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2D Transformations3D TransformationsOpenGL Transformation
The most basic ones Translation Scaling Rotation Shear Shear And others, e.g., perspective transform, projection, etc
Basic types of transformations
Rigid-body: preserves length and angle
Affine: preserves parallel lines, not angles or lengths
Free-form: anything goes
Basic Transformations Homogeneous coordinate system Composition of transformations Composition of transformations
(4,5) (7,5)
Y Y
XBefore Translation
1
*
100
10
01
1
y
x
d
d
y
x
TPPd
dT
y
xP
y
xP
y
x
y
x
Form sHomogeniou
x’ = x + dx
y’ = y + dy
(7,1) (10,1)
XTranslation by (3,-4)
(4,5) (7,5)
Y
X(2,5/4) (7/2,5/4)
X
Y
Before Scaling Scaling by (1/2, 1/4)
Types of Scaling:
Differential ( sx != sy )Uniform ( sx = sy )
X XBefore Scaling Scaling by (1/2, 1/4)
y
x
y
x
y
x
sy
sx
y
x
s
s
PPS
ysy
xsx
*
**
0
0
*
*
*
1
*
100
00
00
1
Form sHomogeniou
y
x
s
s
y
x
y
x
cosr
v
sin
cos
r
rv
sin
cos
r
rv
cossinsincos
sinsincoscos
rry
rrx expand
cossin
sincos
sin
cos
yxy
yxx
ry
rx
but
Y
(2.1,4.9)
(4.9,7.8)
YBefore Rotation Rotation of 45 deg. w.r.t. origin
(5,2) (9,2)X X
1
*
100
0cossin
0sincos
1
Form sHomogeniou
y
x
y
x
cos*sin*
sin*cos**
cossin
sincos
*
yx
yx
y
x
PPR
yyx
xyx
cos*sin*
sin*cos*
(1,1)
Y
X
(-1,1) (1,1)
X
Y
100
010
001
axis-Xabout Reflection
xM
yyxx
100
010
001
axis-Yabout Reflection
yM
yyxx
(1,-1)
100
01
01
100
01
001
100
010
01
b
a
SHbSH
a
SH xyyx
100100100
unit cubeSheared in X
directionSheared in Y
directionSheared in both X
and Y direction
-
(-θ-(θ
(-dx,-dy)-
(dx,dy)
SS
RR
TT
1
)1)
1
: Sclaing
: Rotation
:nTranslaito
y-y
x-x
),(-(sx,sy)
MM
MM
SSsysx
1
1
1
:RefMirror
: Sclaing 11
Translation, scaling and rotation are expressed (non-homogeneously) as: translation: P = P + T
Scale: P = S · PScale: 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 3 matrices
W is 1 for affine transformations in graphics
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)
x
y
w Ph(x,y,w)
P2d(x,y,1)
w=1
1. Rigid-body Transformation
Preserves parallelism of lines
Preserves angle and length
e.g. any sequence of R() and T(dx,dy)
2. Affine Transformation2. 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
1002221
1211
y
x
trr
trr
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
100
0cossin
0sincos
• Orthogonal Transformation Rigid-Body Transformation• For any orthogonal matrix B B-1 = BT
• In general matrix multiplication is not commutative• For the following special cases commutativity holds i.e.
M1.M2 = M2.M1
M1 M2
Translate TranslateTranslate Translate
Scale Scale
Rotate Rotate
Uniform Scale Rotate
• Some non-commutative Compositions: Non-uniform scale, Rotate Translate, Scale Rotate, Translate
OriginalTransitional
Final
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.
R =
Step 1: Translate P(h,k) to origin
T(-h ,-k)
Step 2: Rotate w.r.t to origin
R *
Step 3: Translate (0,0) to P(h,k0)
T(h ,k) *
P3(h,k)
Q3(x’+h, y’ +k)
R,P=
Q(x,y)
P(h,k)
T(-h ,-k)
Q1(x’,y’)
P1 (0,0)
R*
Q2(x’,y’)
P2 (0,0)
T(h ,k) *
S =
Step 1: Translate P(h,k) to origin
Step 2: Scale S(sx,sy) w.r.t origin
(7,1)
Step 3: Translate (0,0) to P(h,k)(7,2)
(1,1)T(-h ,-k)S(sx,sy)*T(h ,k) *Ssx,sy,P=
(4,3)
(1,1) (4,1)
S3/2,1/2,(1,1)
(4,2)
(0,0) (4,0)
T(-1,-1)
(6,1)
(6,0)(0,0)
S(3/2,1/2)
(7,1)(1,1)
T(1,1)
Step 1: Translate (0,b) to origin
Step 2: Rotate - degrees
YYYYYY
T(0 ,-b)ML =
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() *
(0,b)
X
t
O XO XO XO XO
(0,b)
X
t
O
Schaum’s outline series:
Problems:
4.1
4.2
4.3, 4.4, 4.5 => R,P
4.6, 4.7, 4.8 => S 4.6, 4.7, 4.8 => Ssx,sy,P
4.9, 4.10, 4.11, 4.21 => ML
4.12 => Shearing
1. www.willamette.edu/~gorr/classes/GeneralGraphics/Transforms/transforms2d.htm2. http://www.cs.sfu.ca/~torsten/Teaching/Cmpt361/LectureNotes/PDF/06_2Dtrans.pdf
Basics of 3D geometryBasic 3D TransformationsComposite Transformations
Thumb points to +ve Z-axis Fingers show +ve rotation from X to Y
axis
Y YY
X
Z (out of page)
Y
X
Z (larger z areaway from viewer)
Right-handed orentation Left-handed orentation
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
y
x
baaa
baaa
baaa
z
y
x
yyzyyyx
xxzxyxx
z’ = azx x + azy y + azz z + bz ,
is called a 3D affine transformation.
11000
' zbaaa
w
z zzzzyzx
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.
z
y
x
dzz
dyy
dxx
PPdddT
dz
dy
dx
z
y
x
d
d
d
zyx
z
y
x
z
y
x
*),,(
11
*
1000
100
010
001
Original scale Y axiszsz
ysy
xsx
z
y
x
*
*
*
1
*
*
*
1
*
1000
000
000
000
*),,(
z
y
x
z
y
x
zyx
sz
sy
sx
z
y
x
s
s
s
PPsssS
scale all axes
For 3D-Rotation 2 parameters are needed
Angle of rotation
Axis of rotation
Rotation about z-axis:
1
cos*sin*
sin*cos*
1
*
1000
0100
00cossin
00sincos
*,
z
yx
yx
z
y
x
PPR k
cos*sin*
sin*cos*
*0cos0sin
0010
0sin0cos
*,
zx
y
zx
z
y
x
PPR jAbout y-axis
111000
1
cos*sin*
sin*cos*
1
*
1000
0cossin0
0sincos0
0001
*,
zy
zy
x
z
y
x
PPR iAbout x-axis
*),( PPshshSH yxxy
y
z
x
1
*
*
1
*
1000
0100
010
001
z
shzy
shzx
z
y
x
sh
sh
y
x
y
x
Some of the composite transformations to be studied are:
AV,N = aligning a vector V with a vector N R = rotation about an axis L( V, P ) R,L = rotation about an axis L( V, P ) Ssx,sy,P= scaling w.r.t. point P
z
22λ
cos
λsin
by axis-about x Rotate :1 Step
cbc
b
b
z
( 0, b,c)b
z
( a, 0, )
( 0, 0, )
( 0, b,c)
Av = R,i
V = aI + bJ + cK
x
yb
a
c
k
λcos
|V|
x
yb
a
k
|V|
x
y
a
k
|V|
( a, 0, )
22λ
cos
λsin
by axis-about x Rotate :1 Step
cbc
b
b
z
( a, 0, )( 0, b,c)b
z( 0, 0, |V|)
( 0, b,c)a
Av = R,iR-,j *
λcos
222
|V|
|V|
λ)cos(
|V|)sin(
-by axis-yabout V Rotate :2 Step
cba
a
P( a, b, c)
x
yb
a
c
k
|V|
( a, 0, )
x
yb
a
c
|V|
AV-1 = AV
T
AV,N = AN-1 * AV
0--λ acab
1000
0
00
0
λλ
λ-
λ-λ
Vc
Vb
Va
bc
Vac
Vab
V
VA
Let the axis L be represented by vector V and passing through point P
1. Translate P to the origin P
Q
L
z
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
-1 *
V
Q
Q'
x
y
k
Let the plane be represented by plane normal N and a point P in that plane
z
x
y
Let the plane be represented by plane normal N and a point P in that plane
1. Translate P to the origin z
MN,P = T-P
x
y
Let the plane be represented by plane normal N and a point P in that plane
1. Translate P to the origin z
2. Align N with vector k
MN,P = T-PAN *
x
y
Let the plane be represented by plane normal N and a point P in that plane
1. Translate P to the origin z
2. Align N with vector k
3. Reflect w.r.t xy-plane
MN,P = T-PAN *S1,1,-1 *
x
y
z
Let the plane be represented by plane normal N and a point P in that plane
1. Translate P to the origin
x
y
2. Align N with vector k
3. Reflect w.r.t xy-plane
4. Reverse step 2
MN,P = T-PAN *S1,1,-1 *AN-1 *
Let the plane be represented by plane normal N and a point P in that plane
1. Translate P to the origin z
2. Align N with vector k
3. Reflect w.r.t xy-plane
4. Reverse step 2
5. Reverse step 1
MN,P = T-PAN *S1,1,-1 *AN-1 *T-P
-1 *
x
y
The Composite Transform must have
Translate points in fig. 1 into points in fig 2 such that:
– P1 is at Origin
– P1P2 is along positive z-axis
– P1P3 lies in positive y-axis half of yz plane
– Translation of P1 to Origin T
z
x
y
3P
1P
2PT
– Some Combination of Rotations R
R
x
y
z 2P
3P 1P
z
x
y3P
1P2P
Fig. 1 Fig. 2
zzz
yyy
xxx
zRyRxR
zRyRxR
zRyRxR
rrr
rrr
rrr
R
R
Transform body-Rigid is R
be Let
...
...
...
333231
232221
131211
xx
zyx
zyx
Rx.x R
RRR
RRR
vextor of component :Note
other each to
larperpendicu are ii)
vectors unit are i)
Transform body-Rigid is R
,,
,,
TT RRkR
21 PPˆ 1
x
y
3P
1P
2P Rz
kPP
PPR
axis-z along PP aligns R
21
21
21
ˆ
z
z
z
z
zyx
zyx
zyx
TT
R
zR
yR
xR
zRzRzR
yRyRyR
xRxRxR
RRkR
21
21
21
21
21
21
PP
PP
PP
PP
PP
PP
.
.
.
1
0
0
...
...
...
ˆ 1
R
z1
x
y
z 2P
3P 1P
k
x
y
3P
1P 2P
T2131T RRPPPP
iR
ˆ 1 Rx
iR
R
ˆ
2131
2131
2131
PPPP
PPPP
axis-x along PPPP aligns
Rz
R
x
y
z 2P
3P 1P
z
x
2131
2131
x
x
x
2131
2131
zyx
zyx
zyx
T
2131
2131T
RPPPP
PPPP
zR
yR
xR
PPPP
PPPP
zRzRzR
yRyRyR
xRxRxR
RRPPPP
iR
.
.
.
0
0
1
...
...
...
ˆ 1 Rx
ik
jR
R
ˆ
xz
xz
RR
axis- yalong RR aligns
x
y
3P
1P 2P
R
Rz
Ry
yxz
y
y
y
xz
zyx
zyx
zyx
Txz
T
RRR
zR
yR
xR
RR
zRzRzR
yRyRyR
xRxRxR
RRRRjR
.
.
.
0
1
0
...
...
...
ˆ 1R
x
y
z 2P
3P 1P
z Rx
ik
j
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 M 6.6, 6.7, 6.8 MN,P
OpenGL transformation commandsOpenGL transformation commandsTransformation OrderHierarchical Modeling
OpenGL uses 3 stacks to maintain transformation matrices:
Model & View transformation matrix stack
Projection matrix stack
Texture matrix stack Texture matrix stack
You can load, push and pop the stack The top most matrix from each stack is applied to all graphics primitive until it is changed
M N
Model-ViewMatrix Stack
ProjectionMatrix Stack
GraphicsPrimitives(P)
OutputN•M•P
Specify current matrix (stack) : void glMatrixMode(GLenum mode)
▪ Mode : GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE
Initialize current matrix. void glLoadIdentity(void)
ABC
glL
oad
Mat
rix(
M) 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
ABI
ABM
glL
oad
Mat
rix(
M)
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.column major order.
Best use utility function glTranslate, glRotate, glScale for common transformation tasks.
161284
151173
141062
13951
mmmm
mmmm
mmmm
mmmm
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);
CTRvv
Note: v is any vertex placed in rendering pipeline v’ is the transformed vertex from v.
glMatrixMode(GL_MODELVIEW);glLoadIdentity();glTranslatef(...);glRotatef(...);glScalef(...);gluCylinder(...);glScalef(...);gluCylinder(...);
There is a World Coordinate System where: All objects are defined
Transformations are in World Coordinate space
Two Different Views
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
Local View
Translate Object
Then Rotate
glLoadIdentity();
glMultiMatrixf( T);
glMultiMatrixf( R);
draw_ the_ object( v);
v’ = ITRv
Global View
Rotate Object
Then Translate
Effect is same, but perception is different
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
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 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
– void glPushMatrix(void);
– void glPoipMatrix(void);
Some of the OpenGL functions helpful for hierarchical modeling are:
ABCC
AB
– void glPoipMatrix(void);
ABC C
m
glGetFloatv
– void glGetFloatv(GL_MODELVIEW_MATRIX, *m);
ABC
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 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]; // Transformation
void (*f) ( ); // Draw function
struct treenode *sibling;
struct treenode *child;
} treenode;
Scene
Sun Star X
Earth Venus Saturn
Moon Ring
Initializing transformation matrix for nodetreenode torso, head, ...;/* in init function */glLoadIdentity();glRotatef(...);glGetFloatv(GL_MODELVIEW_MATRIX, torso.m);glGetFloatv(GL_MODELVIEW_MATRIX, torso.m);
Initializing pointerstorso.f = drawTorso;torso.sibling = NULL;torso.child = &head;
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
Recursive definitionvoid traverse (treenode *root) {
if (root == NULL) return;glPushMatrix();glMultMatrixf(root->m);root->f();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 !!
