CSE328 Fundamentals of Computer Graphics: …cse328/2012-lecture-notes/...ST NY BR K STATE...

STNY BRKSTATE UNIVERSITY OF NEW YORK

Department of Computer Science

Center for Visual Computing

CSE328 Fundamentals of Computer Graphics: Theory, Algorithms, and Applications

Hong Qin


State University of New York at Stony Brook (Stony Brook University)

Stony Brook, New York 11794-4400

Tel: (631)632-8450; Fax: (631)632-8334

[email protected]

http://www.cs.sunysb.edu/~qin




2D-3D Transformations

• From local, model coordinates to global, world

coordinates




Modeling Transformations

• 2D-3D transformations

• Specify transformations for objects

– Allows definitions of objects in their own coordinate

systems

– Allows use of object definition multiple times in a

scene

– Please pay attention to how OpenGL provides a

transformation stack because they are so frequently

reused




Overview

• 2D Transformations

– Basic 2D transformations

– Matrix representation

– Matrix composition



– Same as 2D




From Model Coordinates to World Coordinates (Local to Global)

CSE528 Lectures

x

yModel coordinates (local)

World coordinates (global)




Basic 2D Transformations• Translation:

– x’ = x + tx

– y’ = y + ty

• Scale:

– x’ = x * sx

– y’ = y * sy

• Shear:

– x’ = x + hx*y

– y’ = y + hy*x

• Rotation:

– x’ = x*cosQ - y*sinQ

– y’ = x*sinQ + y*cosQ




Scaling• Scaling a coordinate means multiplying each of

its components by a scalar

• Uniform scaling means this scalar is the same for all components:

• Non-uniform scaling: different scalars per component:

• How can we represent scaling in matrix form?




Scaling Operation in Matrix Form

by

ax

y

x

'

'

y

x

b

a

y

x

0

0

'

'




Scaling

• Scaling operation:

• Or, in matrix form:

by

ax

y

x

'

'

y

x

b

a

y

x

0

0

'

'

scaling matrix




Rotation




2-D Rotation

x’ = x cos() - y sin()

y’ = x sin() + y cos()




2D Rotation Derivation• x = r cos (f)

• y = r sin (f)

• x’ = r cos (f + )

• y’ = r sin (f + )

• x’ = r cos(f) cos() – r sin(f) sin()

• y’ = r sin(f) sin() + r cos(f) cos()

• x’ = x cos() - y sin()

• y’ = x sin() + y cos()




2-D Rotation

• It is straightforward to see this procedure in

matrix form:

Even though sin() and cos() are nonlinear

functions of ,

– x’ is a linear combination of x and y

– y’ is a linear combination of x and y

y

x

y

x

cossin

sincos

'

'




2D Rotation

y

x

y

x

cossin

sincos

'

'





– x’ = x + tx

– y’ = y + ty

• Scale:– x’ = x * sx

– y’ = y * sy

• Shear:– x’ = x + hx*y

– y’ = y + hy*x

• Rotation:– x’ = x*cosQ - y*sinQ






– x’ = x + tx

– y’ = y + ty

• Scale:– x’ = x * sx

– y’ = y * sy

• Shear:– x’ = x + hx*y

– y’ = y + hy*x

• Rotation:– x’ = x*cosQ - y*sinQ


Transformations can be combined (with simple algebra)




Combining Transformations• Transformations can be combined (with

simple algebra)




Composite Transformations

Transformations can be combined

(with simple algebra)




Matrix Representation

• Represent 2D transformation by a matrix

• Multiply matrix by column vector

apply transformation to point




Overview







– Same as 2D





dcba

yx

dcba

yx''

dycxy

byaxx

'

'





• Represent 2D transformation by a matrix

• Multiply matrix by column vector

apply transformation to point

yx

dcba

yx''

dcba

dycxy

byaxx

'

'





yx

lk

ji

hg

fedcba

yx''





• Transformations can be combined by

multiplication

– Matrices are a convenient and efficient way to

represent a sequence of transformations!





• Transformations combined by multiplication

yx

lk

ji

hg

fedcba

yx''

Matrices are a convenient and efficient way

to represent a sequence of transformations!




2x2 Matrices

• What types of transformations can be

represented with a 2x2 matrix?

2D Identity?

yyxx

''

yx

yx

1001

''

2D Scale around (0,0)?

ysy

xsx

y

x

*'

*'

y

x

s

s

y

x

y

x

0

0

'

'




2x2 Matrices



2D Rotate around (0,0)?

yxyyxx

*cos*sin'*sin*cos'

QQQQ

QQ

QQ

y

x

y

x

cossin

sincos

'

'

2D Shear?

yxshy

yshxx

y

x

*'

*'

y

x

sh

sh

y

x

y

x

1

1

'

'




2x2 Matrices



2D Mirror about Y axis?

yyxx

''

yx

yx

1001

''

2D Mirror over (0,0)?

yyxx

''

yx

yx

1001

''




2x2 Matrices



2D Translation?

y

x

tyy

txx

'

'

Only linear 2D transformations

can be represented with a 2x2 matrix

NO!




Linear Transformations• Linear transformations are combinations of …

– Scale,

– Rotation,

– Shear, and

– Mirror

• Properties of linear transformations:

– Satisfies:

– Origin maps to origin

– Lines map to lines

– Parallel lines remain parallel

– Ratios are preserved

– Closed under composition

)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x

'

'




Homogeneous Coordinates

• Q: How can we represent translation as a 3x3

matrix?

y

x

tyy

txx

'

'





• Homogeneous

coordinates

– represent coordinates in 2

dimensions with a 3-

vector

1

coords shomogeneou y

x

y

x

Homogeneous coordinates appear to be far

less intuitive, but they indeed make graphics

operations much easier





• Q: How can we represent translation as a 3x3

matrix?

• A: Using the rightmost column:

100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx

'

'




Translation

• Example of translation

11100

10

01

1

'

'

y

x

y

x

ty

tx

y

x

t

t

y

x





• Add a 3rd coordinate to every 2D point

– (x, y, w) represents a point at location (x/w, y/w)

– (x, y, 0) represents a point at infinity

– (0, 0, 0) is not allowed

• Note that, (6,3,1); (12,6,2); and (18,9,3) represent the SAME

POINT in 2D

Convenient coordinate system to

represent many useful transformations




Basic 2D Transformations

• Basic 2D transformations as 3x3 matrices

QQ

QQ

1100

0cossin

0sincos

1

'

'

y

x

y

x

1100

10

01

1

'

'

y

x

t

t

y

x

y

x

1100

01

01

1

'

'

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

'

'

y

x

s

s

y

x

y

x

Scale




Affine Transformations

• Affine transformations are combinations of …

– Linear transformations, and

– Translations

• Properties of affine transformations:

– Origin does not necessarily map to origin


– Parallel lines remain parallel

– Ratios are preserved


w

yx

fedcba

w

yx

100

''




Projective Transformations

• Projective transformations …

– Affine transformations, and

– Projective warps

• Properties of projective transformations:

– Origin does not necessarily map to origin


– Parallel lines do not necessarily remain parallel

– Ratios are not preserved


w

yx

ihg

fedcba

w

yx

'

''




Overview







– Same as 2D




Matrix Composition

• Transformations can be combined by

matrix multiplication

QQQQ

w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001

'

''

p’ = T(tx,ty) * R(Q) * S(sx,sy) * p




Matrix Composition

• Matrices are a convenient and efficient way to

represent a sequence of transformations

– General purpose representation

– Hardware matrix multiply

p’ = (T * (R * (S*p) ) )

p’ = (T*R*S) * p




Matrix Composition

• From local coordinates to global coordinates

• Be aware: order of transformations matters• Matrix multiplication is not commutative

p’ = T * R * S * p




Matrix Composition

• A more complicated example: rotating 90

degrees around the mid-point of a line segment

(whose coordinates are (3,2))

• Can we change the order between rotation and

translation?




Will this sequence of operations work?

Matrix Composition

1

'

'

1100

210

301

100

0)90cos()90sin(

0)90sin()90cos(

100

210

301

y

x

y

x

a

a

a

a




Matrix Composition

• After correctly ordering the matrices

• Multiply matrices together

• What results is one matrix – store it (on stack)!

• Multiply this matrix by the vector of each vertex

• All vertices easily transformed with one matrix

multiply




Overview







– Same as 2D




3D Transformations

• Same idea as 2D transformations

– Homogeneous coordinates: (x,y,z,w)

– 4x4 transformation matrices

wz

yx

ponm

lkji

hgfedcba

wz

yx

''

''





wz

yx

wz

yx

1000010000100001

'

''

w

z

y

x

t

t

t

w

z

y

x

z

y

x

1000

100

010

001

'

'

'

w

z

y

x

s

s

s

w

z

y

x

z

y

x

1000

000

000

000

'

'

'

wz

yx

wz

yx

1000010000100001

'

''

Identity Scale

Translation Mirror about Y/Z plane





QQQQ

wz

yx

wz

yx

1000010000cossin00sincos

'

''

Rotate around Z axis:

QQ

QQ

w

z

y

x

w

z

y

x

1000

0cos0sin

0010

0sin0cos

'

'

'

Rotate around Y axis:

QQQQ

wz

yx

wz

yx

10000cossin00sincos00001

'

''

Rotate around X axis:




Reverse Rotations• Q: How do you undo a rotation of , R()?

• A: Apply the inverse of the rotation… R-1() = R(-)

• How to construct R-1() = R(-)

– Inside the rotation matrix: cos() = cos(-)

• The cosine elements of the inverse rotation matrix are

unchanged

– The sign of the sine elements will flip

– This is because the rotation matrix is orthogonal matrix

• Therefore… R-1() = R(-) = RT()




Summary• Coordinate systems are the basis for computer graphics

– World vs. model coordinates (Global vs. Local)

• 2-D and 3-D transformations

– Trigonometry and geometry

– Matrix representations

– Linear, affine, and projective transformations

• Matrix operations


CSE328 Fundamentals of Computer Graphics: …cse328/2012-lecture-notes/...ST NY BR K STATE...

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