1 Modeling Transformations 2D Transformations 3D Transformations OpenGL Transformation

Modeling Transformations

2D Transformations3D TransformationsOpenGL Transformation

2D-Transformations

Basic TransformationsHomogeneous coordinate systemComposition of transformations

Translation – 2D

(4,5) (7,5)

XBefore Translation

Form sHomogeniou

x’ = x + dx y’ = y + dy

(7,1) (10,1)

Translation by (3,-4)

Scaling – 2D

(4,5) (7,5)

X(2,5/4) (7/2,5/4)

Before Scaling Scaling by (1/2, 1/4)

Types of Scaling:

Differential ( sx != sy )Uniform ( sx = sy )

Form sHomogeniou

Rotation – 2D

cossinsincos

sinsincoscos

rrx expand

cossin

sincos

original

Rotation – 2D

(5,2) (9,2)

(2.1,4.9)

(4.9,7.8)

YBefore Rotation Rotation of 45 deg. w.r.t. origin

0cossin

0sincos

Form sHomogeniou

cos*sin*

sin*cos**

cossin

sincos

cos*sin*

sin*cos*

Mirror Reflection

axis-Xabout Reflection

axis-Yabout Reflection

(1,-1)

(-1,1) (1,1)

Shearing Transformation

SH xyyx

unit cubeSheared in X

directionSheared in Y

directionSheared in both X

and Y direction

Inverse 2D - Transformations

),(-(sx,sy)

(-θ-(θ

(-dx,-dy)-

(dx,dy)

:RefMirror

: Sclaing

: Rotation

:nTranslaito

Homogeneous Co-ordinates

Translation, scaling and rotation are expressed (non-homogeneously) as:– translation: P = P + T– Scale: P = S · P– Rotate: P = R · P

Composition is difficult to express, since translation not expressed as a matrix multiplication

Homogeneous coordinates allow all three to be expressed homogeneously, using multiplication by 3 matrices

W is 1 for affine transformations in graphics

Homogeneous Co-ordinates

P2d is a projection of Ph onto the w = 1 plane

So an infinite number of points correspond to : they constitute the whole line (tx, ty, tw)

w Ph(x,y,w)

P2d(x,y,1)

Classification of Transformations

1. Rigid-body Transformation Preserves parallelism of lines Preserves angle and length e.g. any sequence of R() and T(dx,dy)

2. Affine Transformation Preserves parallelism of lines Doesn’t preserve angle and length e.g. any sequence of R(), S(sx,sy) and T(dx,dy)

unit cube 45 deg rotaton Scale in X not in Y

Properties of rigid-body transformation

1002221

The following Matrix is Orthogonal if the upper left 2X2 matrix has the following properties

1.A) Each row are unit vector. sqrt(r11* r11 + r12* r12) = 1

B) Each column are unit vector. sqrt(c11* c11 + c12* c12) = 1

2.A) Rows will be perpendicular to each other (r11 , r12 ) . ( r21 , r22) = 0

B) Columns will be perpendicular to each other (c11 , c12 ) . (c21 ,c22) = 0

e.g. Rotation matrix is orthogonal

0cossin

0sincos

• Orthogonal Transformation Rigid-Body Transformation

• For any orthogonal matrix B B-1 = BT

Commutativity of Transformation Matrices

• In general matrix multiplication is not commutative• For the following special cases commutativity holds

i.e. M1.M2 = M2.M1

Translate Translate

Scale Scale

Rotate Rotate

Uniform Scale Rotate

• Some non-commutative Compositions: Non-uniform scale,

Rotate Translate, Scale Rotate, Translate

OriginalTransitional

Associativity of Matirx Multiplication

Create new affine transformations by multiplying sequences of the above basic transformations.

q = CBAp

q = ( (CB) A) p = (C (B A))p = C (B (Ap) ) etc.

matrix multiplication is associative.

But to transform many points, best to do

M = CBA

then do q = Mp for any point p to be rendered.

To transform just a point, better to do q = C(B(Ap))

For geometric pipeline transformation, define M and set it up with the model-view matrix and apply it to any vertex subsequently defined to its setting.

Rotation of about P(h,k): R,P

Q(x,y)

P(h,k)

Step 1: Translate P(h,k) to origin

T(-h ,-k)

Q1(x’,y’

P1 (0,0)

Step 2: Rotate w.r.t to origin

Q2(x’,y’

P2 (0,0)

Step 3: Translate (0,0) to P(h,k0)

T(h ,k) *P3(h,k)

Q3(x’+h, y’

Scaling w.r.t. P(h,k): Ssx,sy,p

T(-h ,-k)S(sx,sy)*T(h ,k) *Ssx,sy,P=

S3/2,1/2,(1,1)

Step 1: Translate P(h,k) to origin

(4,0)T(-1,-1)

Step 2: Scale S(sx,sy) w.r.t origin

(0,0) S(3/2,1/2)

Step 3: Translate (0,0) to P(h,k)(7,2)

(1,1) T(1,1

Reflection about line L, ML

Step 1: Translate (0,b) to origin

T(0 ,-b)ML =

Step 2: Rotate - degrees

Step 3: Mirror reflect about X-axis

R(-) *T(0 ,b) *

Step 4: Rotate degrees

Step 5: Translate origin to (0,b)

M x*R() *

Problems to be solved:

Schaum’s outline series:

Problems: 4.1 4.2

4.3, 4.4, 4.5 => R,P

4.6, 4.7, 4.8 => Ssx,sy,,P

4.9, 4.10, 4.11, 4.21 => ML

4.12 => Shearing Pg-281(1.24), Pg-320(5.19)

=> Circular view-port

3D Transformations

Basics of 3D geometryBasic 3D TransformationsComposite Transformations

Orientation

Thumb points to +ve Z-axisFingers show +ve rotation from X to Y

Z (out of page)

Z (larger z areaway from viewer)

Right-handed orentation Left-handed orentation

Vectors in 3D

Have length and direction

V = [xv , yv , zv]Length is given by the Euclidean Norm

||V|| = ( xv2 + yv

2 + zv2 )

Dot Product

V • U = [xv, yv, zv]•[xu, yu, zu]

= xv*xu + yv*yu + zv*zu = ||V|| || U|| cos ß

Cross Product V U

= [yv*zu - zv yu , -xv*zu + zv*xu , xv*yu – yv*xu ]= ||V|| || U|| sin ßV U = - ( U x V)

(xv,yv,zv)V=aI+bJ+cK

3D Equation of Curve & Line

Parametric equations of Curve & LineCurve

VtPPPtPL

P0(x0,y0,z0)

P1(x1,y1,z1)

Vt < 0

t = 00 < t < 1

3D Equation of Surface & Plane

Parametric equations of Surface & PlaneSurface

Plane : with Normal, N

kCjBiAN

DCzByAx

ˆˆˆ

3D Plane

Ways of defining a plane

1. 3 points P0, P1, P2 on the plane

2. Plane Normal N & P0 on plane

3. Plane Normal N & a vector V on the plane

Plane Passing through P0, P1, P2

0ˆ)(ˆ)(ˆ)()ˆˆˆ(

ˆˆˆ

CzByAxD

DCzByAx

kzzjyyixxkCjBiA

kCjBiAPPPPN

plane the on is z) y,P(x, if

Affine Transformation

Transformation – is a function that takes a point (or vector) and maps that point (or vector) into another point (or vector).

A coordinate transformation of the form:

x’ = axx x + axy y + axz z + bx ,

y’ = ayx x + ayy y + ayz z + by ,

z’ = azx x + azy y + azz z + bz ,

is called a 3D affine transformation.

zzzzyzx

yyzyyyx

xxzxyxx

The 4th row for affine transformation is always [0 0 0 1]. Properties of affine transformation:

– translation, scaling, shearing, rotation (or any combination of them) are examples affine transformations.

– Lines and planes are preserved.– parallelism of lines and planes are also preserved, but not angles and

length.

Translation – 3D

PPdddT

Scaling – 3D

PPsssS

Original

scale all axes

scale Y axiszsz

Rotation – 3D

cos*sin*

sin*cos*

00cossin

00sincos

For 3D-Rotation 2 parameters are needed

Angle of rotation

Axis of rotation

Rotation about z-axis:

Rotation about Y-axis & X-axis

cos*sin*

sin*cos*

0cos0sin

0sin0cos

cos*sin*

sin*cos*

0cossin0

0sincos0

PPR iAbout x-axis

About y-axis

Shear along Z-axis

PPshshSH

Object Transformation

Line: Can be transformed by transforming the end points

Plane:(described by 3-points) Can be transformed by transforming the 3-points

Plane:(described by a point and Normal) Point is transformed as usual. Special treatment is needed for transforming Normal

Composite Transformations – 3D

Some of the composite transformations to be studied are:

AV,N = aligning a vector V with a vector N

R,L = rotation about an axis L( V, P )

Ssx,sy,P= scaling w.r.t. point P

AV : aligning vector V with k

Av = R,i

V = aI + bJ + cK

by axis-about x Rotate :1 Step

( 0, b,c )b

( a, 0, )

( 0, 0, )

( 0, b,c )

Av = R,i R-,j *

by axis-about x Rotate :1 Step

λ)cos(

|V|)sin(

-by axis-yabout V Rotate :2 Step

P( a, b, c)

( a, 0, )( 0, b,c )b

( 0, 0, |V|)

( 0, b,c )

AV-1 = AV

AV,N = AN-1 * AV

R,L : rotation about an axis L

Let the axis L be represented by vector V and passing through point P

1.Translate P to the origin

2. Align V with vector k

3. Rotate about k

4. Reverse step 2

5. Reverse step 1

R,L = T-PAV *R,k *AV-1 *T-P

MN,P : Mirror reflection

Let the plane be represented by plane normal N and a point P in that plane

MN,P = T-P

2. Align N with vector k

MN,P = T-PAN *

3. Reflect w.r.t xy-plane

MN,P = T-PAN *S1,1,-1 *

4. Reverse step 2

MN,P = T-PAN *S1,1,-1 *AN-1 *

4. Reverse step 2

5. Reverse step 1

MN,P = T-PAN *S1,1,-1 *AN-1 *T-P

Further Composition

The Composite Transform must have

– Translation of P1 to Origin T

– Some Combination of Rotations R

Fig. 1 Fig. 2

Translate points in fig. 1 into points in fig 2 such that:– P3 is moved to yz plane

– P2 is on z-axis

– P1 is at Origin

Finding R

Rx.x R

zRyRxR

vextor of component :Note

other each to

larperpendicu are ii)

vectors unit are i)

Transform body-Rigid is R

be Let

333231

232221

131211

Finding Rz

zRzRzR

yRyRyR

xRxRxR

axis-z along PP aligns

Finding Rx

zRzRzR

yRyRyR

xRxRxR

ˆ1 Rx

axis-x along PPPP aligns

Finding Ry

zRzRzR

yRyRyR

xRxRxR

axis- yalong RR aligns

Problems to be solved:

Schaum’s outline series:

Problems: 6.1

6.2, 6.5, 6.9, 6.10, 6.11, 6.12 Av

6.3, 6.4 R,L

6.6, 6.7, 6.8 MN,P

Transformations in OpenGL

OpenGL transformation commandsTransformation OrderHierarchical Modeling

Transformations in OpenGL

OpenGL uses 3 stacks to maintain transformation matrices:– Model & View transformation matrix stack

– Projection matrix stack

– Texture matrix stack

You can load, push and pop the stackThe top most matrix from each stack is applied to

all graphics primitive until it is changed

Model-ViewMatrix Stack

ProjectionMatrix Stack

GraphicsPrimitives

OutputN•M•P

General Transformation Commands

Specify current matrix (stack) :– void glMatrixMode(GLenum mode)

• Mode : GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE

Initialize current matrix.– void glLoadIdentity(void)

• Sets the current matrix to 4X4 identity matirx

– void glLoadMatrix{f|d}(cost TYPE *M)• Sets the current matrix to 4X4 matrix specified by M

Note: current matrix Top most matrix of the current matrix stack

General Transformation Commands

Concatenate Current Matrix:– void glMultMatrix(const TYPE *M)

• Multiplies current matrix C, by M. i.e. C = C*M

– Caveat: OpenGL matrices are stored in column major order.

– Best use utility function glTranslate, glRotate, glScale for common transformation tasks.

161284

151173

141062

Transformations and OpenGL®

Each time an OpenGL transformation M is called the current MODELVIEW matrix C is altered:

Cvv CMvv

glTranslatef(1.5, 0.0, 0.0);glRotatef(45.0, 0.0, 0.0, 1.0);

Note: v is any vertex placed in rendering pipeline v’ is the transformed vertex from v.

Sample Instance Transformation

glMatrixMode(GL_MODELVIEW);glLoadIdentity();glTranslatef(...);glRotatef(...);glScalef(...);gluCylinder(...);

Thinking About Transformations

As a Global System Objects moves but

coordinates stay the same Think of transformation

in reverse order as they appear in code

As a Local System Objects moves and

coordinates move with it Think of transformation

in same order as they appear in code

There is a World Coordinate System where: All objects are defined Transformations are in World Coordinate space

Two Different Views

Local View Translate Object Then Rotate

Order of Transformation T•R

glLoadIdentity();glMultiMatrixf( T);

glMultiMatrixf( R);

draw_ the_ object( v);v’ = ITRv

Global View Rotate Object

Then Translate

Effect is same, but perception is different

Order of Transformation R•T

glLoadIdentity();glMultiMatrixf( R);

glMultiMatrixf( T);

draw_ the_ object( v);v’ = ITRv

Local View Rotate Object Then Translate

Global View Translate Object

Then Rotate

Effect is same, but perception is different

Hierarchical Modeling

Many graphical objects are structured Exploit structure for

– Efficient rendering– Concise specification of model parameters– Physical realism

Often we need several instances of an object– Wheels of a car– Arms or legs of a figure– Chess pieces

Encapsulate basic object in a function Object instances are created in “standard” form Apply transformations to different instances Typical order: scaling, rotation, translation

OpenGL & Hierarchical Model

glPushMatrix

– void glPushMatrix(void);

glPushMatrix

– void glPoipMatrix(void);

glGetFloatv

– void glGetFloatv(GL_MODELVIEW_MATRIX, *m);

Some of the OpenGL functions helpful for hierarchical modeling are:

Scene Graph

A scene graph is a hierarchical representation of a scene We will use trees for representing hierarchical objects

such that:– Nodes represent parts of an object

– Topology is maintained using parent-child relationship

– Edges represent transformations that applies to a part and all the subparts connected to that part

typedef struct treenode {GLfloat m[16]; // Transformationvoid (*f) ( ); // Draw functionstruct treenode *sibling;struct treenode *child;

} treenode;

Sun Star X

Earth Venus Saturn

Moon Ring

Example - Torso

Initializing transformation matrix for node

treenode torso, head, ...;

/* in init function */

glLoadIdentity();

glRotatef(...);

glGetFloatv(GL_MODELVIEW_MATRIX, torso.m);

Initializing pointers

torso.f = drawTorso;

torso.sibling = NULL;

torso.child = &head;

Generic Traversal

To render the hierarchy:– Traverse the scene graph depth-first– Going down an edge:

• push the top matrix onto the stack

• apply the edge's transformation(s)

– At each node, render with the top matrix– Going up an edge:

• pop the top matrix off the stack

Generic Traversal : Torso

Recursive definitionvoid traverse (treenode *root) {

if (root == NULL) return;

glPushMatrix();

glMultMatrixf(root->m);

root->f();

if (root->child != NULL) traverse(root->child);

glPopMatrix();

if (root->sibling != NULL) traverse(root->sibling);

C is really not the right language for this !!

Viewing Transformation

Viewing Pipeline Revisited

Modeling Transform

WorldCoordinates

GraphicsPrimitives Po

ObjectCoordinates

Viewing Transform Pe

EyeCoordinates

Viewing Transformation in OpenGL To setup the modelview matrix, OpenGL provides the

following function:

eye(eyex, eyey, eyez)

center(centerx, centery, centerz)

up(upx, upy, upz)

gluLookAtgluLookAt(( eyex, eyey, eyezeyex, eyey, eyez, , centerx, centery, centerzcenterx, centery, centerz, , upx, upy, upzupx, upy, upz ))

Implementation

We want to construct an Orthogonal FrameOrthogonal Frame such that,

eCeCeC

puvnormalizeeC

eyecenternormalizev

C.O(eye)

center

up(upx, upy, upz)

(1)(1) its origin is the point eyepoint eye

(2)(2) its -z basis vector points towards the point point centercenter(3)(3) the up vectorup vector projects to the up direction (+ve y-axis)

Let CC (for camera) denote this frame. Clearly,

Thank You

1 Modeling Transformations 2D Transformations 3D Transformations OpenGL Transformation

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