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. 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 o E5 o E4 o E3 E2 o o E1 o P
Assume no pixel lies on boundary of any triangle

5. 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 O
Ray from Q
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