201707 SER332 Lecture 13

SER332
Introduction to Graphics and
Game Development
Lecture 13:
Linear Algebra
Javier Gonzalez-Sanchez
javiergs@asu.edu
PERALTA 230U
Office Hours: By appointment

Javier Gonzalez-Sanchez | SER332 | Spring 2017 | 1
Goal

Goal

Goal
3D Maze
( IJKS 1982)

Points and Vectors
Points specify locations in
space (or in the plane)
Vectors have a
magnitude and direction
x
)4,2(y
)2,4(
q
v
1v
2v

Vectors
2
2
1
R
v
v
v Îú
û
ù
ê
ë
é
=
2
2
2
1 vvv +=
Vector
Magnitude or
Length
v
÷÷
ø
ö
çç
è
æ
= -
1
21
tan
v
v
qOrientation
(angle)

Vectors
ú
û
ù
ê
ë
é
=
0
0
vZero Vector
Unit Vector or
Normalized Vector
1=v

Addition and Subtraction
v
u
u+v
u
v
u-v

Multiplication with a Scalar
v
av

Test yourself
§ Point + Vector = ?
§ Vector + Vector = ?
§ Point – Point = ?
§ Point + Point = ?

Point or Vector
Point + Vector = Point Vector + Vector = Vector
Point - Point = Vector
v
u
u-v
v
u
v-u Point + Point = Not Defined

Angles

Uses
// mouse motion callback function
void motion(int x, int y) {
if (mouse_button == GLUT_LEFT_BUTTON) {
y_angle += (float(x - mouse_x) / width) *360.0;
x_angle += (float(y - mouse_y) / height)*360.0;
}
if (mouse_button == GLUT_RIGHT_BUTTON) {
scale += (y - mouse_y) / 100.0;
if (scale < 0.1) scale = 0.1;
if (scale > 7) scale = 7; }
mouse_x = x;
mouse_y = y;
glutPostRedisplay();
}

Dot Product
• The dot product or inner product measures to what
degree two vectors are aligned
• Notation:
• Computation:
wvwvwv T
×==>< ,
[ ] 332211
3
2
1
321, wvwvwv
w
w
w
vvvwv ×+×+×=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=><

Dot Product
q
v=[5,5]
u=[7,3]
v * u = (5*7) + (5*3) = 50
v * w = (5*5) + (5*0) = 25
u * w = (7*5) + (3*0) = 35
u * u = (7*7) + (3*3) = 58
t * w = (0*5) + (5*0) = 0
w=[5,0]
t=[0,5]

Dot Product
• Two vectors v and w are perpendicular or normal iff
0, =>< wv

Dot Product
• Geometric Interpretation:
w
v
a
l
wv
wv
w
l
COS
×
==
,
)(a

Dot Product
q
v=[5,5]
u=[7,3]
v * u = (5*7) + (5*3) = 50
v * w = (5*5) + (5*0) = 25
u * w = (7*5) + (3*0) = 35
u * u = (7*7) + (3*3) = 58
t * w = (0*5) + (5*0) = 0
w=[5,0]
t=[0,5]
|t | = 5.00
|v| = 7.07
|u| = 7.61
|w| = 5.00
q = cos-1 (25/7.07*5) = 45º

Dot Product Calculation
Length = Magnitude
vvvvv T
== ,

Dot Product Laws
• Commutative law:
• Distributive law:
><=>< vwwv ,,
><+><=>+< uvwvuwv ,,,

Uses

Cross Product
• Notation: u = v x w
• The cross product is a vector!
• Magnitude of u: proportional to
the sine of the angle between v
and w
• u is perpendicular to v and w
• The direction of u follows the
right hand rule
w
v
a
u
asin|||||||||||||u|| wvwv =´=

Cross Product Computation
Computation:
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
-
-
==
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
´
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
=´=
2121
1313
3232
321
321
3
2
1
3
2
1
vwwv
vwwv
vwwv
www
vvv
kji
w
w
w
v
v
v
wvu

Cross Product Laws
• Distributive law:
• Commutative law:
• Associative law:
vwwv ´-=´
uvwvuwv ´+´=+´ )(
uwvuwv ´´¹´´ )()(

Cross Product and Triangle Area
The cross product is related to the area of a triangle
2
PRPQ
area
´
=
Q
R
P
asin|||||||||||||u|| wvwv =´=(parallelogram) area =
(triangle)
v
w

Test Yourself
Given a Triangle with
P[0,0,0], Q[0,5,0] and R[5,5,0]
What is the area?
Q R
P

Test Yourself
Given a Triangle with
P[0,0,0], Q[0,5,0] and R[5,5,0]
What is the area?
Q R
P
area (triangle) = [0,5,0] x [5,5,0] = |[0, 0, -25]|/2 = 25/2 = 12.5
area (parallelogram) = 25
area (parallelogram) = (5)(7.07) sin (45) = (5)(7.07...)(0.70...) = 25

Distance point to line
n
0p
p
//Compute the distance from AB to C
double LineToPointDistance2D
(double[] pA, double[] pB, double[] pointC) {
double dist = CrossProduct(pA, pB, pointC) / distance(pA, pB);
return Math.Abs(dist);
}

Usage: normal calculation (lighting)

Usage: collision detection

Matrix
• A matrix is a set of elements, organized into rows
and columns
ú
û
ù
ê
ë
é
dc
ba
rows
columns

Matrix Addition
• Addition
• Example
• Substraction
ú
û
ù
ê
ë
é
++
++
=ú
û
ù
ê
ë
é
+ú
û
ù
ê
ë
é
hdgc
fbea
hg
fe
dc
ba
ú
û
ù
ê
ë
é
=ú
û
ù
ê
ë
é
+ú
û
ù
ê
ë
é
64
78
51
26
13
52
ú
û
ù
ê
ë
é
--
--
=ú
û
ù
ê
ë
é
-ú
û
ù
ê
ë
é
hdgc
fbea
hg
fe
dc
ba

Matrix Multiplication
• Multiplication
ú
û
ù
ê
ë
é
++
++
=ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
dhcfdgce
bhafbgae
hg
fe
dc
ba Multiply each row by
each column

Matrix Multiplication
pmmnpn BAC ´´´ =
Multiplication:
ú
û
ù
ê
ë
é
=ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
1119
2917
51
26
.
13
52
Examples:
ú
û
ù
ê
ë
é
=ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
1017
3218
13
52
.
51
26
A and B must have
compatible dimensions

Transpose
• Notation:
• Definition:
• Examples: )()( ji
T
ij AA =
ú
û
ù
ê
ë
é
=ú
û
ù
ê
ë
é
52
16
51
26
T
ú
û
ù
ê
ë
é
=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
852
316
83
51
26
T
T
A

Rules
TTT
ABAB =)(
TTT
BABA +=+ )(

Symmetric Matrices
• A is symmetric iff AAT
=

Determinants
• Determinant (Note A must be square)
• Example
13152
13
52
det -=-=ú
û
ù
ê
ë
é
3231
2221
13
3331
2321
12
3332
2322
11
333231
232221
131211
det
aa
aa
a
aa
aa
a
aa
aa
a
aaa
aaa
aaa
+-=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
12212211
2221
1211
2221
1211
det aaaa
aa
aa
aa
aa
-==ú
û
ù
ê
ë
é

Inverse Matrices
• Notation
• I is the Identity (1s on the diagonal, 0s elsewhere)
• Inverse Computation:
IAAAA == -- 11
ú
û
ù
ê
ë
é
-
-
-
=ú
û
ù
ê
ë
é
-
1121
1222
12212211
1
2221
1211 1
aa
aa
aaaaaa
aa

Test Yourself
• Example
ú
û
ù
ê
ë
é
-
-
=ú
û
ù
ê
ë
é
-
61
25
28
1
51
26
1
ú
û
ù
ê
ë
é
=ú
û
ù
ê
ë
é
=ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
-
-
=ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
-
10
01
280
028
28
1
51
26
.
61
25
28
1
51
26
.
51
26
1

Usage: transformations

Homework
Review Linear Algebra

SER332 Advanced Graphics
Javier Gonzalez-Sanchez
javiergs@asu.edu
Spring 2017
Disclaimer. These slides can only be used as study material for the class SER332 at ASU. They cannot be distributed or used for another purpose.

201707 SER332 Lecture 13

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201707 SER332 Lecture 13