PMC and Booleans

1. PMC and Booleans • Point-Membership Classification • Regularized Booleans between polygons

2. Point-in-polygon test in 2D • Given a polygon P and a point q in the plane of P, how would you test whether q lies inside P? • This is called “Point-Membership Classification” (abbreviated PMC)

3. Triangle-based point-in-polygon test • Algorithm for testing whether point Q is inside polygon P Boolean PinPoly(Q,P) { pt O = origin or some arbitrary point; Boolean in:= false; for (each edge (A,B) of P) if (PinT(O,A,B,Q)) in := ! in; return in; } boolean PinT(A,B,C,P) {return (right(A,B,P)== right(B,C,P)) && (right(A,B,P)== right(C,A,P)) ;} Why does it work?

4. P E1 E2 o o o E5 E4 E3 o o o XOR shading of a polygon in 2D • AB is the set of points that lie in A or in B, but not in both • AB := {p: pA  pB } • Let A, B, C … N be primitives, then ABC… N is the set of points contained in an odd number of these primitives • AB := {p: (pA) XOR (pB) XOR (pC) XOR … XOR (pN) } • How to shade a polygon A in 2D: • Polygon P has edges Ei. Triangle Ti = convex hull of O+Ei. • P =T1T2T3… Tn • To fill P:fill each Ti while toggling status of visited pixel Assume no pixel lies on boundary of any triangle

5. P E1 E1 E2 o o o o E5 E4 E3 E3 o o o o Example of XOR polygon filling

6. Relation to ray-casting approach? • A point Q lies in a set P if a ray from Q intersects the boundary of P an odd number of times • If ray hits a vertex or edge or is tangent to a surface, pick another ray • We do the same thing. Look at an example in 2D. Here: • Only edges E1, E2, E3 intersect ray from Q to O • Thus only triangles T1, T2, T3 contain Q Q Ray from Q O Q is in because it is contained in an odd number of triangles

7. Computing polygon area Two methods: • Sum of signed areas of triangles, each joining an arbitrary origin o to a different edge(a,b) • SUM oaR(ob) for each edge (a,b) • Sum of signed areas between each oriented edge the x-axis y by b ay a x ax bx area(a,b):=(ay+by)(bx–ax)/2

8. Area of the symmetric difference • The symmetric difference AB measures the discrepancy between the two solids A and B • It is 0 when A==B • Assume that A and B are bounded by consistently oriented polygonal loops. • Can I compute AB by summing areas of triangles?

9. Boolean operation on polygons • Diminishing boundary principle: The Boundary of a Boolean combination of shapes is a subset of the union of their boundaries • Strategy: Generate-Split-Select • Generate a sufficient set of candidates: edges of polygons • Split them at their pairwise intersections • Select the edge-segments on the boundary of the result • How to identify a good edge segment? • Must separate in from out t s

10. Boolean • A–B (also written A \ B) A A B B

11. Regularization • AB ? B A