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Lecture 3 Hyper-planes, Matrices, and Linear Systems Scott Russell
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Lecture 3Hyper-planes, Matrices, and
Linear SystemsScott Russell
Guarding Art Gallery
Visibility Problem
Art Gallery Problem
To learn more about this problem, you can google “Art Gallery Problem” or google “Art Gallery Problems”
Visibility Problems:Intersection of Ray with Line or Plane
How to describe a line passing a point along a direction?How to describe a line and a plane?How to find their intersection?
Line in 2D
• By linear equation
12 yx
12 yx x=3y=1
Line in 2D
• By a point and a vector: passing (3,1) along vector (2,1)
ty
tx
ttyx
1
23
:)1,2()1,3(),(
set edParametriz
12
have we, geliminatinBy
yx
t
x=3y=1
Line in 2D
• By two points: passing (3,1) and (0,-1/2)
ttty
x
tty
x
:5.0
0
1
3)1(
have we term,same theCombine
:1
3
5.0
0
1
3
set edParametriz
12
have we, geliminatinBy
yx
t
(3,1)
(0,-1/2)
Line and Affine Combination in 2D
• The line passing two points or the affine combination of two points is given by
tb
at
b
at
b
a
b
a
b
a
b
a
:)1(
,affine,line
points twoofn Combinatio Affineor Line
2
2
1
1
2
2
1
1
2
2
1
1
System of Linear Equations (2D)
• Row Picture[conventional view]: two lines meets at a point
1123
12
yx
yx
1123 yx
12 yx x=3y=1
System of Linear Equations (2D)
• Column Picture: linear combination of the first two vectors produces the third vector
1123
12
yx
yx
11
1
2
2
3
1yx
And geometrically• Column Picture: linear combination of the
first two vector produce the third vector
11
1
2
2
3
1yx
2
2
11
1
3
1
3
13
x=3y=1
Coefficient Matrix and Matrix-Vector Product
1123
12
yx
yx
11
1
2
2
3
1yx
11
1
23
21
y
x
A 2 by 2 matrix is a square table of 4 numbers, two per row and two per column
System of Linear Equations (3D)
• Row Picture[conventional view]: Three planes meet at a single point
236
4252
632
zyx
zyx
zyx
2
0
0
• Row Picture[conventional view]: Two planes meet at a single line A line and a plane meet at a single point
Intersection of Planes
System of Linear Equations (3D)
• Column Picture: linear combination of the first three vectors produces the fourth vector
2
4
6
1
2
3
3
5
2
6
2
1
zyx
2
0
0
236
4252
632
zyx
zyx
zyx
Coefficient Matrix and Matrix-Vector Product
2
4
6
136
252
321
z
y
x
A 3 by 3 matrix is a square table of 9 numbers, three per row and three per column
236
4252
632
zyx
zyx
zyx
2
4
6
1
2
3
3
5
2
6
2
1
zyx
Matrix Vector Product (by row)
• If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture
xrow
xrow
xrow
Ax
)2 (
)2 (
)1 (
Matrix Vector Product (by column)
• If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture
3
2
1
321
where
)3 ()2 ()1 (
x
x
x
x
columnxcolumnxcolumnxAx
More about 3D Geometry
• Points and distance, Balls and Spheres– 0 dimension in 3 dimensions
• Lines– 1 dimension in 3 dimensions
• Plane– 2 dimensions in 3 dimensions
Line in 3D
• 2D– By linear equation – A point and a vector– Two points
• Affine combination
• 3D– A point and a vector– Two points
• Affine combination
Line in 3D
• By a point and a vector: passing p along vector v
zz
yy
xx
tvpz
tvpy
tvpx
ttvp
:
Line and Affine Combination in 3D• The line passing two points or the affine combination of
two points is given by
t
v
vv
t
u
uu
t
v
vv
u
uu
v
vv
u
uu
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
:)1(
,affine,line
points twoofn Combinatio Affineor Line
Plane in 3D
• Line in 2D– By linear equation – Affine combination of two points
• “Every” two points determine a line
• 3D– By linear equation– Affine combination of three points
• “Every” three points determine a plane
Linear Equation and its Normal
3
2
1
so
032
implying
632
632
have, we,, and ,, :on points any twofor
632:),,(
21
21
21
212121
222
111
222111
zz
yy
xx
zzyyxx
zyx
zyx
zyxzyxP
zyxzyxP
Normal of a Plane
Plane and Affine Combination in 3D
)1(
,,affine
,,plane
321
321
321
ppp
ppp
ppp
1p
2p
3p
u
v
s
tsst
psptsstppssuv
pttpu
)1(,
)1()1( )1(
)1(
3213
21
High Dimensional Geometric Extension• Points and distance, Balls and Spheres
– 0 dimension in n dimensions
• Lines– 1 dimension in n dimensions
• Plane– 2 dimensions in n dimensions
• k-flat – k-dimensions in n dimensions
• Hyper-plane – (n-1)-dimensions in n dimensions
Affine Combination in n-D
1
,,,affine
1
12211
21
k
k
j j
k
ppp
ppp
Hyper-Planes in d-D
• Line in 2D– By linear equation – Affine combination of two points
• 3D– By linear equation– Affine combination of three points
• n-D– By linear equation– Affine combination of n-1 points
Linear Equation and its Normal
n
n
k kkk
n
k kk
n
k kk
nn
n
k kkn
aaaa
where
ayx
yxa
bya
bxa
yyyxxxP
bxaxxxP
,,,
)(
so
0)(
implying
, ,,,y and ,,, x:on points Any two
:),,,(
21
1
1
1
2121
121
Matrix (Uniform Representation for Any Dimension)
• An m by n matrix is a rectangular table of mn numbers
ji
nmmm
n
n
ajiA
aaa
aaa
aaa
A
,
,2,1,
,22,21,2
,12,11,1
),( write weSometime
...
...
...
...
