Ray Tracing Polyhedra

1. Ray Tracing Polyhedra ©Anthony Steed 1999-2005

2. Overview • Barycentric coordinates • Ray-Polygon Intersection Test • Affine Transformations

3. Line Equation • Recall that given p1 and p2 in 3D space, the straight line that passes between is: for any real number t • This is a simple example of a barycentric combination

4. A Barycentric Combination is • a weighted sum of points, where weight sum to 1. • Let p1,p2,…,pn be points • Let a1,a2,…,an be weights

5. Implications • If p1,p2,…,pn are co-planar points then p as defined will be inside the polygon defined by the points • Proof of this is out of scope, but a few diagrams should convince you of the outline of a proof …

6. Ray Tracing a Polygon • Three steps • Does the ray intersect the plane of the polygon? • i.e. is the ray not orthogonal to the plane normal • Intersect ray with plane • Test whether intersection point lies within polygon on the plane

7. Does the ray intersect the plane? • Ray eq. is p0 + t.d • Plane eq. is n.(x,y,z) = k • Then test is n.d !=0

8. Where does it intersect? • Substitute line equation into plane equation • Solve for t • Find pi

9. Is this point inside the polygon? • If it is then it can be represented in barycentric co-ordinates • There are potentially several barycentric combinations • Many tests are possible • Winding number (can be done in 3D) • Infinite ray test (done in 2D) • Half-space test (done in 2D for convex poylgons)

10. Winding number test • Sum the angles subtended by the vertices. If sum is zero, then outside. If sum is +/-2, inside. • With non convex shapes, can get +/-4, +-6, etc… p2 1 p1 n-1 pn-1

11. Infinite Ray Test • Draw a line from the test point to the outside • Count +1 if you cross an edge in an anti-clockwise sense • Count -1 if you cross and edge in a clockwise sense • For convex polygons you can just count the number of crossings, ignoring the sign • If total is even then point is outside, otherwise inside -1 +1

12. Half-Space Test (Convex Polygons) • A point p is inside a polygon if it is in the negative half-space of all the line segments

13. Note • That the winding angle and half-space tests only tell you if the point is inside the polygon, they do not get you a barycentric combination • With some minor extensions, its easy to show that the infinite ray test finds a barycentric combination.

14. Transforming Polygons • Although its sort of “obvious” that transformations of object preserve flatness and shape, we need to convince ourselves of something specific: that barycentric coordinates are preserved under affine transformations • If they weren’t it would be extremely hard to shade and texture polygons later on in the course • If a rotation, scale, translation are affine then barycentricity is preserved • If barycentricity is preserved then polygons are still “flat” after transformation

15. Transformations Revisited • Homogenous transform as described is affine • Preserves barycentric co-ordinates • f is affine iff

16. To Show Transformation is Affine • Unit vectors e1=(1 0 0), e2=(0 1 0), e3=(0 0 1) and e4=(0 0 0) • p=(x1 x2 x3) • Let x4=1-x1-x2-x3 • Thus

17. ... • But for unit vectors, mapping is easy to derive

18. … • So combine those two

19. ... • But we know x4=1-x1-x2-x3

20. ... • Expand that, remembering what ei is • But this is the matrix transformation

21. Recap • Lines and polygons (and volumes) can be determined in terms of barycentric coordinates • We have shown that homogenous transformations are, by definition, affine • This polygons remain “flat” • Thus we now can use arbitrary transformations on polyhedra