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Geometric Objects and Transformations

Coordinate systems and framesWorking with representation

Object transformation

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Introduction

• Mathematical of Object– Euclidean vector spaces

• Vector space with measure of size– Independent of coordinate system

• Parametric form system

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Scalars, Points, and Vectors

• Geometric object description in space– By length, angle– With 3 fundamental types scalars, points and

vector

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The geometric View

• Point : a location in the space– Mathematical for point

• Neither a size nor shape– Properties

• location– What for :

• Specify an object

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Scalars

• Quantity of object or relation objects• Ex. Distance between object• Specify with real / complex number• Useful rule for scalar

– communitivity and Associativity in multiplicity, additivity

– Ex. a+b = b+a, (a+b)+c = a+(b+c)

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Vector

• Quantity with direction and magnitude ex. velocity, force– Does not have a fixed position in space– Synonymously to line segment

• Computer graphics often connect points with directed line segment– Line segment: a segment of line which

has both magnitude and direction

Directed line segment that connects points

Identical vectors

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Vector properties• Its lengths changed by real number• B = 2A

– B is double in size to vector A with the same direction• Vector combining: (addition)

– Use head to tail combining, the result is the sum of the vector– Any vector in space is able to do addition, independent of its location– Scalar-vector addition make sense: ex. A + 2B – 3C

• Inverse vector:– The vector that has opposite direction to the original vector

Parallel line segments

Addition of line segments Inverse vectors

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Vector operation• Scalar multiplying

– Result change in length (magnitude) • Point-vector addition

– Result change in displacement to the new point position

• Point-point subtraction– Result is a vector between 2 points

• Note: Some expression involving scalars, vectors and

points make sense ex. P+3v, or P-2Q+3v while P+3Q-v do not

Point-vector addition

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Coordinate-free geometry

• For graphics system let the object relate to each other but independent of coordinate system

• Let object relate to an arbitrary location and orientation of the original axis

Object and coordinate system

Object without coordinate system

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The mathematical view: Vector and Affine Spaces

• Scalar operation– Addition , multiplication

• If operation obey closure, associativity, commutivity and inverse properties, the element form a scalar field

– Ex. field . Real number, complex number, rational function

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Vector space

• 2 distinct type of entities in vector space– Scalar and vector

• Scalar-vector multiplication– Vector and vector

• Vector – vector addition

• Euclidean space– Extension of vector space– Add a measure of size or distance to define object– Ex. Length of line segment

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Affine space (space of transformation)

• Extension of vector space– Include point to vector space– Have vector-point addition and point-point

operation

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Computer Science View• Prefer to see object as

abstract data type (ADT)• Operations and data are

defined independently• Fundamental to modern

computer science• Like C++ language features :

class and overloading

Computational point of view declaration (Independent of data type declaration)

Operation that independent of data type

vector u,v;point p,q;scalar a, b;

q = p+a*v;

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Geometric ADT

• Learn how to perform geometric operation and forming geometric object

• Let Greek letter a, b, g,... : scalars

Upper-case letter P, Q, R,… : points

Bold lower-case letters u, v, w,… : vector

• The operation of vector-scalar multiplication

• Subtraction of two points, P and Q -> vector v

• Add vector with point get point

• Vector-vector operation can be in point form

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Point-point subtraction

Use of the head-to-tail rule(a) For vectors, (b)For points

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Lines

• General term definition• Called parametric form of line• Point generating by varying alpha• Line is infinity in length• One we see just line segment

Parametric line equation

Line in an affine space

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Affine sumIf P is a point on line

P=Q+avv = R – Q;

ThusP = Q + a(R-Q)

= aR + (1-a)QLet

a = a1 and (1-a) = a2

Thus a1 + a2 = 1

Then P=a1R+a2Q

Points on line can be found between point Q and P(a)

Affine line addition

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Convexity

• Convex object – Any point that lying on

line segment connect any two points in the object also in the object

• Affine sum definition

...1 1 2 2 n nP = P P Pa a a When

... 11 2 n =a a a

Objects defined by n points P1, P2,…,Pn. Consider the form

Line segment that connects two points

• An object is convex iff for any two points in the object all points on the line segment between these points are also in the object

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P

Q Q

P

convexnot convex

Convexity

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is called the convex hull of the set of pointsconvex hull includes all line segments connecting pairs of points i[P1,P2,…,Pn]The notion of convexity is extreamly important in the design of curves and surfaces;

Convex hull

Convex hull

The set of points formed by the affine sum of n points, under the additional restriction

iα 0, i = 1,2,...,n

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Dot and Cross product

• Dot (inner product)– u.v: result is magnitude of 2 vectors– If u.v = 0, u and v are orthogonal vector– Unit product is

Dot product and projection

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Cross product• Result is vector• Forming from right hand coordinate system

Note: right-handed coordinate systemu points in the direction of the thumb v points in the direction of the index fingern points in the direction of the middle finger

Cross product

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Plane• Infinite flat area• Direct extension of parametric line• Define with 3 non-co-linear points• Suppose P, Q and R are points in plane• The plane equation

and

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The plane can have vector (normal vector) formed from u and v

Let

n : plane vector

Thus

General plane equation

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3D primitives

Object are not lying on plane

Curves in three dimension Surfaces in three dimensions Volumetric objects

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2 problems when incorporate in 3D

• Complex mathematic– Not all object that has 3D efficient

implementation• Approximated method may be used• 3 features characteristic to describe object

– Hollow– Vertices– Flat convex polygon composition

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• Represents Vector in 3D space with 3 basis vector

thus

Coordinate system and frame

Vector derived from three basis vectors

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Point and vector representation in affine space• It is not enough to use only vector to represent

point in space• Frame: fix point (origin) and basis vector

– Vector: represent with 3 basis– Point: fix point and 3 basis

Coordinate system (a) with vector emerging from a common point, (b) with vector moved

A Dangerous representation of vector

Representations and N-tuples• Any vector v representation

Where

• In column matrix form

• Known as Euclidean :

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Coordinate System ChangingWorld /User frame Camera frame

– Done by MODELVIEW matrix– Use 2 basis set of vector and – Representation of by

Let

and

represent matrix from to

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Let

Equivalent to

Where

Assume : the representation of with respect to

or

Where

Then using our representation of the second basis in terms of the first, we find that

Thus

The matrix (MT)-1 takes us from a to b:

We can transfer coefficient matrix from one to the other

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• This changing let us to work with different coordinate system but origin unchanged

• Able to use them to represent rotation and scaling but not translation

Rotation and scaling of a basis Translation of a basis

2 examples 2 different frames

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Example: Suppose that we have a vector whose representation in some basis is

1 a 2

3

We can denote the three basis vectors and , , and hence, = +2 +3Now suppose that we want to

1 2 3

1 2 3

w

=

v v vw v v v

1 1

2 1 2

3 1 2 3

make a new basis from the three vectors , , and The matrix M is

1 0 0 M= 1 1 0

1 1 1

1 2 3v v vu vu v vu v v v

The matrix that converts a representation in , and to one in which the basis vector are and is

In the new system, the representation of w is

That is

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Homogeneous Coordinate3 basis vector is not enough to make point and vector differentFor any point:

In the frame specified by (v1, v2, v3, P0), any point P can be written uniquely as

The we can express this relation formally, using a matrix product, as

The yellow matrix called homogeneous-coordinate representation of point P

In the same frame, any vector w can be written

w can be represented by the column matrix

If and , can be expressed as

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Can be written in the form

M : the matrix representation of the change of frames.

Suppose a and b are the homogeneous-coordinate representations, then

Hence,

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When we work with representations, as is usually the case, we are interested in , which is of the form

Only 12 coefficients

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Advantage of using homogeneous representationo use 4D matrix instead of 3D matrix

o less calculationo modern hardware support the representation

o parallelism for high speed calculation

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Example

1 1

2 1 2

3 1 2 3

If we start with the basic vector , and and convert to a basis determined by the same , and then the three equations are the same

,

The reference point does not c

1 2 3

1 2 3

v v vu u u

u vu v vu v v v

hange, so we add the equation =

Thus, the matrices in which we are interested are the matrix1 0 0 01 1 0 0

M=1 1 1 00 0 0 1

its transpose, and their inverses.

0 0Q P

Suppose move the point to (1, 2, 3, 1)

with displacement vector

And move from point P0 to Q0 then

The matrix becomes

Its inverse is

We can move back and forth between representation

Thus

to

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Original vector

b= (MT)-

1a<-

The origin in the new system is represented as 12

a30

11

b30

is transformed to

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Working with Representation• Represent object from a frame to another such as

world frame to camera frame• Form of representation

a = Cb ; a, b object in two representation frame• Object:

– 3 vector, u, v, n, and 1 new frame origin, p -> (u, v, n, p)– 4 entities -> 4-tuples -> R4

• The solution is inverse matrix form of C, let’s say DD = C-1

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1 1 1 1

2 2 2 2

3 3 3 3

11 1 1 1

1 2 2 2 2

3 3 3 3

DI D

0 0 0 1or

C

0 0 0 1

u v n pu v n p

u v n pu v n p

u v n pu v n p

u v n pu v n p

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Frames in OpenGL• 2 frames:

– Camera: regard as fix– World:

• The model-view matrix position related to camera• Convert homogeneous coordinates representation of object to camera

frame

• OpenGL provided matrix stack for store model view matrices or frames

• By default camera and world have the same origin• If moving world frame distance d from camera the

model-view matrix would be

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Camera and world frames

1 0 0 00 1 0 0

A=0 0 10 0 0 1

-d

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Suppose camera at point (1, 0, 1, 1)

• World frame center point of camera

• Representation of world frame for camera

• Camera orientation: up or down

• Forming orthogonal product for determining v let’s say u

Camera at (1,0,1) pointing toward the origin

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The model view matrix M

Result: Original origin is 1 unit in the n direction from the origin in the camera frame which is the point (0, 0, 1, 1)

OpenGL model view matrixOpenGL set a model-view matrix by send an array of 16 elements to glLoadMatrixWe use this for transformation like rotation, translation and scales etc.

1

T 1

1 10 01 0 1 1 2 20 1 0 0 0 1 0 0

(M )1 0 1 1 1 10 1

2 20 0 0 10 0 0 1

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Modeling a Color Cube• A number of distinct task that we

must perform to generate the image– Modeling– Converting to the camera frame– Clipping– Projecting– Removing hidden surfaces– Rasterizing One frame of cube animation

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Modeling a Cube• Model as 6 planes intersection or six polygons as cube facets• Ex. of cube definition

// object may defined as

void polygon(int a, int b, int c , int d) {/* draw a polygon via list of vertices */ glBegin(GL_POLYGON);

glVertex3fv(vertices[a]);glVertex3fv(vertices[b]);glVertex3fv(vertices[c]);glVertex3fv(vertices[d]);

glEnd();}

GLfloat vertices[8][3] = {{-1.0,-1.0,-1.0}, {1.0,-1.0,-1.0}, {1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0}, {1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}

};// ortypedef point3[3];

// then may define aspoint3 vertices[8] = {

{-1.0,-1.0,-1.0}, {1.0,-1.0,-1.0}, {1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0}, {1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}

};

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Inward and outward pointing faces

• Be careful about the order of vertices

• facing outward: vertices order is 0, 3, 2, 1 etc., obey right hand rule

Traversal of the edges of a polygon

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Data Structure for Object Representation

• Topology of Cube description– Use glBegin(GL_POLYGON); six times– Use glBegin(GL_QUADS); follow by 24

vertices• Think as polyhedron

– Vertex shared by 3 surfaces– Each pair of vertices define edges– Each edge is shared by two faces

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Vertex-list representation of a cube

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The color cube• Color to vertex list -> color 6 faces• Define function “quad” for drawing quadrilateral polygon• Next define 6 faces, be careful about define outwarding

typedef GLfloat point3[3];

point3 vertices[8] = {{-1.0,-1.0, 1.0},{-1.0, 1.0, 1.0}, {1.0,1.0, 1.0}, {1.0,-1.0, 1.0}, {-1.0,-1.0,-1.0}, {1.0,-1.0,-1.0}, {1.0,1.0,-1.0},

{-1.0,1.0,-1.0}};GLfloat colors[8][3] = {{0.0,0.0,0.0},{1.0,0.0,0.0}, {1.0,1.0,0.0}, {0.0,1.0,0.0},

{0.0,0.0,1.0}, {1.0,0.0,1.0}, {1.0,1.0,1.0}, {0.0,1.0,1.0}};

void quad(int a, int b, int c , int d) {glBegin(GL_QUADS); glColor3fv(colors[a]); glVertex3fv(vertices[a]); glColor3fv(colors[b]); glVertex3fv(vertices[b]);

glColor3fv(colors[c]); glVertex3fv(vertices[c]); glColor3fv(colors[d]); glVertex3fv(vertices[d]); glEnd();

}void colorcube() { quad(0, 3, 2, 1); quad(2, 3, 7, 6); quad(0, 4, 7, 3); quad(1, 2, 6, 5); quad(4, 5, 6, 7); quad(0, 1, 5, 4);}

Color the cube surface

•  

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Projection of polygon

Bilinear interpolation

Vertex Array• Use encapsulation, data structure & method together

– Few function call• 3 step using vertex array

– Enable functionality of vertex array: part of initialization– Tell OpenGL where and in what format the array are: part of

initialization– Render the object :part of display call back

• 6 different type of array– Vertex, color, color index, normal texture coordinate and edge flag– Enabling the arrays by

glEnableClientState(GL_COLOR_ARRAY);glEnableClientState(GL_VERTEX_ARRAY);

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// The arrays are the same as before and can be set up as globals:

GLfloat vertices[] = {{-1,-1,-1},{1,-1,1},{1,1,-1},{-1,1,-1},{-1,-1,1}, {1,-1,1}, {1,1,1}, {-1,1,1}};

GLfloat colors[] = {{0,0,0},{1,0,0},{1,1,0},{0,1,0},{0,0,1},{1,0,1}, {1,1,1},{0,1,1}};

// Next identify where the arrays are by

glVertexPointer(3, GL_FLOAT, 0, vertices);

glColorPointer(3, GL_FLOAT, 0, colors);

// Define the array to hold the 24 order of vertex indices for 6 faces

GLubyte cubeIndices[24] = {0,3,2,1, 2,3,7,6, 0,4,7,3, 1,2,6,5, 4,5,6,7, 0,1,5,4};

glDrawElements(type, n, format, pointer);

for (i=0; i<6; i++)

glDrawElement(GL_POLYGON, 4, GL_UNSIGNED_BYTE, &cubeIndex[4*i]};

// Do better by seeing each face as quadrilateral polygon

glDrawElements(GL_QUADS, 24, GL_UNSIGNED_BYTE, cubeIndices);

// GL_QUADS starts a new quadrilateral after each four vertices

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Affine transformation

Transformation:A function that takes a point (or vector) and maps that points (or vector) in to another point (or vector)Point transform

Vector transform

Transformation

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Transformation form of 4D matrix

Transformation function must be linearity function

For vector has 12 degree of freedom for point and vector

f p q f p f qa b a b

v Au

11 12 13 14

21 22 23 24

31 32 33 34

A

0 0 0 1

a a a aa a a aa a a a

1

2

3

u

0

aaa

1

2

3

1

v

bbb

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Rigid body transformation

• Shape doesn’t change• Only change in position and orientation• They are: Translation and rotation.

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Translation

• For all point P of the object– No reference point to frame or representation– 3 degree of freedom

Translation. (a) Object in original position. (b) Object translated

Is an operation that displaces points by a fixed distance in a given directionTo specify a translation, just specify a displacement vector d thus

P P d

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Rotation

• Need axis and angle as input parameter

2D Point rotation θ radian around origin (Z axis) Two-dimensional rotation

x = r.cosf ; y = r.sinf

x’ = r.cos(q f= r.cosf.cosq r.sinf.sinq= x.cosq – y.sinq

y’ = r.sin(qf)= r.cosf.sinq +r.sinf.cosq = x.sinq+y.cosq

These equations can be written in matrix form ascos sin

sin cos

x xy y

q qq q

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3 features for rotation(3 degrees of freedom)

• Around fixed point (origin)• Direction ccw is positive• A line

Rotation arount a fixed point.

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3D rotation• Must define 3 input parameter

– Fixed Point (Pf)– Rotation angle (θ)– A line or vector (rotation axis)

• 3 degrees of freedom• 2 angle necessary specified

orientation of vector• 1 angle for amount of rotation

Three-dimensional rotation

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Scaling• Non rigid-body transformation• 2 type of scaling

– Uniform: scale in all direction -> bigger, smaller– Nonuniform: scale in single direction

• Use in modeling and scaling

Non rigid body transformations

Uniform and nonuniform scaling

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Scaling: input parameter

• 3 degrees of freedom– Point (Pf) – Direction (v)– Scale factor (a)

• a > 1 : get bigger• 0 £ a < 1 : smaller• a < 0 : reflected

Reflection

Effect of scale factor

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Transformation in homogeneous coordinateMost graphics APIs force us to work within some reference system. Although we can alter this reference system – usually a frame – we cannot work with high-level representations, such as the expression.

Q = P + αv.

Instead, we work with representations in homogeneous coordinates, and with expressions such as

q = p + αv.

Within a frame, each affine transformation is represented by a 4x4 matrix of the form

TranslationIf we move the point p to p by displacing by a distance d then

p = p+dDefine

p = , p= , d=

1 1 0 may be written for each of components

x

y

z

x xy yz z

It

aaa

Let T is a translation matrix, we may written in matrix form as: And

1 0 00 1 0

T=0 0 10 0 0 1

Sometimes write it as

x

y

z

x

y

z

x xy y

z z

p Tp

aa

a

aaa

T , ,x y za a a

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ScalingUse fixed point parameter as referenceScaling in one direction means scale in each direction elementThe scaling equation are:

For the homogeneous form

where

We may say that the inverse form of scaling is :

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Rotation2D rotation is actual 3D rotation around Z axisFor general equations:

.cos .sin

.sin .cosx x yy x yz z

q qq q

z

z

z

For the Homogeneous equation, let denotes R is a rotation matrix around z axisThen the homogeneous form for rotation is p =R pwhere

cos sin 0 0sin cos 0 0

R0 0 1 00 0 0 1

q qq q

q

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x

For rotation about x-axis the rotation matrix R

1 0 0 00 cos sin 0

R R0 sin cos 00 0 0 1

and for rotation about y-axis R

cos 0 sin 00 1 0 0

R Rsin 0 cos 00 0 0 1

x

x

y

y y

q qq

q q

q q

qq q

-1 T

For inverse rotation transformation in each axis: Rotate to the same axis with equivalent angle backward:

R = R R

We call a matrix whose inverse is equal to its transpose is call an or

q q q

thogonal matrix

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Shear

• Let pull in top right edge and bottom left edge– Neither y nor z are changed– Call x shear direction

Shear

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Computation of the shear matrix

cot xx x yy yz z

q

Leading to the shearing matrix

1 cot 0 00 1 0 0

H0 0 1 00 0 0 1

x

x x

q

q

Inverse of the shearing matrix: shear in the opposite direction

1H Hx x x xq q

x

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Shear in other direction

cot

1 0 0 0cot 1 0 0

H0 0 1 00 0 0 1

y

yy y

x xy y x

z z

q

qq

Shear in y-axis direction when look on xy-plane

cot1 0 0 00 1 0 0

Hcot 0 1 0

0 0 0 1

z

z zz

x xy yz z x q

qq

Shear in z-axis direction when look on zx-plane

What about Hy and Hz when look in yz plane ?

Shearing general form

• shear with reference line

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(0,0)

(0,1) (1,1)

(1,0) (0,0) (1,0)

(2,1) (3,1)

x x

y y

(0,0)

(0,1) (1,1)

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

(1,1) (2,1)

x x

y y

yref =-1

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Concatenation of transformation

Pipeline transformation

Application of transformations one at a time

The sequence of transformation isq=CBAp

It can be seen that one do first must be near most to inputWe may proceed in two stepsCalculate total transformation matrix:

M=CBA

Matrix operation : q=Mp

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Derive example of M: rotation about a fixed point

<- This is what we try to do.

Transformation processes

Sequence of transformation

Rotation of a cube about its center

 

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General rotation

Rotation of a cube about the x-axis

Rotation of a cube about the z-axis. The cube is show (a) before rotation, and (b) after rotation

Rotation of a cube about the y-axis

R R R Rx y z

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The Instance transformation• From the example object, 2 options to do with

object– Define object to its vertices and location

with the desire orientation and size– Define each object type once at a

convenience size, place and orientation , next move to its place

• Object in the scene is instance of the prototype

• If apply transformation, called instance transformation

• We can use database and identifier for eachScene of simple objects

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Instance transformation

M=TRS //instant transformation equation of object

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Rotation around arbitrary axis

• Input parameter– Object point– Vector (line segment or 2 points)– Angle of rotation

• Idea– Translate to origin first T(-P0)– Rotate

• q-axis component • q around 1 component axis, let say z

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<- Convert u to unit length vector

<- Shift to origin (T(-P)), end shift back (T(P))

<- Individual axis rotation z first

Rotation of a cube about an arbitrary axis

R = Rx(-qx).Ry(-qy).Rz(q).Ry(qy).Rx(qx)

Let rotation axis vector is u and

u = P2 – P1

Movement of the fixed point to the origin

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Problem: How we fine θx and θy ?

<- From v, unit vector

<-<-

f = angle of vector respect to origin

Sequence of rotations

Direction angles

2 2 2x y z 1a a a

cos cos cos2 2 2x y z 1f f f

coscos

cos

x x

y y

z z

f af a

f a

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<- For rotation around X, project vector to plane XZ (X=0) meanwhile projection on YZ plane with length d

->

Computation of the x rotation

Computation of the y rotation

2 2y zd a a

1 0 0 0

0 0

0 0

0 0 0 1

yz

x xy z

d dR

d d

aa

qa a

0 0

0 1 0 00 0

0 0 0 1

x

y yx

d

Rd

a

qa

Finally we calculate all the matrices to find

0 0M T p T px x y y z y y x xR R R R Rq q q q q

• Example: rotate around origin to point (1,2,3) 45 degree

Normalized vector (1, 2, 3)

Z axis rotation (0,0,1,0) 45 degree => θz = 45o

X axis (1,0,0,0) rotation angle is Y axis rotation

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Last take back to point

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OpenGL transformation matrices

• Use glMatrixMode function for selected operation to apply

• Most graphic system use Current Transformation Matrix (CTM) for any transformation

Current transformation matrix (CTM)

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CTM step

• Applied Identity matrix unless operate every round– C <- I

• Applied Translation– C <- CT

• Applied Scaling– C <- CS

• Applied Rotation– C <- CR

• Or postmultiply with M– C <- M or C <- CM

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CTM: Translation, rotation and scaling

• CTM may be viewed as the part of all primitive product GL_MODELVIEW and GL_PROJECTION

• From function of glMatrixMode

Model-view and projection matrices

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Rotation about a fixed point matrix step

glMatrixMode(GL_MODELVIEW);glLoadIdentity();glTranslatef(4.0, 5.0, 6.0);glRotatef(45.0, 1.0, 2.0, 3.0);glTranslatef(-4.0, -5.0, -6.0);

//rotate 45 deg. around a vector line 1,2,3// with fixed point of (4, 5, 6)

Modeling more than 1 independent object

• use paradigm of glPushMatrix() and glPopMatrix()

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The quaternion• Extension of complex number• Useful for animation and hardware

implementation of rotation• Consider the complex number

• Rotate point (a, b) around z axis with θ radian• quaternion is equivalent to rotate in 3D space

ic a ib re q

2 2 1; tan

ic a ib reWhere

br a ba

q

q

(a, b)

θ

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Quaternion form

• q0 define rotation angle• q define point

0 1 2 3 0

1 2 3 1 2 3

2 2 2

, , , ,q

q , ,

1

a q q q q q

whereq q q q i q j q k

and

i j k i j k

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Multiplication properties

• Identity– (1, 0)

• Inverse

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Rotation with quaternion• Let p=(0,p) // point in space with p= (x,y,z)• And unit quaternion r and its inverse r-1

// v is a unit vector // inverse of quaternion

r : rotation of θ around a unit vector, thus rotate of point p around v with θ and p’ is

if v= (0,0,1) //rotate around z axis, p’ will be

Quaternion to Rotation MatrixReplace:

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Using quaternion in OpenGL

• can use only in the rotation transformation• the other transformations still use the matrix

manipulation.• when use in OpenGL

– multiplay the object first and transform the result to matrix form

– use at the display state

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Case study Trackball

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Exercise4.1Consider the solution of either constant-coefficient linear differential

or difference equations (recurrences). Show that the solutions of the homogeneous equations form a vector space. Relate the solution for a particular in homogeneous equation to an affine space.

4.2Show that the following sequences commute:a. A rotation and a uniform scaling b. Two rotations about the same axis c. Two translations

4.3Write a library of functions that will allow you to do geometric programming. Your library should contain functions for manipulating the basic geometric types (points, lines, vectors) and operations on those types, including dot and cross products. It should allow you to change frames. You can also create functions to interface with OpenGL, so that you can display the results of geometric calculations.

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4.4 If we are interested in only two-dimensional graphics, we can use three-dimensional homogeneous coordinates by representing a point as p = [x y 1]T and a vector as v =[a b 0]T. Find the 3x3 rotation, translation, scaling, and shear matrices. How many degrees of freedom are there in an affine transformation for transforming two-dimensional points?

4.5 We can specify an affine transformation by considering the location of a small number of points both before and after these points have been transformed. In three dimensions, how many points must we consider to specify the transformation uniquely? How does the required number of points change when we work in two dimensions?

4.6 How must we change the rotation matrices if we are working in a left-handed system and we retain our definition of a positive rotation?

4.7 Show that any sequence of rotations and translations can be replaced by a single rotation about the origin, followed by a translation.

4.8 Derive the shear transformation from the rotation, translation, and scaling transformations.

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4.9 In two dimensions, we can specify a line by the equation y = mx + b. Find an affine transformation to reflect two- dimensional points about this line. Extend your result to reflection about a plane in three dimensions,

4.10 In Section 4.8 we showed that an arbitrary rotation matrix could be composed from successive rotations about the three axes. How many ways can we compose a given rotation if we can do only three simple rotations? Are all three of the simple rotation matrices necessary?

4.11 Add shear to the instance transformation. Show how to use this expanded instance transformation to generate parallelepipeds from a unit cube.

4.12 Find a homogeneous-coordinate representation of a plane.4.13 Determine the rotation matrix formed by glRotate. That is, assume that

the fixed point is the origin and that the parameters are those of the function.

4.14 Write a program to generate a Sierpinski gasket as follows. Start with a white triangle. At each step, use transformations to generate three similar triangles that are drawn over the original triangle, leaving the center of the triangle white and the three corners black.

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4.15 Start with a cube centered at the origin and aligned with the coordinate axes. Find a rotation matrix that will orient the cube symmetrically, as shown in Figure 4.65.

4.16 We have used vertices in three dimensions to define objects such as three-dimensional polygons. Given a set of vertices, find a test to determine whether the polygon that they determine is planar.

4.17 Three vertices determine a triangle if they do not lie in the same line. Devise a rest for co-linearity of three vertices.

4.18 We defined an instance transformation as the product of a translation, a rotation, and a scaling. Can we accomplish the same effect by applying these three types of transformations in a different order?

4.19 Write a program that allows you to orient the cube with one mouse button, to translate it with a second, and to zoom in and out with a third.

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4.20 Given two nonparallel three-dimensional vectors u and r, how can we form an orthogonal coordinate system in which u is one of the basis vectors?

4.21 An incremental rotation about the z-axis is determined by the matrix

What negative aspects are there if we use this matrix for a large number of steps? Hint: consider points a distance of 1 from the origin. Can you suggest a remedy?

4.22 Find the quaternion for 90-degree rotations about the x- and y-axes. Determine their product.

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4.23 Determine the rotation matrix

find the corresponding quaternion.4.24 Redo the trackball program using quaternion instead of

rotation matrices.4.25 Implement a simple package in C++ that can do the required

mathematical operations for transformations. Such a package might include matrices, vectors, and frames