06 Geometric Transformations

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Geometric Transformations
Lecture 6
1107190 - Introdução à Computação Gráfica – Turma 01
Prof. Christian Azambuja Pagot
CI / UFPB
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Transformation Types 
(Perception)
● Scale.
● Shear.
● Rotation.
● Translation.
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Transformation Classification
● Linear transformations.
● Affine transformations.
● Projective transformations.
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Transformation Classification
● Linear transformations.
– Maps straight lines to straight lines.
– Can be expressed in matrix form.
– Examples:
● Scale, rotation and shear.
● Affine transformations.
● Projective transformations.
Linear
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Transformation Classification
● Linear transformations.
● Affine transformations.
– Include all linear transformations.
– Preserves parallelism between lines.
– Examples:
● Scale, rotation, shear and translation.
● Projective transformations.
Affine
Linear
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Transformation Classification
● Linear transformations.
● Affine transformations.
● Projective transformations.
– Include all affine transformations.
– Keep straight lines straight.
– Do not preserve parallelism between lines.
– Examples:
● Scale, roration, shear, translation and homogeneous 
projection.
Projective
Affine
Linear
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Scale
y
x
y
x
x '=x⋅sx
y '= y⋅s y [ x 'y ' ]=[sx 00 s y ][ xy ]
Matrix Form
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Types of Scaling
● Isotropic:
● Anisotropic:
y
x
y
x s x≠s yOriginal
Scaled
y
x
y
x s x=s yOriginal
Scaled
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Mirroring
sx=1
s y=−1
sx=−1
s y=−1
y
x
Original
y
x
y
x
Scaled
Scaled
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Scaling off the Origin
y
x
Original
y
x
Scaled
Implicit
translation
s x=2, s y=2
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Shear
● In 2D:
– Shear keeps coordinate U fixed while changes the 
coordinate V along its axis. The amount of change 
to V is linear with respect to U.
y
x
y
x
Original Shear
x '=x+mx y
y '= y
[ x 'y ' ]=[1 mx0 1 ] [ xy ]
Matrix form
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Shear
● In 2D:
– Can be applied in both directions.
[ x 'y ' ]=[ 1 m xm y 1 ][ xy ]
y
x
y
x
Original Shear
?
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Shear
● Most of the current graphics APIs do not 
implement shear.
● However, shear can be reproduced through 
the combination of a rotation, scaling, and 
another rotation.
y
x
Original
y
x
Scale
y
x
Rotation
y
x
Rotation
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Rotation
y
x
r
w
α
x=r cos(α)
y=r sin (α)
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Rotation
y
x
r
α
w
x=r cos(α)
y=r sin (α)
w'
θ
y'
x'
x '=r cos(α+θ)
y '=r sin (α+θ)
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Rotation
x '=r cos(α)cos (θ)−r sin(α)sin(θ)
y '=r sin (α)cos (θ)+r cos (α)sin(θ)
x '=r cos(α+θ)
y '=r sin (α+θ)
x=r cos(α)
y=r sin (α)
x '=x cos (θ)− y sin(θ)
y '= ycos (θ)+ x sin (θ) [ x 'y ' ]=[cos(θ) −sin (θ)sin(θ) cos (θ) ] [ xy ]
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Rotation “at” and “off” Origin
y
x
Original
y
x
Rotated
y
x
Original
y
x
Rotated
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Translation
y
x
Original
y
x
Translated
x '=x+d x
y '= y+d y
Translation is an affine transformation
and cannot be represented 
by a matrix!
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Homogeneous Coordinates (HC)
● In HC, n-dimensional vectors are represented with n+1 
components: 
– Example: (x,y) → (xw, yw, w), where w is the homogeneous coordinate.
● To transform from HC to Euclidean space, divide coordinates by 
the homogeneous coordinate:
– Example: (xw, yw, w) → (xw/w, yw/w)
● Allows the representation of translation by means of a matrix.
● The use of HC gives rise to a Projective Space.
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Translation in HC
(x , y)
Euclidean
space
(xw , yw , w)
Homogeneous
space w=1
(x , y ,1)
Homogeneous
space
x '=x+dx
y '= y+d y
1=1 [
x '
y '
1 ]=[1 0 d x0 1 d y0 0 1 ] [ xy1 ]
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All Other Transformations can be 
Rewritten in Terms of HC
[ x 'y '1 ]=[sx 0 00 s y 00 0 1 ][ xy1 ]
[ x 'y '1 ]=[1 mx 00 1 00 0 1] [ xy1 ]
[ x 'y '1 ]=[cos(θ) −sin (θ) 0sin(θ) cos (θ) 00 0 1] [ xy1 ]
[ x 'y ' ]=[sx 00 s y ][ xy ]
Scale
[ x 'y ' ]=[1 mx0 1 ] [ xy ]
Shear along X axis.
[ x 'y ' ]=[cos(θ) −sin (θ)sin(θ) cos (θ) ] [ xy ]
Rotation
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How about Transformations in 
3D Space?
● 3D transformations can also be represented 
through matrices.
● As in the 2D case, 3D transformations are 
expressed with the help of HC to allow for the 
representation of translation in matrix form.
● 3D vectors in Euclidean space are 
represented through 4-element vectors in HC:
– Example: (x, y, z) → (xw, yw, zw, w)
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3D Scale
● The scale can be computed along x, y and z 
axis.
x '=x⋅sx
y '= y⋅s y
z '= z⋅s z
Matrix Form in HC
[ x 'y 'z '1 ]=[
s x 0 0 0
0 s y 0 0
0 0 sz 0
0 0 0 1
] [ xyz1 ]
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3D Shear
● Extra possibilities due the z axis.
● The example below changes x and y by 
amounts proportional to z:
x '=x+mx z
y '= y+m y z
z '= z [ x 'y 'z '1 ]=[
1 0 mx 0
0 1 m y 0
0 0 1 0
0 0 0 1][
x
y
z
1 ]
Matrix Form in HC
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3D Rotation
● Rotation is always ...
– … around the coordinate system origin.
– … about an axis.
● There are three possibilities for a rotation in 
3D space:
– About X, Y, or Z axis.
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3D Rotation about X
x
y
z
x '=x
y '= y cos (θ)− zsin (θ)
z '= y sin (θ)+ zcos (θ) [ x 'y 'z '1 ]=[
1 0 0 0
0 cos (θ) −sin(θ) 0
0 sin (θ) cos(θ) 0
0 0 0 1 ] [
x
y
z
1 ]
Matrix Form in HC
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3D Rotation about Y
x
y
z
x '=x cos (θ)+ zsin (θ)
y '= y
z '=−x sin(θ)+z cos (θ) [ x 'y 'z '1 ]=[
cos (θ) 0 sin (θ) 0
0 1 0 0
−sin(θ) 0 cos (θ) 0
0 0 0 1 ] [
x
y
z
1 ]
Matrix Form in HC
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3D Rotation about Z
x
y
z
x '=x cos (θ)− y sin(θ)
y '=x sin (θ)+ y cos(θ)
z '= z [ x 'y 'z '1 ]=[
cos (θ) −sin(θ) 0 0
sin (θ) cos(θ) 0 0
0 0 1 0
0 0 0 1 ] [
x
y
z
1 ]
Matrix Form in HC
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Gimbal Lock
● Loss of degree of freedom in rotations in 3D 
space due the parallelism between two of the 
three gimbal.
3 DOFs
Images by MathsPoetry (Wikipedia)
2 DOFs due Gimbal lock
(Green and blue gimbals aligned)
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Gimbal Lock
● Animated Gimbal lock
Gimbal lock (video)
Video by GuerrillaCG (YouTube)
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How to Solve Gimbal Lock?
● Two common approaches:
– Change the precedence order of the rotations in 
advance.
– Use quaternions, which is a Gimbal lock-free 
mathematical rotation representation.
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3D Translation
● An additional dimension creates one additional 
DOF regarding translation:
x '=x+dx
y '= y+d y
z '= z+dz [
x '
y '
z '
1 ]=[
1 0 0 dx
0 1 0 d y
0 0 1 dz
0 0 0 1
][ xyz1 ]
Matrix Form in HC
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Composing Transformations
● The representation of geometric 
transformations with matrices allows for the 
composition of transformations through 
matrix product!Universidade Federal da Paraíba
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Composing Transformations
● Example:
– Suppose that one wishes to apply a scale S and then a rotation 
R on a 3D vertex v
1
, generating, as a result, the transformed 
vertex v
2
:
● Approach 1:
– vtmp = Sv1
– v2 = Rvtmp
● Approach 2:
– Matrix product is associative:
● v
2
 = R(Sv
1
) → v
2
 = (RS)v
1
– Thus, we can say that M = RS
● v
2
 = Mv
1
Be aware that the order 
of the product matters!
M will first apply the scaling S and,
after that, the rotation R!
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Composing Transformations
● Example:
– Supposing a translation matrix T and a rotation 
matrix R:
● M
1
 = TR (applies rotation first, and then translation.)
● M
2
 = RT (applies translation first, and then rotation.)
M
1
M
2y
x
Original
y
x
Original
y
x
Transformed
y
x
Transformed
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Transformations in OpenGL
● Transformation functions in OpenGL:
– glRotatef(<degrees>, <x>, <y>, <z>)
– glScalef(sx, sy, sz)
– glTranslatef(dx, dy, dz)
● Those transformations will affect the current 
active matrix!
● OBS: The last transformation called is the 
first to be applied!
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Transformations in OpenGL
● Code example:
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glTranslatef(0.0f, 1.0f, 2.0f);
glRotatef(45.0f, 0, 1, 0);
glScalef(0.2f, 4.0f, ­3.0f);
glutSolidSphere(1.0f, 20, 20);
Set active 
matrix
Reset the 
active matrix
3rd. transformation
to be applied
2nd. transformation
to be applied
1st. transformation
to be applied
Draws a sphere
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