PPT&Programs&Labcourse

http://211.87.235.32/share/ComputerGraphics/

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Transformations

Shandong University Software College

Instructor: Zhou YuanfengE-mail: yuanfeng.zhou@gmail.com

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Objectives

• Introduce standard transformations Rotation

Translation

Scaling

Shear

•Derive homogeneous coordinate transformation matrices

•Learn to build arbitrary transformation matrices from simple transformations

Why transformation?

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Before

After

After

Before

Why transformation?

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Snow construction

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General Transformations

A transformation maps points to other points and/or vectors to other vectors

Q=T(P)

v=T(u)

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Affine Transformations

•Linear transformation + translation•However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations

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Affine Transformations

•Line preserving•Characteristic of many physically important transformations

Rigid body transformations: rotation, translation

Scaling, shear

• Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

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Pipeline Implementation

transformation rasterizer

u

v

u

v

T

T(u)

T(v)

T(u)T(u)

T(v)

T(v)

vertices vertices pixels

framebuffer

(from application program)

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Notation

We will be working with both coordinate-free representations of transformations and representations within a particular frame

P,Q, R: points in an affine space u, v, w: vectors in an affine space , , : scalars p, q, r: representations of points

-array of 4 scalars in homogeneous coordinates u, v, w: representations of vectors

-array of 4 scalars in homogeneous coordinates

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The World and Camera Frames

• When we work with representations, we work with n-tuples or arrays of scalars

• Changes in frame are then defined by 4 x 4 matrices

• In OpenGL, the base frame that we start with is the world frame

• Eventually we represent entities in the camera frame by changing the world representation using the model-view matrix

• Initially these frames are the same (M=I)

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Moving the Camera

If objects are on both sides of z=0, we must move camera frame

1000

d100

0010

0001

M =

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Translation

•Move (translate, displace) a point to a new location

•Displacement determined by a vector v Three degrees of freedom P’=P+v

P

P’

v

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How many ways?

Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way

object translation: every point displaced by same vector

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Translation Using Representations

Using the homogeneous coordinate representation in some frame

p=[ x y z 1]T

p’=[x’ y’ z’ 1]T

v=[vx vy vz 0]T

Hence p’ = p + v or x’=x+ vx y’=y+ vy z’=z+ vz

note that this expression is in four dimensions and expressespoint = vector + point

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Translation Matrix

We can also express translation using a 4 x 4 matrix T in homogeneous coordinatesp’=Tp where

T = T(vx, vy, vz) =

This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

1 0 0

0 1 0

0 0 1

0 0 0 1

x

y

z

v

v

v

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Rotation (2D)

Consider rotation about the origin by degrees radius stays the same, angle increases by

PRy

x

y

x

rrry

rrrx

ryrx

cossin

sincos

'

'

cossinsincos)sin('

sinsincoscos)cos('

sin,cos

x’=x cos –y sin y’ = x sin + y cos

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Rotation about the z axis

• Rotation about z axis in three dimensions leaves all points with the same z

Equivalent to rotation in two dimensions in planes of constant z

or in homogeneous coordinates

p’=Rz()p

x’=x cos –y sin y’ = x sin + y cos z’ =z

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Rotation Matrix

1000

0100

00 cossin

00sin cos

R = Rz() =

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Rotation about x and y axes

• Same argument as for rotation about z axis For rotation about x axis, x is unchanged

For rotation about y axis, y is unchanged

R = Rx() =

R = Ry() =

1000

0 cos sin0

0 sin- cos0

0001

1000

0 cos0 sin-

0010

0 sin0 cos

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Scaling

1000

000

000

000

z

y

x

s

s

s

S = S(sx, sy, sz) =

x’=sxxy’=syyz’=szz

p’=Sp

Expand or contract along each axis (fixed point of origin)

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Reflection

corresponds to negative scale factors

originalsx = -1 sy = 1

sx = -1 sy = -1 sx = 1 sy = -1

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Inverses

• Although we could compute inverse matrices by general formulas, we can use simple geometric observations

Translation: T-1(dx, dy, dz) = T(-dx, -dy, -dz)

Rotation: R -1() = R(-)• Holds for any rotation matrix• Note that since cos(-) = cos() and sin(-)=-sin()

R -1() = R T()

Scaling: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)

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Concatenation

• We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices

• Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p

• The difficult part is how to form a desired transformation from the specifications in the application

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Order of Transformations

•Note that matrix on the right is the first applied

•Mathematically, the following are equivalent

p’ = ABCp = A(B(Cp))•Note many references use column matrices to represent points. In terms of column matrices ((A B) T = BT AT)

p’T = pTCTBTAT

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General Rotation About the Origin

x

z

yv

A rotation by about an arbitrary axiscan be decomposed into the concatenationof rotations about the x, y, and z axes

R() = Ry(-y) Rx(-x) Rz(z) Ry(y) Rx(x)

x y z are called the Euler angles

Note that rotations do not commuteWe can use rotations in another order butwith different angles

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Rotation About a Fixed Point other than the Origin

Move fixed point to origin

Rotate

Move fixed point back

M = T(pf) R() T(-pf)

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Instance transformation

• In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size

•We apply an instance transformation to its vertices to

Scale

Orient

Locate

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Shear

• Helpful to add one more basic transformation

• Equivalent to pulling faces in opposite directions

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Shear Matrix

Consider simple shear along x axis

x’ = x + y cot y’ = yz’ = z

1000

0100

0010

00cot 1

H() =

Rotation around arbitrary vector

•R(Ax,Ay,Az)

• If then

•v=w×u

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2 2 231 32 33, , , ,x y z x y zw A A A A A A a a a

2 211 12 13, ,0 , ,y x x yu A A A A a a a

022 yx AA 131211 ,, aaa(1,0,0)u

1

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1 1

x

yT T

z

R = Rz() =

21 22 23, ,a a a

11 12 13

21 22 23

31 32 33

0

0

0

1 0 0 0 1 1

u a a a x

v a a a y

w a a a z

T

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OpenGL Matrices

• In OpenGL matrices are part of the state•Multiple types

Model-View (GL_MODELVIEW) Projection (GL_PROJECTION) Texture (GL_TEXTURE) (ignore for now) Color(GL_COLOR) (ignore for now)

•Single set of functions for manipulation•Select which to manipulated by­glMatrixMode(GL_MODELVIEW);­glMatrixMode(GL_PROJECTION);

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Current Transformation Matrix (CTM)

• Conceptually there is a 4 x 4 homogeneous coordinate matrix, the current transformation matrix (CTM) that is part of the state and is applied to all vertices that pass down the pipeline

• The CTM is defined in the user program and loaded into a transformation unit

CTMvertices vertices

p p’=CpC

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CTM operations

• The CTM can be altered either by loading a new CTM or by postmutiplication

Load an identity matrix: C ILoad an arbitrary matrix: C M

Load a translation matrix: C TLoad a rotation matrix: C RLoad a scaling matrix: C S

Postmultiply by an arbitrary matrix: C CMPostmultiply by a translation matrix: C CTPostmultiply by a rotation matrix: C C RPostmultiply by a scaling matrix: C C S

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Rotation about a Fixed Point

Start with identity matrix: C IMove fixed point to origin: C CTRotate: C CRMove fixed point back: C CT -1

Result: C = TR T –1 which is backwards.

This result is a consequence of doing postmultiplications.Let’s try again.

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Reversing the Order

We want C = T –1 R T so we must do the operations in the following order

C IC CT -1

C CRC CT

Each operation corresponds to one function call in the program.

Note that the last operation specified is the first executed in the program

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CTM in OpenGL

•OpenGL has a model-view and a projection matrix in the pipeline which are concatenated together to form the CTM

•Can manipulate each by first setting the correct matrix mode

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Rotation, Translation, Scaling

glRotatef(theta, vx, vy, vz)

glTranslatef(dx, dy, dz)

glScalef(sx, sy, sz)

glLoadIdentity()

Load an identity matrix:

Multiply on right:

theta in degrees, (vx, vy, vz) define axis of rotation

Each has a float (f) and double (d) format (glScaled)

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Example

• Rotation about z axis by 30 degrees with a fixed point of (1.0, 2.0, 3.0)

• Remember that last matrix specified in the program is the first applied

• Demo

glMatrixMode(GL_MODELVIEW);glLoadIdentity();glTranslatef(1.0, 2.0, 3.0);glRotatef(30.0, 0.0, 0.0, 1.0);glTranslatef(-1.0, -2.0, -3.0);

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Arbitrary Matrices

•Can load and multiply by matrices defined in the application program

•The matrix m is a one dimension array of 16 elements which are the components of the desired 4 x 4 matrix stored by columns

• In glMultMatrixf, m multiplies the existing matrix on the right

glLoadMatrixf(m)glMultMatrixf(m)

Matrix multiply

• //沿 Y轴向上平移 10个单位glTranslatef(0,10,0);//画第一个球体DrawSphere(5);//沿 X轴向左平移 10个单位glTranslatef(10,0,0);//画第二个球体DrawSphere(5);

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procedure RenderScene();    begin      glMatrixMode(GL_MODELVIEW);      //沿 Y轴向上平移 10个单位       glTranslatef(0,10,0);      //画第一个球体       DrawSphere(5);      //加载单位矩阵       glLoadIdentity();      //沿 X轴向上平移 10个单位       glTranslatef(10,0,0);      //画第二个球体       DrawSphere(5);    end;

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Matrix Stacks

• In many situations we want to save transformation matrices for use later

Traversing hierarchical data structures

Avoiding state changes when executing display lists

•OpenGL maintains stacks for each type of matrix

Access present type (as set by glMatrixMode) by

glPushMatrix()glPopMatrix()

GL_PROJECTIONGL_MODEVIEW

Matrix Stacks

• procedure RenderScene();    begin      glMatrixMode(GL_MODELVIEW);     //push matrix stack      glPushMatrix;       //translate 10 along Y axis      glTranslatef(0,10,0);      //draw the first sphere      DrawSphere(5);      //come back to the last saved state      glPopMatrix;      // translate 10 along X axis       glTranslatef(10,0,0);      //draw the second sphere      DrawSphere(5);    end;

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Reading Back Matrices

• Can also access matrices (and other parts of the state) by query functions

• For matrices, we use as

glGetIntegervglGetFloatvglGetBooleanvglGetDoublevglIsEnabled

float m[16];glGetFloatv(GL_MODELVIEW, m);

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Using Transformations

• Example: use idle function to rotate a cube and mouse function to change direction of rotation

• Start with a program that draws a cube (colorcube.c) in a standard way

Centered at origin

Sides aligned with axes

Will discuss modeling in next lecture

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main.c

void main(int argc, char **argv) { glutInit(&argc, argv); glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB | GLUT_DEPTH); glutInitWindowSize(500, 500); glutCreateWindow("colorcube"); glutReshapeFunc(myReshape); glutDisplayFunc(display); glutIdleFunc(spinCube); glutMouseFunc(mouse); glEnable(GL_DEPTH_TEST); glutMainLoop();}

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Idle and Mouse callbacks

void spinCube() {theta[axis] += 2.0;if( theta[axis] > 360.0 ) theta[axis] -= 360.0;glutPostRedisplay();

}void mouse (int btn, int state, int x, int y){ char *sAxis [] = { "X-axis", "Y-axis", "Z-axis" }; /* mouse callback, selects an axis about which to rotate */ if (btn == GLUT_LEFT_BUTTON && state == GLUT_DOWN) { axis = (++axis) % 3; printf ("Rotate about %s\n", sAxis[axis]); }}

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Display callback

void display(){ glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); glLoadIdentity(); glRotatef(theta[0], 1.0, 0.0, 0.0); glRotatef(theta[1], 0.0, 1.0, 0.0); glRotatef(theta[2], 0.0, 0.0, 1.0); colorcube(); glutSwapBuffers();}

Note that because of fixed from of callbacks, variables such as theta and axis must be defined as globals

Demo

Polygonal Mesh

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Polygonal mesh

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Object Point cloud

•Surface reconstruction

Polygonal mesh

PhD thesis of Hugues Hoppe1994

•Mesh smoothing

Polygonal mesh

Non-iterative, feature preserving mesh smoothingACM Transactions on Graphics, 2003

•Mesh simplification

Polygonal mesh

CGAL, manual

•Parameterization

Polygonal mesh

Polygonal mesh

•Mesh morphing

Polygonal mesh

Mean Value Coordinates for Closed Triangular Meshes Ju T., Schaefer S. and Warren J.ACM SIGGRAPH 2005

•Remeshing&Optimization

Polygonal mesh

SGP 2009

Polygonal mesh

•Polygonal mesh in OpenGL

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f=x^4-10*r^2*x^2+y^4-10*r^2*y^2+z^4-10*r^2*z^2r=0.13

Representation

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Representation

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