Transforms (1)

7/31/2019 Transforms (1)

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Geometric Transformations

Jehee Lee

Seoul National University


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Transformations

Linear transformations

Rigid transformations

Affine transformations

Projective transformations

T

Global reference frame

Local moving frame


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Linear Transformations

A linear transformationT is a mapping betweenvector spaces

Tmaps vectors to vectors

linear combination is invariant under T

In 3-spaces, T can be represented by a 3x3 matrix

)()()()( 11000

NN

N

i

ii TcTcTccT vvvv

major)(Row

major)(Column)(

3331

1333

Nv

vMvT


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Examples of Linear Transformations

2D rotation

2D scaling

y

x

y

x

cossin

sincos

ysxs

yx

ss

yx

y

x

y

x

00

yx, yx ,


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2D shear Along X-axis

Along Y-axis

y

dyx

y

xd

y

x

10

1

dxy

x

y

x

dy

x

1

01


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2D reflection Along X-axis

Along Y-axis

y

x

y

x

y

x

10

01

y

x

y

x

y

x

10

01


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Properties of Linear Transformations

Any linear transformationbetween 3D spacescan be represented by a 3x3 matrix

Any linear transformationbetween 3D spacescan be represented as a combination of rotation,shear, and scaling

Rotationcan be represented as a combinationof scalingand shear


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Changing Bases

Linear transformations as a change of bases

yx, yx ,

0v

1v1v

0v

y

x

y

x

yxyx

1010

1010

vvvv

vvvv


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Changing Bases

Linear transformations as a change of bases yx, yx ,

0

v

1v1v

0v

11001

11000

vvv

vvv

bb

aa

y

xT

y

x

ba

ba

y

x

y

x

ba

ba

y

x

11

00

11

00

1010 vvvv


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Affine Transformations

An affine transformationT is an mappingbetween affine spaces

Tmaps vectors to vectors, and points to points

T is a linear transformation on vectors

affine combination is invariant under T

In 3-spaces, T can be represented by a 3x3matrix together with a 3x1 translation vector

)()()()( 11000

NN

N

i

ii TcTcTccT pppp

131333)( TpMpT


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Homogeneous Coordinates

Any affine transformation between 3D spacescan be represented by a 4x4 matrix

Affine transformation is linear in homogeneouscoordinates

110)(

131333 pTMpT


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Examples of Affine Transformations

2D rotation

2D scaling

1100

0cossin

0sincos

1

y

x

y

x

yx, yx ,

11100

00

00

1

ys

xs

y

x

s

s

y

x

y

x

y

x


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2D shear

2D reflection

11100

010

01

1

y

dyx

y

xd

y

x

11100

010001

1

yx

yx

yx


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2D translation

11100

10

01

1y

x

y

x

ty

tx

y

x

t

t

y

x


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2D transformation for vectors Translationis simply ignored

00100

1001

0

yx

yx

tt

yx

y

x


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3D rotation

11000

0cossin0

0sincos0

0001

1

z

y

x

z

y

x

11000

0100

00cossin

00sincos

1

z

y

x

z

y

x

11000

0cos0sin

0010

0sin0cos

1

z

y

x

z

y

x


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Composite Transformations

Composite 2D Translation

),(

),(),(

2121

2211

yyxx

yxyx

tttt

ttttT

T

TT

100

10

01

100

10

01

100

10

01

21

21

1

1

2

2

yy

xx

y

x

y

x

tt

tt

t

t

t

t


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Composite 2D Scaling

),(

),(),(

2121

2211

yyxx

yxyx

ssss

ssssT

S

SS

100

00

00

100

00

00

100

00

00

21

21

1

1

2

2

yy

xx

y

x

y

x

ss

ss

s

s

s

s


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Composite 2D Rotation

)(

)()(

12

12

R

RRT

100

0)cos()sin(

0)sin()cos(

100

0cossin

0sincos

100

0cossin

0sincos

1212

1212

11

11

22

22


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Composing Transformations

Suppose we want,

We have to compose two transformations

)90( R )3,(xT


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Matrix multiplication is not commutative

)3,()90()90()3,( xx TRRT

Translationfollowed by

rotation

Translationfollowed by

rotation


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R-to-L : interpret operations w.r.t. fixed coordinates

L-to-R : interpret operations w.r.t local coordinates

)90()3,( RT xT

)90(R

)3,(xT

)90( R)3,(xT

(Column major convention)


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Pivot-Point Rotation

Rotation with respect to a pivot point (x,y)

100

sin)cos1(cossin

sin)cos1(sincos100

10

01

100

0cossin

0sincos

100

10

01

),()(),(

xy

yx

y

x

y

x

yxTRyxT


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Fixed-Point Scaling

Scaling with respect to a fixed point (x,y)

100

)1(0

)1(0

100

10

01

100

00

00

100

10

01

),(),(),(

yss

xss

y

x

s

s

y

x

yxTssSyxT

yy

xx

y

x

yx


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Scaling Direction

Scaling along an arbitrary axis

)(),()(1 RssSR yx

)(1 R),( yx ssS)(R


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Properties of Affine Transformations

Any affine transformationbetween 3D spaces can berepresented as a combination of a linear transformationfollowed by translation

An affine transf. maps linesto lines

An affine transf. maps parallel linesto parallel lines

An affine transf. preserves ratios of distancealong a line

An affine transf. does not preserve absolute distancesand angles


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Review of Affine Frames

A frameis defined as a set of vectors {vi | i=1, , N}and a point o

Set of vectors {vi} are bases of the associate vectorspace

o is an origin of the frame N is the dimension of the affine space

Any point p can be written as

Any vector v can be written as

NNccc vvvop 2211

NNccc vvvv 2211


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Changing Frames

Affine transformations as a change of frame

yx,

0v

1v

11

1010

1010

y

x

y

x

yxyx

ovvovv

ovvovv

0v1v

o

o


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Changing Frames

Affine transformations as a change of frame

ovvo

vvv

vvv

100

11001

11000

cc

bb

aa

11001

11001

111

000

111

000

1010

y

x

cba

cba

y

x

y

x

cba

cba

y

x

ovvovv

yx,

0v

1v0v

1v

o

o


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Changing Frames

In case the xyz system has standard bases


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Rigid Transformations

A rigid transformationT is a mapping betweenaffine spaces

Tmaps vectors to vectors, and points to points

Tpreserves distances between all points

Tpreserves cross product for all vectors (to avoidreflection)

In 3-spaces, T can be represented as

1detand

where,)( 131333

RIRRRR

TpRp

TT

T


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Rigid Body Rotation

Rigid body transformations allow only rotationand translation

Rotation matrices form SO(3)

Special orthogonal group

IRRRR TT

1det R

(Distance preserving)

(No reflection)


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Rigid Body Rotation

R is normalized The squares of the elements in any row or column

sum to 1

R is orthogonal The dot product of any pair of rows or any pair

columns is 0

The rows (columns) of R correspond to thevectors of the principle axes of the rotatedcoordinate frame

IRRRR TT


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3D Rotation About Arbitrary Axis


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1. Translation : rotation axis passes through theorigin

2. Make the rotation axis on the z-axis

3. Do rotation

4. Rotation & translation

),,( 111 zyxT

)()( yx RR

)(zR

)()(111

xy RRT


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Rotate u onto the z-axis


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u: Project u onto the yz-plane to compute angle

u: Rotate u about the x-axis by angle



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Rotate u about the x-axis onto the z-axis Let u=(a,b,c) and thus u=(0,b,c)

Let uz=(0,0,1)

22cos

cbc

z

z

uu

uu

bx

zxz

u

uuuuu sin22

sincb

bb

z

uu


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Rotate u about the x-axis onto the z-axis Since we know both cos and sin , the rotation

matrix can be obtained

Or, we can compute the signed angle

Do not use acos() since its domain is limited to [-1,1]

1000

00

00

0001

)(

2222

2222

cb

c

cb

b cb

b

cb

c

x R

),(atan22222 cb

b

cb

c


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Euler angles

Arbitrary orientation can be represented by threerotation along x,y,z axis

1000

0

00CS-SSCCC

)()()(),,(

CCSCS

SCCSSCCSSSCSSSCSC

RRRR xyzXYZ


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Gimble

Hardware implementation of Euler angles Aircraft, Camera


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Euler Angles

Rotation about threeorthogonal axes 12 combinations

XYZ, XYX, XZY, XZX

YZX, YZY, YXZ, YXY

ZXY, ZXZ, ZYX, ZYZ

Gimble lock

Coincidence of inner mostand outmost gimbles rotation

axes Loss of degree of freedom


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Euler Angles

Euler angles are ambiguous Two different Euler angles can represent the same

orientation

This ambiguity brings unexpected results of animationwhere frames are generated by interpolation.

),2

,0and)0,2

,21

(R(),r,r(rRzyx


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Taxonomy of Transformations

Linear transformations 3x3 matrix

Rotation + scaling + shear

Rigid transformations

SO(3) for rotation 3D vector for translation

Affine transformation 3x3 matrix + 3D vector or 4x4 homogenous matrix

Linear transformation + translation

Projective transformation 4x4 matrix

Affine transformation + perspective projection


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Taxonomy of Transformations

Projective

Affine

Rigid


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Composition of Transforms

What is the composition of linear/affine/rigidtransformations ?

What is the linear (or affine) combination oflinear (or affine) transformations ?

What is the linear (or affine) combination of rigid

transformations ?


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OpenGL Geometric Transformations

glMatrixMode(GL_MODELVIEW);


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OpenGL Geometric Transformations

Construction glLoadIdentity();

glTranslatef(tx, ty, tz);

glRotatef(theta, vx, vy, vz);

(vx, vy, vz)is automatically normalized

glScalef(sx, sy, sz);

glLoadMatrixf(Glfloat elems[16]);

Multiplication glMultMatrixf(Glfloat elems[16]);

The current matrix is postmultiplied by the matrix

Row major


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Hierarchical Modeling

A hierarchical model is created by nesting thedescriptions of subparts into one another to form a treeorganization


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OpenGL Matrix Stacks

Four matrix modes Modelview, projection, texture, and color

glGetIntegerv(GL_MAX_MODELVIEW_STACK_DEPTH,

stackSize);

Stack processing

The top of the stack is the current matrix

glPushMatrix(); // Duplicate the current matrix at the top glPopMatrix(); // Remove the matrix at the top


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Programming Assignment #1

Create a hierarchical model using matrix stacks

The model should consists of three-dimensionalprimitives such as polygons, boxes, cylinders,spheres and quadrics.

The model should have a hierarchy of at leastthree levels

Animate the model to show the hierarchicalstructure

Eg) a car driving with rotating wheels Eg) a runner with arms and legs swing

Make it aesthetically pleasing or technicallyillustrative

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Transforms (1)