval [t1, t2] along the ray that is inside the primitive • Note: for planes, the interval is either [a,inf] or [inf,b]

11. CSG Operations on Intervals • Assume we have intervals A = [a1,b1] and B = [a2,b2], and we want to combine them with a CSG operation • There are 5 cases to check for each operation: • No Overlap: • Partial Overlap: • Full Overlap:

12. CSG Union on 2 Intervals • Remember, Union is A or B • In the no-overlap case, we return the two intervals • The rest of the cases produce one interval, [min(a1,a2), max(b1,b2)]

13. CSG Intersection on 2 Intervals • Remember, Intersection is A and B • In the no-overlap case, we return no intervals • The rest of the cases produce one interval, [max(a1,a2), min(b1,b2)]

14. CSG Difference on 2 Intervals • Remember, Difference is A and not B • The order is important! • Here we have 5 cases: Return { [a1,b1] } Return { [0,0] } Return { [a1,a2], [b2,b1] } Return { [a1,a2] } Return { [b2,b1] }

15. Lists of intervals • Complex volumes are not convex • A ray may intersect the volume more than once • Instead of a single interval, we get a set of them • We want to combine interval sets S1and S2 with a Boolean CSG operation • Brute force algorithm: • For each interval Ai in S1, apply the appropriate 2-interval operation with each interval Bj from S2 • Return the list of new intervals • This algorithm is O(N2). O(NlogN) algorithms do exist • I’m not certain the list of new intervals will be unique. You may have to check if any overlap, and combine them.

16. Binary CSG Tree Traversal • Depth-first traversal of the tree • At each primitive node, pass back the list of intervals along the ray that intersect the primitive • At each Boolean node, combine the two child lists and pass back the new set of intervals • The intersection point is at (r+td), where t is the lower value of the first interval in the list returned from the root node of the tree

17. Materials and Normals for CSG • Need to determine normal at intersection point • Need tertiary information with intervals • Either object pointers or normals themselves • Note that normals are reversed for subtracted objects • Correct material properties at intersection point are debatable… • Some people take material properties at intersection point • Others prefer to define material properties for the entire CSG object as a whole

18. Additional CSG Bits • It is possible to partially implement CSG without doing a full CSG tree • This is how I did difference objects in my raytracer • Blob’s notes have a different CSG algorithm • His algorithm looks like it might be more efficient, but it requires a point inside/outside test • Very difficult for arbitrary triangle meshes

19. General Raytracing Bits

20. Common bugs • Wrong direction for eye rays • Make sure they are going into the scene • Taking the wrong intersection point • Ray intersects with a sphere twice, make sure you use the closer intersection! • Incorrect direction for reflected rays • Sending the reflected ray into the object • Total Internal Reflection in refraction • Get a sqrt(< 0) in the formula for refracted ray direction • In this case, ray reflects instead of refracting

21. Numerical Error • Self-Intersection due to Numerical Error • Produces ‘surface dirt’ – black speckles on otherwise smooth objects • Throw away intersections close to intersection point • Bad if you have very thin objects • Throw away any other intersections with current object • Assumes objects are convex!!! (not so for CSG, torus) • Avoid by moving a small distance along the normal at the intersection point before casting secondary rays • Bad for thin objects again, but probably the simplest method

22. Transforming Normals • I’m assuming you are implementing object transformations as described in Blob’s Notes • He explains how to transform the intersection point from object space to world space • He doesn’t explain how to transform normals • Tutorial on my website: • http://www.unknownroad.com/rtfm/graphics/rt_normals.html

23. Shadow Cache • Assumption: the last object that was hit by a shadow feeler is likely to be the one hit the next time • So test it first • For shadow feelers, it doesn’t matter if it’s the closest object (not so for reflection!) • Shadow feelers are usually a large part of the cost of rendering a scene, so this is a big speedup that is relatively simple to implement

24. WWW Material • http://fuzzyphoton.tripod.com/index.htm • http://www.scs.leeds.ac.uk/cuddles/hyperbks/Raytracer/contents.htm • http://www.scs.leeds.ac.uk/cuddles/hyperbks/Rendering/introduction.html • http://www.cs.wisc.edu/~schenney/courses/cs559-s2001/lectures/lecture-26-online.ppt