1. 1 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Practice  Quiz next time –Geometry –Topology  One quiz will not count towards your grade –The one with lowest grade –If you missed a class, you don’t need to make up

2. 2 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Topology in 1D  We consider subsets S of the real line, such as S=[2,3[, which includes all real numbers x: 2≤x<3 S=]2,3], which includes all real numbers x: 2<x≤3 S=[5], which contains only the number 5 S=[2,3[ + [5], which includes real numbers x: 2≤x<3 and also the number 5  The operators complement (!), boundary (.b), interior (.i), exterior (.e), closure (.k), regularization (.r) are relative to the real line  Hence if S=[2,3[, then S.i=]2,3[, excluding the value 2.

3. 3 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q1  S = [2,3[ + ]3,4[ + [5], What is S.i (interior)?

4. 4 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R1  S = [2,3[ + ]3,4[ + [5], S.i = ]2,3[ + ]3,4[

5. 5 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q2  S = [2,3[ + ]3,4[ + [5], What is S.b (boundary)?

6. 6 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R2  S = [2,3[ + ]3,4[ + [5], S.b = [2] + [3] + [4] + [5]

7. 7 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q3  S = [2,3[ + ]3,4[ + [5], What is S.k (closure)?

8. 8 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R3  S = [2,3[ + ]3,4[ + [5], S.k = [2,4] + [5]

9. 9 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q4  S = [2,3[ + ]3,4[ + [5], What is S.r (regularization)?

10. 10 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R4  S = [2,3[ + ]3,4[ + [5], S.r = S.i.k = [2,4]

11. 11 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q5  You are given 2 regularized sets, A and B, in the plane. Use.i,.b,.k,.e…, and Boolean operators to express the condition that A and B touch, even though they do not interfere.

12. 12 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R5  You are given 2 regularized sets, A and B, in the plane. A and B touch, when A.i B.i=  AND A.b B.b≠ 

13. 13 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q6  Let vector V have coordinates What are the coordinates of R(V)?

14. 14 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R6  Let vector V have coordinates R(V) = V R(V) Assuming Y goes down

15. 15 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008  Write the geometric formulation of a test for establishing whether point C lies to the right of edge from A to B boolean right(A,B,C) { return … } Q7 AB C

16. 16 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008  Write the geometric formulation of a test for establishing whether point C lies to the right of edge from A to B boolean right(A,B,C) { return R(AB)  AC>0;} R7 A R(AB) B C AC

17. 17 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q8  Using right(A,B,C), formulate a test to establish whether point P lines inside triangle(A,B,C) boolean PinT(A,B,C,P) {return … ;} A B C P

18. 18 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R8  Using right(A,B,C), formulate a test to establish whether point P lines inside triangle(A,B,C) boolean PinT(A,B,C,P) {return (right(A,B,P)== right(B,C,P)) && (right(A,B,P)== right(C,A,P)) ;} A B C P A B C P

19. 19 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q9  Write the geometric formulation of a test to assess whether point C lies in the relative interior of edge (A,B). boolean PinE(A,B,C) {return … ;} AB C

20. 20 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R9  Write the geometric formulation of a test to assess whether point C lies in the relative interior of edge (A,B). boolean PinE(A,B,C) {return (R(AB)  AC==0) && (0 < AB  AC) && (AB  AC < AB  AB) ; } AB C

21. 21 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q10  Write the geometric formulation of a test to assess whether the closed edges (A,B) and (C,D) are disjoint (no intersection). boolean disjointEdges(A,B,C,D) { return …}

22. 22 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R10  Write the geometric formulation of a test to assess whether the closed edges (A,B) and (C,D) are disjoint (no intersection). boolean disjointEdges(A,B,C,D) { return (right(A,B,C) == right(A,B,D)) || (right(C,D,A) == right(C,D,A)) }

23. 23 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q: Point in polygon test  How to test whether a point Q lies in polygon P? Q P A B C D

24. 24 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R: Point in polygon test  How to test whether a point Q lies in polygon P?  Construct a ray R from Q that does not hit any vertex  Compute the number x of intersections or R and bP –One edge at a time, in any order –Works even if P has holes or several components  Return x %2 == 1 Q P A B C D R

25. 25 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 Q11 (bonus)  Write the test to assess whether ray (Q,T) intersects edge (A,B) boolean rayEdgeHit(Q,T,A,B) {return …} Q B R T A

26. 26 www.gvu.gatech.edu/~jare k Jarek Rossignac,  2008 R11  Write the test to assess whether ray (Q,T) intersects edge (A,B) boolean rayEdgeHit(Q,T,A,B) {return QA  R(T) != QB  R(T) & R(AB)  AQ != R(AB)  T } Q B R T A Is this correct? How to verify???