CS430
Computer Graphics
Vectors Part III
Intersections
Chi-Cheng Lin, Winona State University
Topics
Intersection of Two Line Segments
 Application – Excircle
 Intersection of Line and Plane

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Intersection of Two Line Segments
Problem: Given two line segments AB and
CD, determine whether they intersect,
and if they do, find the intersection point
 Cases
D
B

B
A
D
C
A
D
A
D
C
B
A
C
C
B
A
C
D
B
A
C
B
D
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Parametric Line

AB(t) = A + bt, where b = B -A
- < t < : parent line of AB
0.0  t  1.0: line segment AB
0.0  t < : ray
B
A
t <0
t >1
t =1
t =0
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Intersection of Two Line Segments
AB(t) = A + bt, where b = B -A
CD(u) = C + du, where d = D -C
 Parent lines of AB and CD intersect if
A + bt = C + du
Let c = C - A
 b t = c + du
 d•bt = d•c + d•du
 d•bt = d•c

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Intersection of Two Line Segments
Case I: d•b  0
 AB and CD are not parallel (why?)
t = (d•c)/(d•b)
Similarly, (how?)
u = (b•c)/(d•b)
 Q: When do two AB and CD intersect?
 A: When 0.0  t  1.0 and 0.0  u  1.0

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Intersection of Two Line Segments

Case II: d•b = 0
 AB and CD are parallel
If (bc) = 0, the line segments are colinear (why?)  have to test for overlap
If (bc)  0, the line segments are not colinear  they do not intersect

Examples
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Application – Circle through 3 Points
Given 3 non-collinear points A, B, and C,
find the circle passing through the 3 points
 excircle of triangle defined by the points
 The center S is the intersection of the
perpendicular bisectors of AB, BC, and CA

A
A
?
B
C
S
B
C
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Circle through 3 Points
Let a = B - A, b = C - B, c = A – C
 Midpoint of A = A + a/2 (why?)
Direction perpendicular to AB = a
 Parametric form of perpendicular
bisector of AB = A + a/2 + at (why?)
 Parametric form of perpendicular
bisector of CA = C + c/2 + cu

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Circle through 3 Points

The intersection of AB and CA is S
 A + a/2 + at = C + c/2 + cu
As a + b + c = 0 (why?)
 at = b/2 + cu (how?)
 t = 0.5(bc)/(ac) (how?)
 center S  A  1 (a  b  c a )

2
a c
a
bc 2
(  ) 1
radius = |S - A| =
2 a c
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Intersection of Line and Plane

Line in parametric form
R(t) = A + ct
where A is a known point on the line
c is the direction vector of the line
R is a point on the line
If t is allowed to vary from 0 to infinity,
R(t) is a ray
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Intersection of Line and Plane

Plane in point normal form
n(P -B) = 0
where n is a normal vector to the plane
B is a known point on the plane
P is any other point on the plane
If P is not on the plane,
n(P -B)  0
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Intersection of Line and Plane

If a ray R(t) intersect the plane, then
n(R(t) -B) = 0 (we want to find t)
 n(A + ct -B) = 0
 n(A -B) + nct = 0
 t (nc) = n(B -A)
P
P
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Intersection of Line and Plane
Case I: nc  0
t = n(B -A)/(nc)
 “Hit point” P = A + ct
 Case II: nc = 0
 ray and plane are parallel

If n(B -A) = 0, the ray lies on the plane
Otherwise, there is no intersection
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Direction Ray Hits the Plane

If nc > 0   < 90o
Ray is aimed along with the normal

If nc = 0   = 90o
Ray is parallel to the normal

If nc < 0   > 90o
Ray is aimed counter to the normal
n
n

c
A
along with
A
c

counter to
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