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Geometric Transformations
CENG 477 Introduction to Computer GraphicsFall 2015-2016
Computer EngineeringMETU
2D Geometric Transformations
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 pointsCircle, 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'=
t
t=
y
x=
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'
cossinsincos)sin(
sinsincoscos)cos(
rrry
rrrx
sin
cos
ry
rx
The original coordinates are:
cossin
sincos
yxy
yxx
Substituting them in the
first equation we get:
In the matrix form we have:
where
PRP
cossin
sincosR
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'
cos)(sin)(
sin)(cos)(
rrr
rrr
yyxxyy
yyxxxx
(xr,yr)
Scaling
● Change the size of an object. Input: scaling factors (sx,sy)
PSP
S
'
=
s
s=
ys=y'xs=x'
y
x
yx
0
0
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.
h
yh
xh
=
h
y
x
=Ph
y=y
h
x=x h
h
hh
● 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
t
t
=t,tT
yx'
y
x
yx
100
10
01
PθR=P
θθ
θθ
=θR
'
100
0cossin
0sincos
Ps,sS=P
s
s
=s,sS
yx
y
x
yx
100
00
00
Composite Transformations
● Application of a sequence of transformations to a point:
PM
PMMP 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=tt
tt
=t
t
t
t
=t,tTt,tT 212121
21
1
1
2
2
1122
100
10
01
100
10
01
100
10
01
PTT
PTTP
)},(),({
}),({),(
1122
1122
yxyx
yxyx
tttt
tttt
yyxxyy
xx
y
x
y
x
yxyx ss,ss=ss
ss
=s
s
s
s
=s,ss,s 212121
21
1
1
2
2
1122
100
00
00
100
00
00
100
00
00
SSS
Composite Transformations
φ+θ=φ+θφ+θ
φ+θφ+θ
=φθ+φθφθ+φθ
φθφθφθφθ
=θθ
θθ
=φθ
R
RR
100
0)cos()sin(
0)sin()cos(
100
0coscossinsinsincoscossin
0cossinsincossinsincoscos
100
0cossin
0sincos
100
0cossin
0sincos
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
100
sincos1cossin
sincos1sincos
100
10
01
100
0cossin
0sincos
100
10
01
θxθyθθ
θy+θxθθ
=y
x
θθ
θθ
y
x
=y,xθy,x
rr
rr
r
r
r
r
rrrr TRT
rr y,x T
R
rr y,xT
Scaling with respect to a fixed point
– Translate to origin
– Scale
– Translate back
100
10
10
100
10
01
100
00
00
100
10
01
yfy
xfx
f
f
y
x
f
f
ffyxff
sys
sxs
=y
x
s
s
y
x
=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 0
0 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,yto 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 positivey' axis:
xx
y
y'
x'
V
yx v,v==
y'
V
Vv
:direction in ther unit vecto
yxxy u,u=v,v=
x'
u
v 90 clockwise rotate direction, in ther unit vecto
● Elements of any rotation matrix can be expressed as elements of a set of orthogonal unit vectors:
100
0
0
100
0
0
yx
xy
yx
yx
vv
vv
=vv
uu
=R
xx
yy'
x'
P
0P
0
0
PP
PPv
=
● Example:
xx'
yy'
P
0P
111
1.220.6
1.610.8
111
121
443
100
20.80.6
10.60.8
100
110
201
100
00.80.6
00.60.8
12,
0.60.8,
0.6,0.82.5
2
2.5
1.5
21.5
1.5,2
3.5,32,1
220
0
=T
=
=,=
=,=+
==
==
M
TvuRM
=u
PP
PPv
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 DIMENSIONALTRANSFORMATIONS
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
11000
100
010
001
1
z
y
x
t
t
t
z
y
x
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
1000
0cossin0
0sincos0
0001
)(
xR PRP )(x
1000
0100
00cossin
00sincos
)(
zR PRP )(z
1000
0cos0sin
0010
0sin0cos
)(
yR PRP )(y
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
x
z
y
x
Translate Rotate Translate back
PTRTP ),,0()(),,0( ppxpp zyzy
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
),,( cbaV
Vuu is the unit vector along V:
First step: Translate P1 to origin:
1000
100
010
001
1
1
1
z
y
x
T
Next step: Align u with the z axiswe need two rotations: rotate around x axis to get u onto 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( cbu
22 cos cbdd
c
z
z
uu
uu
bxzxz uuuuuu sin
sindb
d
b
d
c sin cos
1000
00
00
0001
)(
d
c
d
bd
b
d
c
x RProjection 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
duu
uu
z
z
cos
)(sin ayzyz uuuuuu
1000
00
0010
00
)(
sin cos
da
ad
ad
y
R
),,()()()()()(),,()( 111111 zyxzyx xyzyx TRRRRRTR
β
Rotation, ... Alternative Method
Any rotation around origin can be represented by 3 orthogonal unit vectors:
1000
0
0
0
333231
232221
131211
rrr
rrr
rrr
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* vectors
onto 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
uuu
uu
uuu
uu
Rotation, ... Alternative Method
1000
0
00
0
)/,/,(),,()/,/,0(
)/,/,0(),,0()0,0,1(),,(
),,(
22
cbad
b
d
cd
ca
d
bad
dcadbadcbadbdc
dbdccb
bccba
cba
zyx
xx
xy
z
R
uuu
uuuu
uuu
uu
Check if this is equal to
)()( xy RR
Scaling
● Change the coordinates of the object by scaling factors.
x 'y 'z '1
sx 0 0 0
0 s y 0 0
0 0 sz 0
0 0 0 1
xyz1
P ' S P
z
y
x
z
y
x
PSP
P
P
Scaling with respect to a Fixed Point
● Translate to origin, scale, translate back
z
y
x
z
y x
z
y
x
z
y
x
Translate Scale Translate back
PTSTP ),,(),,( ffffff zyxzyx
Scaling with respect to a Fixed Point
1000
)1(00
)1(00
)1(00
),,(),,(
1000
00
00
00
1000
100
010
001
1000
100
010
001
1000
000
000
000
1000
100
010
001
),,(),,(
zfz
yfy
xfx
ffffff
fzz
fyy
fxx
f
f
f
f
f
f
z
y
x
f
f
f
ffffff
szs
sys
sxs
zyxzyx
zss
yss
xss
z
y
x
z
y
x
s
s
s
z
y
x
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
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