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Geometric Geometric TransformationsTransformations
Ceng 477 Introduction to Computer GraphicsFall 2007
Computer EngineeringMETU
2D Geometric 2D Geometric TransformationsTransformations
Basic Geometric Transformations
● Geometric transformations are used to transform the objects and the camera in a scene (for animation or modelling) and are also used to transform World Coordinates to View Coordinates
● Given the shape, transform all the points of the shape? Transform the points and/or vectors describing it.
● For example: Polygon: corner points Circle, Ellipse: center point(s), point at angle 0
● Some transformations preserves some of the attributes like sizes, angles, ratios of the shape.
Translation
● Simply move the object to a relative position.
TPP
PTP
+=
y'x'
=tt
=yx
=
t+y=y't+x=x'
y
x
yx
'
' ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
T
P
P'
Rotation
● A rotation is defined by a rotation axis and a rotation angle.
● For 2D rotation, the parameters are rotation angle (θ) and the rotation point (xr ,yr ).
● We reposition the object in a circular path arround the rotation point (pivot point)
θ
xr
yr
Rotation
● When (xr ,yr )=(0,0) we have
r P
P'
θφθφθφθφθφθφ
cossinsincos)sin(sinsincoscos)cos(
rrryrrrx
+=+=′−=+=′
φφ
sincos
ryrx
==The original coordinates are:
θθθθ
cossinsincos
yxyyxx
+=′−=′Substituting them in the
first equation we get:
In the matrix form we have:
where
PRP ⋅=′
⎥⎦
⎤⎢⎣
⎡ −=
θθθθ
cossinsincos
R
Rotation
● Rotation around an arbitrary point (xr ,yr )
● This equations can be written as matrix operations (we will see when we discuss homogeneous coordinates).
r
P
P'
θθθθ
cos)(sin)(sin)(cos)(
rrr
rrr
yyxxyyyyxxxx
−+−+=′−−−+=′
(xr ,yr )
Scaling
● Change the size of an object. Input: scaling factors (sx ,sy )
PSP
S
' ⋅
⎥⎦
⎤⎢⎣
⎡
=
ss
=
ys=y'xs=x'
y
x
yx
00
PP'
non-uniform vs.uniform scaling
Homogenous Coordinates● All transformations can be represented by matrix
operations.
● Translation is additive, rotation and scaling is multiplicative (+ additive if you rotate around an arbitrary point or scale around a fixed point); making the operations complicated.
● Adding another dimension to transformations make translation also representable by multiplication. Cartesian coordinates vs homogenous coordinates.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
hyhxh
=hyx
=Phy=y
hx=x h
hhh
● Many points in homogenous coordinates can represent the same point in Cartesian coordinates.
● In homogenous coordinates, all transformations can be written as matrix multiplications.
Transformations in Homogenous C.
● Translation
● Rotation
● Scaling
( )
( ) Pt,tT=P
tt
=t,tT
yx'
y
x
yx
⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1001001
( )
( ) PθR=P
θθθθ
=θR
' ⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
1000cossin0sincos
( )
( ) Ps,sS=P
ss
=s,sS
yx
y
x
yx
⋅′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1000000
Composite Transformations
● Application of a sequence of transformations to a point:
PMPMMP 12
⋅=⋅⋅=′
Composite Transformations
● First: composition of similar type transformations
● If we apply to successive translations to a point:
( ) ( ) ( )yyxxyy
xx
y
x
y
x
yxyx t+t,t+tT=tttt
=tt
tt
=t,tTt,tT 212121
21
1
1
2
2
1122
1001001
1001001
1001001
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
PTT
PTTP
⋅⋅=
⋅⋅=′
)},(),({
}),({),(
1122
1122
yxyx
yxyx
tttt
tttt
( ) ( ) ( )yyxxyy
xx
y
x
y
x
yxyx ss,ss=ssss
=ss
ss
=s,ss,s 212121
21
1
1
2
2
1122
1000000
1000000
1000000
⋅⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅ SSS
Composite Transformations
( ) ( )
( )φ+θ=φ+θφ+θφ+θφ+θ
=φθ+φθφθ+φθφθφθφθφθ
=θθθθ
=φθ
R
RR
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −⋅
1000)cos()sin(0)sin()cos(
1000coscossinsinsincoscossin0cossinsincossinsincoscos
1000cossin0sincos
1000cossin0sincos
ϕϕϕϕ
Rotation around a pivot point– Translate the object so that the
pivot point moves to the origin
– Rotate around origin
– Translate the object so that the pivot point is back to its original position
( ) ( ) ( )
( )( )
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−⋅⋅
100sincos1cossinsincos1sincos
1001001
1000cossin0sincos
1001001
θxθyθθθy+θxθθ
=yx
θθθθ
yx
=y,xθy,x
rr
rr
r
r
r
r
rrrr TRT
( )rr y,x −−T
( )θR
( )rr y,xT
Scaling with respect to a fixed point
– Translate to origin
– Scale
– Translate back
( ) ( ) ( )
( )( )
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−⋅⋅
1001010
1001001
1000000
1001001
yfy
xfx
f
f
y
x
f
f
ffyxff
syssxs
=yx
ss
yx
=y,xs,sy,x TST
( )ff y,x −−T
( )yx s,sS
( )ff y,xT
Order of matrix compositions
● Matrix composition is not commutative. So, be careful when applying a sequence of transformations.
pivot same pivot
Other Transformations
● Reflection
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
100
010
001
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
100
010
001
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
100
010
001
● Shear: Deform the shape like shifted slices.
(1,1)(3,1)
(2,1)
x ' x shx y y ' y
1 shx 00 1 00 0 1
(0,1)
Transformations Between the Coordinate Systems
● Between different systems: Polar coordinates to cartesian coordinates
● Between two cartesian coordinate systems. For example, relative coordinates or window to viewport transformation.
● How to transform from x,y to x',y' ?
● Superimpose x',y' to x,y
● Transformation:
– Translate so that (x0 ,y0 ) moves to (0,0) of x,y
– Rotate x' axis onto x axis
xx
y
y'
x'
x0
y0
( ) ( )00, yxTθR −−⋅−
xx
y
y'x'
● Alternate method for rotation: Specify a vector V for positive y' axis:
xx
y
y'
x'
V( )yx v,v==
y'
VVv
:direction in ther unit vecto
( ) ( )yxxy u,u=v,v=x'
−uv o90 clockwise rotate direction, in ther unit vecto
● Elements of any rotation matrix can be expressed as elements of a set of orthogonal unit vectors:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
10000
10000
yx
xy
yx
yx
vvvv
=vvuu
=R
xx
yy'
x'
P
0P
| |0
0
PPPPv
−−=
● Example:
xx'
yy'
P
0P
( ) ( )
| |( ) ( )
( )
( ) ( )
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
−−⋅
−
⎟⎠⎞
⎜⎝⎛
−−
111
1.220.6
1.610.8111
121
443 100
20.80.6
10.60.8
100
110
201
100
00.80.6
00.60.812,
0.60.8,
0.6,0.82.52
2.51.5
21.51.5,2
3.5,32,1
220
0
=T
==,=
=,=+
==
==
M
TvuRM
=uPPPPv
PP
0
TM T
Let triangle T be defined asthree column vectors:
Affine Transformations
● Coordinate transformations of the form:
● Translation, rotation, scaling, reflection, shear. Any affine transformation can be expressed as the combination of these.
● Rotation, translation, reflection: preserve angles, lengths, parallel lines
yyyyx
xxyxx
b+ya+xa=y'
b+ya+xa=x'
3 DIMENSIONAL3 DIMENSIONAL TRANSFORMATIONSTRANSFORMATIONS
3D Transformations
● x,y,z coordinates. Usual notation: Right handed coordinate system
● Analogous to 2D we have 4 dimensions in homogenous coordinates.
● Basic transformations:
– Translation
– Rotation
– Scaling
x
y
z
z
x
y
y
z
x
Translation
● move the object to a relative position.
z
y
x
z
y
x
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′′′
11000100010001
1zyx
ttt
zyx
z
y
x
PTP ⋅=′
P
P′
Rotation
● Rotation arround the coordinate axes
x axis y axis z axis
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
Counterclockwise when looking along the positive half towards origin
Rotation around coordinate axes
● Arround x
● Arround y
● Arround z
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
10000cossin00sincos00001
)(θθθθ
θxR PRP ⋅=′ )(θx
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
1000010000cossin00sincos
)(θθθθ
θzR PRP ⋅=′ )(θz
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
10000cos0sin00100sin0cos
)(θθ
θθ
θyR PRP ⋅=′ )(θy
Rotation Arround a Parallel Axis
● Rotating the object around a line parallel to one of the axes: Translate to axis, rotate, translate back.
z
y
x
z
y
x
z
y
xz
y
x
Translate Rotate Translate back
PTRTP ⋅−−⋅⋅=′ ),,0()(),,0( ppxpp zyzy θ
Figure from the textbook
Rotation Around an Arbitrary Axis
● Translate the object so that the rotation axis passes though the origin
● Rotate the object so that the rotation axis is aligned with one of the coordinate axes
● Make the specified rotation
● Reverse the axis rotation
● Translate back
z
y
x
Rotation Around an Arbitrary Axis
Rotation Around an Arbitrary Axis
),,( 12121212 zzyyxx −−−=−= PPV),,( cba==
VVuu is the unit vector along V:
First step: Translate P1 to origin:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
=
1000100010001
1
1
1
zyx
T
Next step: Align u with the z axiswe need two rotations: rotate around x axis to get uonto the xz plane, rotate around y axis to get u aligned with z axis.
Rotation Around an Arbitrary Axis
z
x
u
Align u with the z axis 1) rotate around x axis to get u into the xz plane, 2) rotate around y axis to get u aligned with the z axis
y
α
z
xu
y
βz
x
uu'
αuz
y
Dot product and Cross Product
● v dot u = vx * ux + vy * uy + vz * uz. That equals also to |v|*|u|*cos(a) if a is the angle between v and u vectors. Dot product is zero if vectors are perpendicular.
v x u is a vector that is perpendicular to both vectors you multiply. Its length is |v|*|u|*sin(a), that is an area of parallelogram built on them. If v and u are parallel then the product is the null vector.
Rotation Around an Arbitrary Axis
z
x
uu'
α
Align u with the z axis 1) rotate around x axis to get u into the xz plane, 2) rotate around y axis to get u aligned with the z axis
uz
We need cosine and sine of α for rotation
),,0( cb=′u
22 cos cbddc
z
z +==′⋅′
=uuuuα
bxzxz uuuuuu =′=×′ αsin
αsindb =
db
dc
== αα sin cos
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡−
=
1000
00
000001
)(
dc
db
db
dc
x αRProjection of u onyz plane
Rotation Around an Arbitrary Axis
z
x
u
u''= (a,0,d)
Align u with the z axis 1) rotate around x axis to get u into the xz plane,2) rotate around y axis to get u aligned with the z axis
duuuu
z
z =⋅′′⋅′′
=βcos
)(sin ayzyz −⋅=′′=×′′ uuuuuu β
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
−==
100000001000
)(
sin cos
da
adad
y β
ββ
R
),,()()()()()(),,()( 111111 zyxzyx xyzyx −−−⋅⋅⋅⋅−⋅−⋅= TRRRRRTR αβθβαθ
β
Rotation, ... Alternative MethodAny rotation around origin can be represented by 3 orthogonal unit vectors:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
1000000
333231
232221
131211
rrrrrrrrr
R
Define a new coordinate system with the given rotation axis u using:
),,( zyx uuu ′′′
This matrix can be thought of asrotating the unit r1* , r2* , and r3* vectorsonto x, y, and z axes.
So, to align a given rotation axis u onto the z axis,we can define an (orthogonal) coordinate system and form this R matrix
zyx
x
xy
z
uuuuuuuu
uu
′×′=′
××
=′
=′
Rotation, ... Alternative Method
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
⋅−⋅−
=
⋅−⋅−=×−=′×′=′
−=+
−=
××
=××
=′
==′
10000
00
0
)/,/,(),,()/,/,0(
)/,/,0(),,0()0,0,1(),,(),,(
22
cbadb
dc
dca
dbad
dcadbadcbadbdc
dbdccbbccba
cba
zyx
xx
xy
z
R
uuuuuuu
uuu
uu
Check if this is equal to
)()( αβ xy RR ⋅
Scaling
● Change the coordinates of the object by scaling factors.
x 'y 'z '1
sx 0 0 00 s y 0 00 0 sz 00 0 0 1
xyz1
z
y
x
z
y
xPSP ⋅=′
P
P′
Scaling with respect to a Fixed Point
● Translate to origin, scale, translate back
z
y
x
z
y x
z
y
xz
y
x
Translate Scale Translate back
PTSTP ⋅−−−⋅⋅=′ ),,(),,( ffffff zyxzyx
Scaling with respect to a Fixed Point
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
=−−−⋅⋅=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=−−−⋅⋅
1000)1(00)1(00)1(00
),,(),,(
100000
0000
1000100010001
1000100010001
1000000000000
1000100010001
),,(),,(
zfz
yfy
xfx
ffffff
fzz
fyy
fxx
f
f
f
f
f
f
z
y
x
f
f
f
ffffff
szssyssxs
zyxzyx
zssyssxss
zyx
zyx
ss
s
zyx
zyxzyx
TST
TST
Reflection
● Reflection over planes, lines or points
1 0 0 00 1 0 00 0 1 00 0 0 1
z
y
x
z
y
x
z
y
x
1 0 0 00 1 0 00 0 1 00 0 0 1
1 0 0 00 1 0 00 0 1 00 0 0 1
z
y
x
1 0 0 00 1 0 00 0 1 00 0 0 1
Shear
● Deform the shape depending on another dimension
SH z
1 0 a 00 1 b 00 0 1 00 0 0 1
x and y value depends on z value of the shape
SH x
1 0 0 0a 1 0 0b 0 1 00 0 0 1
y and z value depends on x value of the shape
OpenGL Geometric-Transformation Functions
● In the core OpenGL library,
– a separate function is available for each basic transformation (translate, rotate, scale)
– all transformations are specified in 3D
● Parameters
– Translation: translation amount in x, y, z axes
– Rotation: angle, orientation of the rotation axis that passes through the origin
– Scaling: scaling factors for three coordinates
Basic OpenGL Transformations
● glTranslate* (tx, ty, tz);
– For 2D applications set tz = 0
● glRotate* (theta, vx, vy, vz);
– theta in degrees
– The rotation axis is defined by the vector (vx,vy,vz), i.e., P0 = (0,0,0) P1 = (vx,vy,vz)
● glScale* (sx, sy, sz);
– Use negative values to get reflection transformation
OpenGL Matrix Operations
● glMatrixMode (GL_MODELVIEW);
– modelview mode to tell OpenGL that we will be specifying geometric transformations. The command simply says that the current matrix operations will be applied on the 4 by 4 modelview matrix.
– the other mode is the projection mode, which specifies the matrix that is used of projection transformations (i.e., how a scene is projected onto the screen)
– There are also color and texture modes that we will discuss later
OpenGL Matrix Operations
● Once you are in the modelview mode, a call to a transformation routine generates a matrix that is multiplied by the current matrix for that mode
● Whatever object defined is multiplied with the current matrix
● The contents of the current matrix can also be manipulated explicitly
– glLoadIdentity();
– glLoadMatrix* (elements16);
where elements16 is a single subscripted array that specifies a matrix in column-major order
OpenGL Matrix Operations
● Example:
for (int k=0; k<16;k++)
elements16[k]=(float)k;
glLoadMatrixf(elements16);
will produce the matrix
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
0.150.110.70.30.140.100.60.20.130.90.50.10.120.80.40.0
M
OpenGL Matrix composition
● glMultMatrix* (otherElements16)
– The current matrix is postmultiplied with the matrix specified in otherElements16
what does this imply?
MMM ′⋅= currcurr
In a sequence of transformation commands, the lastone specified in the code will be the first transformation to be applied.
OpenGL Matrix Stacks
● OpenGL maintains a matrix stack for all the four matrix modes
● When we apply geometric transformations using OpenGL functions, the 4 by 4 matrix at the top of the matrix stack is modified
● The top is also called the current matrix
● If we want to create multiple transformation sequences and save the composition results we can make use of the OpenGL matrix stack
OpenGL Matrix Stacks
● Initially, there is only the identity matrix in the stack
● To find out how many matrices are currently in the stack:– glGetIntegerv(GL_MODELVIEW_STACK_DEPTH,numMats)
● glPushMatrix ();
– The current matrix is copied and stored in the second stack position
● glPopMatrix ();
– Destroys the matrix at the top and the second matrix in the stack becomes the current matrix
