Matrices en computacion grafica

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Matrices in Computer Graphics

Ting Yip

Math 308A12/3/2001


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Abstract

In this paper, we discuss and explore the basic matrix operation such as translations,rotations, scaling and we will end the discussion with parallel and perspective view.

These concepts commonly appear in video game graphics.

Introduction

The use of matrices in computer graphics is widespread. Many industries like

architecture, cartoon, automotive that were formerly done by hand drawing now are doneroutinely with the aid of computer graphics. Video gaming industry, maybe the earliestindustry to rely heavily on computer graphics, is now representing rendered polygon in 3-

Dimensions.

In video gaming industry, matrices are major mathematic tools to construct andmanipulate a realistic animation of a polygonal figure. Examples of matrix operationsinclude translations, rotations, and scaling. Other matrix transformation concepts like

field of view, rendering, color transformation and projection. Understanding of matricesis a basic necessity to program 3D video games.

Homogeneous Coordinate Transformation

Points (x, y, z) in R3 can be identified as a homogeneous vector ( )

1,,,,,,

h

z

h

y

h

xhzyx with

h 0 on the plane in R4. If we convert a 3D point to a 4D vector, we can represent atransformation to this point with a 4 x 4 matrix.

The last coordinate is a scalar ter .

Graphics

(Screenshots taken from Operation Flashpoint)

Polygon figures like these use many flat or conic surfaces to represent a realistic human soldier.


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Transformation of Points

In general, transformation of points can be represented by this equation:

Transformed Point = Transformation Matrix

Original Point

In a more explicit case, a plane spanned by two vectors can be represented by this

equation:

=

=

y

x

fc

eb

da

MatrixtionTransforma

f

e

d

,

c

b

a

spanMatrixtionTransforma

PlaneOriginalMatrixtionTransforma=PlanedTransforme

Representation of a plane using matrices

EXAMPLE

Point (2, 5, 6) in R3 a Vector (2, 5, 6, 1) or (4, 10, 12, 2) in R4

NOTEIt is possible to apply transformation to 3D points without converting them to 4D

vectors. The tradeoff is that transformation can be done with a single matrixmultiplication after the convertion of points to vectors. (More on this after

Translation.)

x and y are scalars

cb

a

+

f

e

d

y

c

b

a

x

f

e

d


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Translation

A translation basically means adding a vector to a point, making a point transforms to anew point. This operation can be simplified as a translation in homogeneous coordinate

(x, y, z, 1) to (x + tx, y + ty, z + tz, 1). This transformation can be computed using a single

matrix multiplication.

Translation Matrix for Homogeneous Coordinates in R4 is given by this matrix:

=

1000

100

010

001

),,(z

y

x

zyxt

t

t

tttT

Given any point (x, y, z) in R3, the following will give the translated point.

+++

=

111000

100

010

001

z

y

x

z

y

x

tz

ty

tx

z

y

x

t

t

t

For a sphere to move to a new position, we can think of this as all the points on the sphere move to the

translated sphere by adding the blue vector to each point.


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Graphics


In video game, objects like airplane that doesnt change its shape dynamically (rigid body) uses

Translation to move across the sky. All the points that make up the plane have to be translated by the

same vector or the image of the plane will appear to be stretched.

NOTE

++

+

=

+

kz

jy

ix

k

j

i

z

y

x

If we have more than one point, we would have to apply this addition to every point.

++++

++

++

+

=

+

++

+

=

+

kzkz

jyjy

ixix

kz

jy

ix

k

j

i

z

y

x

kz

jy

ix

k

j

i

z

y

x

zz

yy

xx

21

21

21

2

2

2

2

2

2

1

1

1

1

1

1

21

21

21

& aa

With homogeneous coordinate, we can use a single matrix multiplication.

++++

++

++++++

=

kzkzjyjy

ixix

kzkz

jyjy

ixix

zz

yy

xx

k

j

i

21

21

21

21

21

21

21

21

21

11111000

100

010

001

As we can see, linear system is easier to solve with homogenenous coordinate

transformation.


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Scaling

Scaling of any dimension requires one of the diagonal values of the transformation matrix

to equal to a value other than one. This operation can be viewed as a scaling in

homogeneous coordinate (x, y, z, 1) to (sxx, syy, szz, 1). Values for sx, sy, szgreater thanone will enlarge the objects, values between zero and one will shrink the objects, and

negative values will rotate the object and change the size of the objects.

Scaling Matrix for Homogeneous Coordinates in R4 is given by this matrix:

=

1000

000

000

000

),,(z

y

x

zyxs

s

s

sssS

Given any point (x, y, z) in R3, the following will give the scaled point.

=

111000

000

000

000

zs

ys

xs

z

y

x

s

s

s

z

y

x

z

y

x

If we want to scale the hexahedron proportionally, we apply the same scaling matrix to each point that

makes up the hexahedron.


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Rotations

Rotations are defined with respect to an axis. In 3 dimensions, the axis of rotation needs

to be specified.

A rotation about the x axis is represented by this matrix:

+

=

=

=

1

cossin

sincos

11000

0cossin0

0sincos0

0001

1

)(

1000

0cossin0

0sincos0

0001

)(

zy

zy

x

z

y

x

z

y

x

RR xx

A rotation about the y axis is represented by this matrix:

A rotation about the z axis is represented by this matrix:

+

=

=

=

1

cossin

sincos

11000

0100

00cossin

00sincos

1

)(

1000

0100

00cossin

00sincos

)(z

yx

yx

z

y

x

z

y

x

RR zz

+

+

=

=

=

1

cossin

sincos

11000

0cos0sin

0010

0sin0cos

1

)(

1000

0cos0sin

0010

0sin0cos

)(

zx

y

zx

z

y

x

z

y

x

RR yy

3D rotation can be viewed as replacing x1and x2with two axes.


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EXAMPLEThis wire polygon cube is

represented by a matrix that contains

its vertex point in every column.

11111111

9977977933331111

75757755

Rotated Cube

Original Cube

If we want to rotate this cube with respect to the x axis by

3

:

11111111

99779779

33331111

75757755

1000

03

cos3

sin0

03

sin3

cos0

0001

=

5 5 7 7 5 7 5 7

1

2

9

23

1

2

7

23

1

2

7

23

1

2

9

23

3

2

7

23

3

2

7

23

3

2

9

23

3

2

9

23

+9

2

1

2 3 +7

2

1

2 3 +7

2

1

2 3 +9

2

1

2 3 +7

2

3

2 3 +7

2

3

2 3 +9

2

3

2 3 +9

2

3

2 3

1 1 1 1 1 1 1 1


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Projection Transformation

Even though we programmed objects in 3-Dimensions, we have to actually view the

objects as 2-Dimensions on our computer screens. In another word, we want to transformpoints in R3to points in R2.

Parallel Projection

In parallel projection, we simply ignore the z-coordinate. This operation can be viewedas a transformation in homogeneous coordinate (x, y, z, 1) to (x, y, 0, 1).

Parallel Matrix for Homogeneous Coordinates in R4 is given by this matrix:

=

1000

0000

0010

0001

P

Given any point (x, y, z) in R3, the following will give the parallel projected point.

=

1

0

11000

0000

0010

0001

y

x

z

y

x

Perspective Projection

Video game tends to use perspective projections over other projections to represent a realworld, where parallelism is not preserved. Perspective Projections is the way we see

things, i.e. bigger when the object is closer.

http://mane.mech.virginia.edu/~engr160/Graphics/Perspective.html


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Important:1) Translate the eye to the Origin

2) Rotation until direction of eye is toward the negative z-axis

D is the distance of the eye to the view plane

z is the distance of the eye to the object(Note: not your eyes but the eyes of the computer polygon person)

Perspective Matrix for Homogenous Coordinates in R4is given by this matrix:

01

00

0000

00100001

d

Given any point (x, y, z) in R3, the following will give the parallel projected point.

=

1

00

10

1

00

0000

0010

0001

d

zy

d

zx

d

z

y

x

z

y

x

d

(Note: This matrix transformation does not give pixel coordinate on the monitor. The transformed

coordinate is with respect to the objects coordinate. We have to translate the objects coordinate to pixel

coordinate on the monitor.)

Eye

Object

Perspective Projection of the cube


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Conclusion

I chose to do this project to show my curiosity in math and computer science. I had thechance to talk about video games and math that are often overlooked as unrelated. Asshown in this project,Linear Algebrais extremely useful for video game graphics. Using

matrices to manipulate points is a common mathematical approach in video gamegraphics.

Graphics


In sniping mode, the eye moves closer to the object.


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References

Dam, Andries. Introduction to Computer Graphicshttp://www.cs.brown.edu/courses/cs123/lectures/Viewing3.pdf

Holzschuch, Nicola. Projections and Perspectiveshttp://www.loria.fr/~holzschu/cours/HTML/ICG/Resources/Projections/index.html

Lay, David. Linear Algebra and its Applications. Second Edition.

Runwal, Rachana. Perspective Projectionhttp://mane.mech.virginia.edu/~engr160/Graphics/Perspective.html

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Matrices en computacion grafica