PPT&Programs&Labcourse
http://211.87.235.32/share/ComputerGraphics/
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Transformations
Shandong University Software College
Instructor: Zhou YuanfengE-mail: [email protected]
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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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