Advanced Computer Graphics: 2D/3D Transformations

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Advanced Computer Graphics2D/3D Transformations

• Kocaeli Universitesity
• Computer Engineering Department

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Geometric Transformation

• What is geometric transformation?
• Operations that are applied to the geometric
  description of an object to change its position,
  orientation, or size are called geometric
  transformations

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2D Transformations
y
y
x
x
y
x
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2D Transformations
y
y
x

• Applications
• Animation
• Image/object manipulation
• Viewing transformation
• etc.

x
y
x
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2D Transformation

• Given a 2D object, transformation is to change
  the objects
• Position (translation)
• Size (scaling)
• Orientation (rotation)
• Shapes (shear)
• Apply a sequence of matrix multiplications to the
  object vertices

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Point Representation

• We can use a column vector (a 2x1 matrix) to
  represent a 2D point x
• 
  y
• A general form of linear transformation can be
  written as
• x ax by c
• OR
• y dx ey f

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Translation

• Re-position a point along a straight line
• Given a point (x,y), and the translation distance
  (tx,ty)

The new point (x, y) x x tx
y y ty
ty
tx
OR P P T where P x p
x T tx
y y
ty
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3x3 2D Translation Matrix
Use 3 x 1 vector

• Note that now it becomes a matrix-vector
  multiplication

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Translation

• How to translate an object with multiple
  vertices?

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2D Rotation

• Default rotation center Origin (0,0)

• gt 0 Rotate counter clockwise

q

• lt 0 Rotate clockwise

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2D Rotation
(x,y) -gt Rotate about the origin by q
r
How to compute (x, y) ?
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2D Rotation
(x,y) -gt Rotate about the origin by q
r
How to compute (x, y) ?
x r cos (f) y r sin (f)
x r cos (f q) y r sin (f q)
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2D Rotation
x r cos (f) y r sin (f)
x r cos (f q) y r sin (f q)
r
x r cos (f q) r cos(f) cos(q)
r sin(f) sin(q)
x cos(q) y sin(q)
y r sin (f q) r sin(f) cos(q) r
cos(f)sin(q)
y cos(q) x sin(q)
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2D Rotation
x x cos(q) y sin(q)
y y cos(q) x sin(q)
r
Matrix form?
3 x 3 How?
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3x3 2D Rotation Matrix
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2D Rotation

• How to rotate an object with multiple vertices?

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2D Scaling
Scale Alter the size of an object by a scaling
factor (Sx, Sy), i.e.
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2D Scaling

• Not only the object size is changed, it also
  moved!!
• Usually this is an undesirable effect
• We will discuss later (soon) how to fix it

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3x3 2D Scaling Matrix
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Scaling facts
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Put it all together

• Translation
• Rotation
• Scaling

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Or, 3x3 Matrix Representations

• Translation
• Rotation
• Scaling

x cos(q) -sin(q) 0 x y
sin(q) cos(q) 0 y 1
0 0 1 1

x Sx 0 0 x y
0 Sy 0 y 1 0
0 1 1
Why use 3x3 matrices?
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Why Use 3x3 Matrices?

• So that we can perform all transformations using
  matrix/vector multiplications
• This allows us to pre-multiply all the matrices
  together
• The point (x,y) needs to be represented as
• (x,y,1) -gt this is called Homogeneous
• coordinates!
• How to represent a vector (vx,vy)?

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Why Use 3x3 Matrices?

• So that we can perform all transformations using
  matrix/vector multiplications
• This allows us to pre-multiply all the matrices
  together
• The point (x,y) needs to be represented as
• (x,y,1) -gt this is called Homogeneous
• coordinates!

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Shearing

• Y coordinates are unaffected, but x coordinates
  are translated linearly with y
• That is
• y y
• x x y h

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Shearing in Y
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Reflection
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Reflection
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Reflection
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Reflection about X-axis
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Reflection about X-axis
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Reflection about Y-axis
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Reflection about Y-axis
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Whats the Transformation Matrix?
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Whats the Transformation Matrix?
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More about reflection?

• Reflection about yx
• Reflection about y-x

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Rotation Revisit

• The standard rotation matrix is used to rotate
  about the origin (0,0)

cos(q) -sin(q) 0 sin(q)
cos(q) 0 0 0 1
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Arbitrary Rotation Center

• To rotate about an arbitrary point P (px,py) by
  q

(px,py)
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Arbitrary Rotation Center

• To rotate about an arbitrary point P (px,py) by
  q
• Translate the object so that P will coincide with
  the origin T(-px, -py)

(px,py)
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Arbitrary Rotation Center

• To rotate about an arbitrary point P (px,py) by
  q
• Translate the object so that P will coincide with
  the origin T(-px, -py)
• Rotate the object R(q)

(px,py)
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Arbitrary Rotation Center

• To rotate about an arbitrary point P (px,py) by
  q
• Translate the object so that P will coincide with
  the origin T(-px, -py)
• Rotate the object R(q)
• Translate the object back T(px,py)

(px,py)
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Arbitrary Rotation Center

• Translate the object so that P will coincide with
  the origin T(-px, -py)
• Rotate the object R(q)
• Translate the object back T(px,py)
• Put in matrix form T(px,py) R(q) T(-px, -py)
  P

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Scaling Revisit

• The standard scaling matrix will only anchor at
  (0,0)

Sx 0 0 0 Sy 0
0 0 1
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Arbitrary Scaling Pivot

• To scale about an arbitrary fixed point P
  (px,py)

(px,py)
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Arbitrary Scaling Pivot

• To scale about an arbitrary fixed point P
  (px,py)
• Translate the object so that P will coincide with
  the origin T(-px, -py)

(px,py)
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Arbitrary Scaling Pivot

• To scale about an arbitrary fixed point P
  (px,py)
• Translate the object so that P will coincide with
  the origin T(-px, -py)
• Scale the object S(sx, sy)

(px,py)
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Arbitrary Scaling Pivot

• To scale about an arbitrary fixed point P
  (px,py)
• Translate the object so that P will coincide with
  the origin T(-px, -py)
• Scale the object S(sx, sy)
• Translate the object back T(px,py)

(px,py)
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Affine Transformation

• Translation, Scaling, Rotation, Shearing are all
  affine transformation

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Affine Transformation

• Translation, Scaling, Rotation, Shearing are all
  affine transformation
• Affine transformation transformed point P
  (x,y) is a linear combination of the original
  point P (x,y), i.e.

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Affine Transformation

• Translation, Scaling, Rotation, Shearing are all
  affine transformation
• Affine transformation transformed point P
  (x,y) is a linear combination of the original
  point P (x,y), i.e.
• Any 2D affine transformation can be decomposed
  into a rotation, followed by a scaling, followed
  by a shearing, and followed by a translation.
• Affine matrix translation x shearing x
  scaling x rotation

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Composing Transformation

• Composing Transformation the process of
  applying several transformation in succession to
  form one overall transformation
• If we apply transforming a point P using M1
  matrix first, and then transforming using M2, and
  then M3, then we have
• (M3 x (M2 x (M1 x P )))

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Composing Transformation

• Composing Transformation the process of
  applying several transformation in succession to
  form one overall transformation
• If we apply transforming a point P using M1
  matrix first, and then transforming using M2, and
  then M3, then we have
• (M3 x (M2 x (M1 x P ))) M3 x M2 x
  M1 x P

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Composing Transformation

• Matrix multiplication is associative
• M3 x M2 x M1 (M3 x M2) x M1 M3 x (M2 x
  M1)
• Transformation products may not be commutative A
  x B ! B x A
• Some cases where A x B B x A
• A
  B
• translation
  translation
• scaling
  scaling
• rotation
  rotation
• uniform scaling rotation
• (sx sy)
• translation rotation?

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Example1 General Two-Dimensional Pivot-Point
Rotation

• So we can generate a 2D rotation about any other
  pivot point (x, y) by performing the following
  sequence of translate-rotate-translate operations

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Example1 (Cont)

1. Translate the object so that the pivot-point
   position is moved to the coordinate origin
2. Rotate the object about the coordinate origin
3. Translate the object so that the pivot point is
   returned to its original position

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Example2 Scaling without translation

1. Translate the object to so that the fixed point
   coincides with the coordinate origin
2. Scale the object with respect to the coordinate
   origin
3. Use the inverse of the translation in step (1) to
   return the object to its original position

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Example2 (Cont)
Translate
Scale
Translate
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Three-Dimensional Transformations
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Extending From 2d Approach

• Methods for geometric transformations in three
  dimensions are extended from two-dimensional
  methods by including considerations for the z
  coordinate.
• A three-dimensional position, expressed in
  homogeneous coordinates, is represented as a
  four-element column vector. Thus , each geometric
  transformation operator is now 4 by 4 matrix.

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Translation
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Rotation z-axis rotation

• 3D Coordinate-Axis Rotations
• z-axis rotation (counter-clockwise)

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Rotation x-axis rotation

• counter-clockwise

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Rotation y-axis rotation

• counter-clockwise

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Rotating about an axis that is parallel to one of
the coordinates axes

• Translate the object so that the rotation axis
  coincides with the parallel coordinate axis
• Perform the specified rotation about that axis
• Translate the object so that rotation axis is
  moved back to its original
• A coordinate position P is transformed with the
  sequence

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Rotating about an axis that is not parallel to
one of the coordinate axes

• In this case, we also need rotation to align the
  rotation axis with a selected coordinate axis and
  then to bring the rotation axis back to its
  original orientation
• A rotation axis can be defined with two
  coordinate position, or one position and
  direction angles.
• Now we assume that the rotation axis is defined
  by two points, and that the direction of rotation
  is to be counter clockwise when looking along the
  axis from p2 to p1.

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• The components of the rotation axis vector are
  then computed as
• And the unit rotation-axis vector u is
• Where

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Rotating about an axis that is not parallel to
one of the coordinate axes

• Translate the object so that the rotation axis
  passes through the coordinate origin
• Rotate the object so that the axis of rotation
  coincides with one of the coordinate axes
• Perform the specified rotation about the selected
  coordinate axis
• Apply inverse rotations to bring the rotation
  axis back to its original orientation
• Apply the inverse translation to bring the
  rotation axis to its original spatial position.
• we can transform the rotation axis onto any one
  of the three coordinate axes. But the z axis is
  often a convenient choice

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• The first step in the rotation sequence is to set
  up the translation matrix that repositions the
  rotation axis so that it passes through the
  coordinate origin. We move p1 to the origin.

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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
u
u'
u
a
a
x
x
x
uz
u
ß
z
z
z
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• Since rotation calculations involve sine and
  cosine functions, so we can use standard vector
  operation to obtain elements of the two rotation
  matrices. A vector dot product can be used to
  determine the cosine term, and a vector cross
  product can be used to calculate the sine term.
• Firstly, we establish the transformation matrix
  for rotation around the x axis by determining the
  values for the sine and cosine of the rotation
  angle necessary to get u into the x-z plane.

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• We need the angle between vector u and x-z
  plane .
• We directly calculate cosine and sine values of
  .
• If we represent the projection of u in the zy
  plane as the vector
• Then the cosine of the rotation angle can be
  determined from the dot product of and then
  
• unit vector along the z axis.

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• Similarly, we can determine the sine of
  from the cross product of
• So

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• So the matrix elements for rotation of this
  vector about the x axis and into the xz plane

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• The rotation matrix about y axis is

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• The third step we have aligned the rotation axis
  with the positive z axis. The specified rotation
  angle we can now be applied as a rotation
  about the z axis

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• So the transformation matrix for rotation about
  an arbitrary axis can then be expressed as the
  composition of these seven individual
  transformations

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Scaling

• Change the coordinates of the object by scaling
  factors.

y
x
z
y
x
z
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Scaling with respect to a Fixed Point

• Translate to origin, scale, translate back

x
y
y
y
y
x
x
x
z
z
z
z
Translate
Scale
Translate back
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Scaling with respect to a Fixed Point
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Reflection

• Reflection over planes, lines or points

y
y
y
y
x
x
x
x
z
z
z
z
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Shear

• Deform the shape depending on another dimension

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BACKUP
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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
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Rotation Around an Arbitrary Axis
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Rotation Around an Arbitrary Axis
u is the unit vector along V
First step Translate P1 to origin
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.
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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
u
u'
u
a
a
x
x
x
uz
u
ß
z
z
z
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Dot product and Cross Product

• v dot u vx ux vy uy vz uz. That
  equals also to vucos(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 vusin(a), that is
  an area of parallelogram built on them. If v and
  u are parallel then the product is the null
  vector.

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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 a for rotation
u
u'
a
x
uz
z
Projection of u on yz plane
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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
ß
x
u'' (a,0,d)
z
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Rotation, ... Alternative Method
Any rotation around origin can be represented by
3 orthogonal unit vectors
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 with the given
rotation axis u using
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Rotation, ... Alternative Method
Check if this is equal to