Numerical Analysis – Eigenvalue and Eigenvector Hanyang University Jong-Il Park.
Computer Graphics - Transformation - Hanyang University Jong-Il Park.
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Transcript of Computer Graphics - Transformation - Hanyang University Jong-Il Park.
Computer GraphicsComputer Graphics- Transformation -- Transformation -
Hanyang University
Jong-Il Park
Division of Electrical and Computer Engineering, Hanyang University
ObjectivesObjectives
Introduce standard transformations Rotation Translation Scaling Shear
Derive homogeneous coordinate transformation matrices
Learn to build arbitrary transformation matrices from simple transformations
Learn how to carry out transformations in OpenGL Introduce OpenGL matrix modes
Model-view Projection
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General TransformationsGeneral 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 TransformationsAffine 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 ImplementationPipeline 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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NotationNotation
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 points-array of 4 scalars in homogeneous coordinates
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TranslationTranslation
Move (translate, displace) a point to a new location
Displacement determined by a vector d Three degrees of freedom P’=P+d
P
P’
d
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How many ways?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 RepresentationsTranslation Using Representations
Using the homogeneous coordinate representation in some frame
p=[ x y z 1]T
p’=[x’ y’ z’ 1]T
d=[dx dy dz 0]T
Hence p’ = p + d or
x’=x+dx
y’=y+dy
z’=z+dznote that this expression is in four dimensions and expressespoint = vector + point
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Translation MatrixTranslation Matrix
We can also express translation using a 4 x 4 matrix T in homogeneous coordinates
p’=Tp
where
T = T(dx, dy, dz) =
This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together
1000
d100
d010
d001
z
y
x
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Rotation (2D)Rotation (2D)
Consider rotation about the origin by degrees radius stays the same, angle increases by
x’=x cos –y sin y’ = x sin + y cos
x = r cos y = r sin
x’ = r cos (y’ = r sin (
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Rotation about the Rotation about the zz axis 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 MatrixRotation Matrix
1000
0100
00 cossin
00sin cos
R = Rz() =
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Rotation about Rotation about xx and and yy axes 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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Non-rigid-body TransformationsNon-rigid-body Transformations
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ScalingScaling
1000
000
000
000
z
y
x
s
s
s
S = S(sx, sy, sz) =
x’=sxxy’=syxz’=szx
p’=Sp
Expand or contract along each axis (fixed point of origin)
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ReflectionReflection
corresponds to negative scale factors
originalsx = -1 sy = 1
sx = -1 sy = -1 sx = 1 sy = -1
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InversesInverses
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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Linear TransformationsLinear Transformations
x' = Ax + By + Cz + D
y' = Ex + Fy + Gz + H
z' = Ix + Jy + Kz + L
Transform the geometry – leave the topology as is
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Geometry vs. TopologyGeometry vs. Topology
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Transformations in H.C.Transformations in H.C.
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Identity MatrixIdentity Matrix
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Matrix InverseMatrix Inverse
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TranslationTranslation
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2D Rotation2D Rotation
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3D Rotation about Z3D Rotation about Z
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3D Rotation about X & Y3D Rotation about X & Y
About X
About Y
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ConcatenationConcatenation
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 TransformationsOrder 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 p’T = pTCTBTAT
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General Rotation About the OriginGeneral Rotation About the Origin
x
z
y
v
A rotation by about an arbitrary axiscan be decomposed into the concatenationof rotations about the x, y, and z axes
R() = 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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GeneralGeneral RotationRotation
Read Sect. 4.9.4.
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Rotation About a Fixed Point other Rotation About a Fixed Point other than the Originthan the Origin
1. Move fixed point to origin
2. Rotate
3. Move fixed point back
M = T(pf) R() T(-pf)
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InstancingInstancing
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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ShearShear
Helpful to add one more basic transformation Equivalent to pulling faces in opposite
directions
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Shear MatrixShear Matrix
Consider simple shear along x axis
x’ = x + y cot y’ = yz’ = z
1000
0100
0010
00cot 1
H() =
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Matrix Multiplication Is NOT Matrix Multiplication Is NOT CommutativeCommutative
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Combining Translations, RotationsCombining Translations, Rotations
' ( )P TR P MP RP T
11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
1 0 0 0
0 1 0 0
0 0 1 0 0 1
0 0 0 1 0 0 0 1 0 0 0 1
x x
y y
z z
T R R R R R R T
T R R R R R R T R TM
T R R R R R R T
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Combining Translations, RotationsCombining Translations, Rotations
' ( ) ( )P RT P MP R P T RP RT
11 12 13
3 3 3 3 3 121 22 23
1 331 32 33
0 1 0 0
0 0 1 0
0 10 0 0 1
0 0 0 1 0 0 0 1
x
y
z
R R R T
R R TR R R TM
R R R T
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Matrix Multiplication Is Matrix Multiplication Is AssociativeAssociative
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Transformation GameTransformation Game
transformation_game.jar
http://www.cs.brown.edu/exploratories/freeSoftware/catalogs/repositoryApplets.html
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OpenGL MatricesOpenGL 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 be manipulated by
glMatrixMode(GL_MODELVIEW); glMatrixMode(GL_PROJECTION);
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Current Transformation Matrix (CTM)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 verticesp p’=Cp
C
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Concatenation of TransformationsConcatenation of Transformations
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Hardware can do this very quickly!
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CTM operationsCTM 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 CM
Postmultiply 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 PointRotation about a Fixed PointStart 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 OrderReversing 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 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, ScalingRotation, 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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ExampleExample
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
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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Compound TransformationCompound Transformation
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Arbitrary MatricesArbitrary 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)
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Matrix StacksMatrix Stacks
In many situations we want to save transformation matrices for use later Traversing hierarchical data structures (Chapter 10) 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()
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Reading Back MatricesReading Back Matrices
Can also access matrices (and other parts of the state) by query functions
For matrices, we use as
glGetIntegervglGetFloatvglGetBooleanvglGetDoublevglIsEnabled
double m[16];glGetFloatv(GL_MODELVIEW, m);
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Using TransformationsUsing 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
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main.c 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 callbacksIdle 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){ if(btn==GLUT_LEFT_BUTTON && state == GLUT_DOWN) axis = 0; if(btn==GLUT_MIDDLE_BUTTON && state == GLUT_DOWN) axis = 1; if(btn==GLUT_RIGHT_BUTTON && state == GLUT_DOWN) axis = 2;}
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Display callbackDisplay 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 form of callbacks, variables
such as theta and axis must be defined as globals
Camera information is in standard reshape callback
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Using the Model-view MatrixUsing the Model-view Matrix In OpenGL the model-view matrix is used to
Position the camera Can be done by rotations and translations but
is often easier to use gluLookAt Build models of objects
The projection matrix is used to define the view volume and to select a camera lens
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Model-view and Projection MatricesModel-view and Projection Matrices
Although both are manipulated by the same functions, we have to be careful because incremental changes are always made by postmultiplication For example, rotating model-view and projection
matrices by the same matrix are not equivalent operations. Postmultiplication of the model-view matrix is equivalent to premultiplication of the projection matrix
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Smooth RotationSmooth Rotation
From a practical standpoint, we are often want to use transformations to move and reorient an object smoothly Problem: find a sequence of model-view matrices
M0,M1,…..,Mn so that when they are applied successively to one or more objects we see a smooth transition
For orientating an object, we can use the fact that every rotation corresponds to part of a great circle on a sphere Find the axis of rotation and angle Virtual trackball (see text)
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Incremental Rotation Incremental Rotation
Consider the two approaches For a sequence of rotation matrices R0,R1,…..,Rn , find
the Euler angles for each and use Ri= Riz Riy Rix
Not very efficient
Use the final positions to determine the axis and angle of rotation, then increment only the angle
Quaternions can be more efficient than either
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QuaternionsQuaternions Extension of imaginary numbers from two to
three dimensions Requires one real and three imaginary
components i, j, k
Quaternions can express rotations on sphere smoothly and efficiently. Process: Model-view matrix quaternion Carry out operations with quaternions Quaternion Model-view matrix
q=q0+q1i+q2j+q3k
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InterfacesInterfaces One of the major problems in interactive
computer graphics is how to use two-dimensional devices such as a mouse to interface with three dimensional objects
Example: how to form an instance matrix? Some alternatives
Virtual trackball 3D input devices such as the spaceball Use areas of the screen
Distance from center controls angle, position, scale depending on mouse button depressed