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
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 OEi.
• 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
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
a
ay
x
ax
bx
area(a,b)(ayby)(bxax)/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 AB
• 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

• AB (also written A \ B)

A
A
B
B
11
Regularization

• AB ?

B
A