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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
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
1
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.
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
2
Q1
 S = [2,3[ + ]3,4[ + [5],
What is S.i (interior)?
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
3
R1
 S = [2,3[ + ]3,4[ + [5],
S.i = ]2,3[ + ]3,4[
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
4
Q2
 S = [2,3[ + ]3,4[ + [5],
What is S.b (boundary)?
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
5
R2
 S = [2,3[ + ]3,4[ + [5],
S.b = [2] + [3] + [4] + [5]
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
6
Q3
 S = [2,3[ + ]3,4[ + [5],
What is S.k (closure)?
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
7
R3
 S = [2,3[ + ]3,4[ + [5],
S.k = [2,4] + [5]
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
8
Q4
 S = [2,3[ + ]3,4[ + [5],
What is S.r (regularization)?
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
9
R4
 S = [2,3[ + ]3,4[ + [5],
S.r = S.i.k = [2,4]
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
10
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.
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
11
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≠
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
12
Q6
 Let vector V have coordinates <x,y>
What are the coordinates of R(V)?
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
13
R6
 Let vector V have coordinates <x,y>
R(V) = <-y,x>
V
Assuming Y goes down
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
R(V)
14
Q7
 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 … }
A
B
C
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
15
R7
 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;}
A
B
R(AB)
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
AC
C
16
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 … ;}
B
P
A
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
C
17
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)) ;}
B
B
P
P
C
A
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
C
A
18
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 … ;}
A
B
C
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
19
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)
;}
A
B
C
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
20
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 …}
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
21
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))
}
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
22
Q: Point in polygon test
 How to test whether a point Q lies in polygon P?
A
C
Q
P
B
D
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
23
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
A
 Return x %2 == 1
C
Q
R
P
B
D
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
24
Q11 (bonus)
 Write the test to assess whether ray (Q,T) intersects edge (A,B)
boolean rayEdgeHit(Q,T,A,B) {return …}
A
Q
T
R
B
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
25
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)
&&
A
R(AB)AQ != R(AB)T
}
Is this correct?
Q
T
How to verify???
R
B
Jarek Rossignac, 2008
www.gvu.gatech.edu/~jare
26