Geometric
Transformations
CENG 477 Introduction to Computer Graphics
Fall 2015-2016
Computer Engineering
METU
2D Geometric
Transformations
Basic Geometric Transformations
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●
●
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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.
x' = x + t x
P'
y' = y + t y
P
t x 
x 
 x' 
P =   T =   P '=  
 y
 y' 
t y 
P '= P + T
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
yr
xr
Rotation
●
When (xr,yr)=(0,0) we have
P'
x  r cos(   )  r cos  cos   r sin  sin 
y  r sin(    )  r cos  sin   r sin  cos 
The original coordinates are: x  r cos 
y  r sin 
Substituting them in the
first equation we get:
In the matrix form we have:
where
cos 
R
 sin 
 sin  
cos  
x  x cos   y sin 
y  x sin   y cos 
P  R  P
P
Rotation
●
Rotation around an arbitrary point (xr,yr)
P'
x  xr  ( x  xr ) cos   ( y  yr ) sin 
y  yr  ( x  xr ) sin   ( y  yr ) cos 
●
P
(xr,yr)
This equations can be written as matrix
operations (we will see when we discuss
homogeneous coordinates).
Scaling
●
Change the size of an object. Input: scaling
factors (sx,sy)
x' = xsx
sx
S= 
0
y' = ys y
P'
P
0
s y 
P' = S  P
non-uniform vs.
uniform scaling
Homogenous Coordinates
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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.
xh
x=
h
yh
y=
h
 xh   h  x 




P =  yh  = h  y 
 h   h 
●
●
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.
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Translation
1 0 t x 
T t x ,t y = 0 1 t y 
0 0 1 
P' = T t x ,t y  P
●
Rotation
cosθ  sin θ 0
Rθ  =  sin θ cosθ 0
 0
0
1
P' = Rθ   P
●
Scaling
 s x 0 0
S s x , s y =  0 s y 0
 0 0 1
P = S s x , s y  P
Composite Transformations
●
Application of a sequence of transformations
to a point:
P  M 2  M1  P
 MP
Composite Transformations
●
●
First: composition of similar type
transformations
If we apply to successive translations to a
point:
P  T(t2 x , t2 y ) {T(t1x , t1 y )  P}
 {T(t2 x , t2 y )  T(t1x , t1 y )}  P
1 0 t2 x  1 0 t1x  1 0 t1x  t2 x 
T t2 x ,t2 y  T t1x ,t1 y = 0 1 t2 y   0 1 t1 y  = 0 1 t1 y  t2 y  = T t1x + t2 x ,t1 y + t2 y 
0 0 1  0 0 1  0 0
1 
Composite Transformations
cosθ  sin θ
R θ   R φ =  sin θ cosθ
 0
0
cosθcosφ  sin θsin φ  cosθsin φ  sin θcosφ
sin θcosφ + cosθsin φ  sin θsin φ + cosθcosφ


0
0
 s2 x
Ss2 x , s2 y  Ss1x , s1 y =  0
 0
0
s2 y
0
0 s1x
0   0
1  0
0
s1 y
0
0 cos
0   sin 
1  0
 sin 
cos
0
0
0 =
1
0 cos(θ + φ)  sin( θ + φ) 0
0 =  sin( θ + φ) cos(θ + φ) 0 = R θ + φ
1 
0
0
1
0 s1x  s2 x
0 =  0
1  0
0
s1 y  s2 y
0
0
0 = Ss1x  s2 x , s1 y  s2 y 
1
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
T xr , yr   R θ   T xr , yr  =
1 0 xr  cosθ  sin θ 0 1 0  xr 
0 1 y    sin θ cosθ 0  0 1  y  =
r 
r

 
0 0 1   0
0
1 0 0
1 
cosθ  sin θ xr 1  cosθ + yr sin θ 
 sin θ cosθ y 1  cosθ   x sin θ 
r
r


 0

0
1
T xr , yr 
R 
Txr , yr 
Scaling with respect to a fixed point
–
Translate to origin
–
Scale
–
Translate back
T x f , y f 
Tx f , y f  Ss x , s y  T x f , y f =
1 0
0 1

0 0
x f  sx
y f    0
1   0
sx
0

 0
0
sy
0
0
sy
0
0 1 0  x f 
0  0 1  y f  =
1 0 0
1 
x f 1  s x 
y f 1  s y 

1
S s x , s y 
Tx f , y f 
Order of matrix compositions
●
Matrix composition is not commutative. So,
be careful when applying a sequence of
transformations.
pivot
same pivot
Other Transformations
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Reflection
1 0 0


0  1 0 
0 0 1 
  1 0 0


0 1 0 
0 0 1 
  1 0 0


0  1 0 
0 0 1 
●
Shear: Deform the shape like shifted slices.
(0,1) (1,1) (2,1)
(3,1)
x'
x sh x y
1 sh x 0
0 1 0
0 0 1
y'
y
Transformations Between the
Coordinate Systems
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Between different systems: Polar coordinates
to cartesian coordinates
Between two cartesian coordinate systems.
For example, relative coordinates or window
to viewport transformation.
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How to transform from x,y
to x',y' ?
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Superimpose x',y' to x,y
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Transformation:
–
–
y
y0
Translate so that (x0,y0)
moves to (0,0) of x,y
Rotate x' axis onto x axis
x0
x
y
R θ   T  x0,  y0 
x
●
Alternate method for rotation:
Specify a vector V for positive
y' axis:
unit vecto r in the y' direction :
V
v=
= v x ,v y 
V
y
V
unit vecto r in the x' direction, rotate v clockwise 90
u = v y ,vx = u x ,u y 
x
●
Elements of any rotation matrix can be expressed as
elements of a set of orthogonal unit vectors:
u x
R = vx
 0
uy
vy
0
0 v y
0 = vx
1  0
 vx
vy
0
0
0
1
P  P0
v=
P  P0 
y
P
P0
x
●
Example:
P0 = 2,1 P = 3.5,3
1.5,2 =  1.5 , 2  = 0.6,0.8
P  P0
v=
=


P  P0  1.52 + 2 2  2.5 2.5 
u = 0.8,0.6 
M = R u , v   T 2,1 =
0.8  0.6 0  1 0  2 = 0.8  0.6  1 

 
 

0.6
0.8
0
0
1

1
0.6
0.8

2

 
 

0
0
1  0 0 1  0
0
1 
3 4 4
Let triangle T be defined as

1
2
1


three column vectors:
1 1 1 
M  T = 0.8 1 1.6


0.6
2
1.2


1
1 1 
y
P
MT
T
P0
x
Affine Transformations
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Coordinate transformations of the form:
x' = axx x + axy y + bx
y' = a yx x + a yy y + by
●
●
Translation, rotation, scaling, reflection, shear. Any
affine transformation can be expressed as the
combination of these.
Rotation, translation, reflection:
preserve angles, lengths, parallel lines
3 DIMENSIONAL
TRANSFORMATIONS
3D Transformations
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●
●
z
x,y,z coordinates. Usual notation:
Right handed coordinate system
Analogous to 2D we have 4 dimensions
in homogenous coordinates.
x
Basic transformations:
–
Translation
–
Rotation
–
Scaling
y
y
x
z
x
z
y
Translation
●
move the object to a relative position.
 x  1
 y   0
 
 z   0
  
 1  0
0 0 tx   x



1 0 t y   y

0 1 tz   z 
  
0 0 1  1 
y
P
P
x
z
y
P  T  P
x
z
Rotation
●
Rotation arround the coordinate axes
y
y
x
z
x
z
x axis
y
x
z
y axis
Counterclockwise when looking along
the positive half towards origin
z axis
Rotation around coordinate axes
●
●
●
Arround x
Arround y
Arround z
0
1
0 cos 
R x ( )  
0 sin 

0
0
0
 sin 
 cos 
 0
R y ( )  
 sin 

 0
0 sin 
1
0
cos 
 sin 
R z ( )  
 0

 0
cos 
0
0 cos 
0
0
 sin 
cos 
0
0
0
0
0

1
0
0
0

1
0 0
0 0
1 0

0 1
P  R x ( )  P
P  R y ( )  P
P  R z ( )  P
Rotation Arround a Parallel Axis
Rotating the object around a line parallel to one of
the axes: Translate to axis, rotate, translate back.
●
P  T(0, y p , z p )  R x ( )  T(0, y p , z p )  P
y
y
y
y
x
x
z
x
x
z
z
z
Translate
Rotate
Translate back
Figure from the textbook
Rotation Around an Arbitrary Axis
●
●
y
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
x
z
Rotation Around an Arbitrary Axis
Rotation Around an Arbitrary Axis
V  P2  P1  ( x2  x1 , y2  y1 , z2  z1 )
u is the unit vector along V: u 
V
 (a, b, c)
V
First step: Translate P1 to origin:
1
0
T
0

0
0 0  x1 
1 0  y1 
0 1  z1 

0 0 1 
Next step: Align u with the z axis
we 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
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
y
y
α
α
x
x
β
z
u
u'
u
z
uz
u
z
x
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
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
We need cosine and sine of α for rotation
u
u'
α
uz
z
u  (0, b, c)
Projection of u on
yz plane
x
u  u z c
cos  

u u z d
d  b2  c 2
u  u z  u x u u z sin   u xb
b  d sin 
cos  
c
d
sin  
b
d
1

0
R x ( )  
0

0
0
c
d
b
d
0
0
b

d
c
d
0
0

0

0

1
Rotation Around an Arbitrary Axis
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
u  u z
cos  
d
u  u z
β
u  u z  u y u u z sin   u y  (a)
cos   d
d
0
R y ( )  
a

0
x
sin   a
0  a 0
1 0 0
0 d 0

0 0 1
u
u''= (a,0,d)
z
R ( )  T( x1 , y1 , z1 )  R x ( )  R y (  )  R z ( )  R y (  )  R x ( )  T( x1 , y1 , z1 )
Rotation, ... Alternative Method
Any rotation around origin can be represented by
3 orthogonal unit vectors:
 r11 r12
r
r22
21

R
 r31 r32

0 0
r13 0
r23 0
r33 0

0 1
This matrix can be thought of as
rotating 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
Define a new coordinate system (ux , uy , uz )
with the given rotation axis u using:
uz  u
uy 
u ux
u ux
ux  uy  uz
Rotation, ... Alternative Method
uz  u  (a, b, c)
u  u x (a, b, c)  (1,0,0) (0, c,b)


 (0, c / d ,b / d )
2
2
u ux
u ux
b c
ux  uy  uz  (0, c / d ,b / d )  (a, b, c)  (d ,a  b / d ,a  c / d )
uy 

d

R  0

a
0

 a b
d
c
d
b
 ac
d
b
d
c
0
0

0

0

0
1
Check if this is equal to
R y (  )  Rx ( )
Scaling
●
Change the coordinates of the object by scaling
factors.
y
P
sx 0 0 0
P
x'
x
x
0 sy 0 0 y
y'
z'
0 0 sz 0 z
z
1
0 0 0 1 1
y
P' SP
P  S  P
x
z
Scaling with respect to a Fixed Point
Translate to origin, scale, translate back
●
P  T( x f , y f , z f )  S  T( x f , y f , z f )  P
y
x
y
x
z
y
y
x
x
z
z
z
Translate
Scale
Translate back
Scaling with respect to a Fixed Point
1 0
0 1
T( x f , y f , z f )  S  T(  x f ,  y f ,  z f )  
0 0

0 0
1 0 0 x f   s x 0 0  s x x f 
0 1 0 y   0 s

0

s
y
f  
y
y f 


0 0 1 z f   0 0 s z  s z z f 

 

0
0
0
1
0
0
0
1

 

sx
0
 T( x f , y f , z f )  S  T(  x f ,  y f ,  z f )  
0

0
1
0
x f  sx
y f   0


zf 0
 
1  0
0
sy
0
0
0
0
sz
0
0
0
0
sy
0
0
0
0
sz
0
x f (1  s x ) 
y f (1  s y )
z f (1  s z ) 

1

0 1
0 0
0  0

1  0
0 0
1 0
0 1
0 0
xf 
y f 
zf 

1
Reflection
Reflection over planes, lines or points
●
y
y
y
x
z
1
0
0
0
0
0
1
0
0
0
0
1
x
x
z
z
0
1
0
0
y
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
x
z
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
Shear
●
Deform the shape depending on another dimension
SH z
1
0
0
0
0
1
0
0
a
b
1
0
0
0
0
1
x and y value depends on z value of the shape
SH x
1
a
b
0
0
1
0
0
0
0
1
0
0
0
0
1
y and z value depends on x value of the shape