Part III Geometric Transformations

1
Part III Geometric Transformations
2
Vector and Dot Product

• Vector addition
• or (a b)(c d)(ac bd)
• We will use column vectors and row vectors
  interchangeably.
• Dot product v1v2 (x1 y1)(x2 y2) x1x2
  y1y2
• Vector length
• 2nd def. of dot product v1v2 v1v2cos?
• When v1 1, v1v2 is the length of the
  projection of v2 onto v1 .

v2
v1
v2cos?
3
Vector and Dot Product (contd)

• Vector normalization makes a unit vector (of
  length 1) v/v
• e.g. (3 4) ? (3/5 4/5)
• If vivj is 0, vi vj are orthogonal/perpendicul
  ar.
• If vivj is positive, vi is in the same
  half-plane defined by vj.
• Equivalently, vj is in the same half-plane
  defined by vi.
• If vivj is negative, vi vj are in different
  half-planes defined by vj (or vi).
• ? If vi is a unit vector, the dot product of vi
  vi are 1.
• e.g.
• ? So far 2D, but the same for 3D
• e.g. v1v2(x1 y1 z1)(x2 y2 z2)x1x2y1y2z1 z2

y
? (0 2) (5 0) 05200 ? (2 2) (5 0)
252010 gt 0 ? (-3 2) (5 0) -3520-15 lt
0
(0 2)
(2 2)
(-3 2)
(5 0)
x
4
An Example Use of Dot Product

• Lets derive the shortest distance from a point
  (p q r) to a plane axbyczd0.
• A plane P is typically defined by its normal (a b
  c) and a point (l m n) on P.
• Given a normal (a b c), a plane is set to
  axbyczd0, where d is unknown.
• Then, for axbyczd0, albmcnd0 and
  therefore d -al-bm-cn.
• Normalize (a b c)
• Connect (p q r) and (l m n) to make a vector
  C(p-l q-m r-n).
• Note that NC is the length of the projection of
  C onto N because N 1, and therefore NC is
  the distance from a point (p q r) to the plane.

(p q r)
C
(a b c)
(l m n)
axbyczd0
5
Cross Product

• ? v1 x v2 ( y1z2 - z1y2 z1x2 - x1z2
  x1y2 - y1x2 )
• ? v1 x v2 u v1v2 sin? where u is a unit
  vector perpendicular to both v1 v2 based on the
  right hand rule.

?
?
?

v2
u
v1
6
Coordinate System

• A set of vectors vis is linearly independent if
  a1v1a2v2 .. anvn0 (zero vector) only when
  every ai0.
• Any set of n linearly independent vectors in an
  n-dimensional vector space V is a basis.
• A basis v1, v2, .. vn is an orthonormal basis
  if vivj 0 if i? j and vivj 1 if ij.
• An origin and a basis constitute a frame or a
  coordinate system.

7
Matrix

• column and row
• identity matrix

• transpose
• matrix composition
• M-1 is an inverse matrix of M if MM-1I.

3x2
2x3
3x3
8
Translation

• A transformation is a function that maps a
  point/vector into another point/vector.
• For now, we will discuss the basic 4 types of
  transformations
• translation, scaling, rotation, and shearing.
• Translation

( x?, y?)
dy
(x, y)
dx
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Scaling

• Scaling is done about the origin.
• Scale x by sx along x-axis x? sxx Scale y
  by sy along y-axis y? syy
• The above example is called a uniform scaling
  where sx sy.
• If sx ? sy, non-uniform scaling.
• If sx or sy lt 1, objects are moved closer to the
  origin.
• If sx or sy gt 1, objects are moved farther from
  the origin.

10
Rotation

• Rotation by an angle ? about the origin.
• The distance from P to the origin is r.
• Angles are measured counter-clockwise (CCW).
• x r cos?
• y r sin?
• x? r cos(??) r cos? cos? - r sin? sin?
  xcos? - ysin?
• y? r sin(??) r cos? sin? r sin? cos?
  xsin? ycos?
• For CW, use cos(-?)cos? and sin(-?)-sin?.

P? (x?, y?)
r
r
P(x, y)
r
11
Shearing

• Shearing along x
• Used for italic font design (whereas scaling is
  for font size).

y? y (unaffected) x? x ay (in the example,
a ? 0.5)

• Similarly, shearing along y

y
x
12
Affine Transformation

• A mapping T V?V (V is a vector space) is a
  linear transformation if, for all scalars a and
  all vectors x,y, it satisfies the following
  conditions
• T(ax)aT(x)
• T(xy)T(x)T(y)
• Linear transformations include identity
  mapping(I), rotation, scaling, shearing,
  reflection, etc.
• A translation in a vector space V is a mapping
  that associates to each vector x of V another
  vector yx? where ? is a fixed vector.
• Translations are not linear transformations.
• A transformation T in a vector space V is affine
  if there exists a linear transformation T' such
  that, for all x,y of V, T(x)-T(y)T'(x-y).
• Both translations linear transformations are
  affine transformations.
• Affine transformations preserve lines
  parallelism, but not necessarily their lengths
  and angles, and are represented by the 3x4
  matrix. Compare affine transformations with
  (perspective) projection transformation. (later!)

13
Homogeneous Coordinates

• Observe that rotation, scaling shearing are
  done by matrix composition while translation is
  by vector addition. Can we treat them
  uniformly?
• YES!! Use Homogeneous Coordinates.
• Practical motivations for Homogeneous
  Coordinates.
• We can treat linear transformations (rotation,
  scaling, shearing, etc.) translation uniformly.
• We can deal with projections (later!).
• A point (x, y) in Cartesian Coordinates is
  represented as (wx, wy, w) in Homogeneous
  Coordinates where w can be any non-zero value.
• e.g. (2 3) ? (2 3 1) (4 6 2) (5 7.5
  2.5) ... but, typically w1.
• For a vector, w0.
• e.g. (2 3 0) is a vector whereas (2 3 1) is
  a point.
• e.g. (1 3 0) is a vector connecting two points
  (4 5 1) and (5 8 1).

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Transformations in Homogeneous Coordinates

• translation (x? xdx, y? ydy)

• scaling

• rotation

• shearing (along x)

• Note that all transformations are done by matrix
  composition thanks to homogeneous coordinates.
• Transformations which do not alter a bodys shape
  are called rigid-body transformations and consist
  of rotations translations.

15
Rotation about An Arbitrary Point

• How to rotate about an arbitrary point p(x, y),
  not about the origin?

q?(a?,b?)
q?
?
q???
q???
q(a,b)
q
?
q??
q??
p
p (x,y)
p?O
p?O
translate q by -p rotate by ?
back-translate
3x3 3x3
3x3
translate translate rotate translate, rotate
back-translate

• Thanks to Homogeneous Coordinates, three
  transformations can be combined into a single 3x3
  matrix.

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Inverse Transformation

• Note that the first and last translations are
  inverses of each other.
• Such an inverse transformation is represented as
  an inverse matrix, but can be obtained pretty
  easily if its geometry is considered.
• Using your knowledge in linear algebra, show that
  the above holds.

q?
q???
q???
q
?
q??
q??
p
p?O
p?O
translate q by -p rotate by ?
back-translate
inverse transformation
17
Associativity Commutativity

• Matrix multiplication is associative.
• So is transformation composition.
• ABC (AB)C A(BC)
• In contrast, matrix multiplication is not
  commutative in general.
• Neither is transformation composition.
• e.g. combination of rotation (by 90CCW)
  translation along y
• rotation first translation first

18
3D Translation and Scaling

• First of all, Homogeneous Coordinate for 3D
• (x y z) ? (wx wy wz w) e.g (x y z 1)
• Translation
• Scaling

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3D Rotation

• For 3D rotation, we need an axis of rotation.
• Also, by convention, positive angles are for CCW.
• Recall that, in 2D, x? xcos? - ysin? and y?
  xsin? ycos?.
• Add z-axis and make z? z. Then, its 3D
  rotation about the z-axis.

y
(x,y)
(x?, y?)
x
y
(x,y,0)
(x?, y?,0)
x
z
Rz(?)
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3D Rotation (contd)

• Rotation about x-axis
• Note that the coordinate system follows the right
  hand rule.
• Lets do cyclic permutation of the 3 axes, where
  the right hand rule holds.
• Note that rotation about x-axis is the same as
  rotation about the old z-axis except the
  permutation.
• For rotation about y-axis, do one more
  permutation!
• 3D shearing will be discussed when we deal with
  viewing.

x?y?z?x
y
z
y'
y
z'
z
x'
x
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3D Rotation about An Arbitrary Rotation Axis
y
by ?
x
z
Step1 Translate such that the rotation axis
passes through the origin.
Step2 Rotate such that the rotation axis
coincides with a coordinate axis.
Step3 Rotate by ?
Step4 Inverse rotation. (Inverse of Step2)
Step5 Inverse translation. (Inverse of Step1)
22
3D Rotation Step1

• Rotation axis is typically defined by 2 end
  points p1 and p2 (or a point p and a vector v) ?
  Move p1(x1 y1 z1) to the origin.

23
3D Rotation Step2

• Why Step2? Because available rotations are all
  about coordinate axes.
• However, we also cannot directly perform Step2.
  So, divide Step2 into two substeps Step2-1 and
  Step2-2.
• Consider the unit vector u along the rotation
  axis.

Step2-1 Rotation about x onto xz-plane
Step2-2 Rotation about y onto yz-plane
p2?(x2? y2? z2?)
p2 (x2 y2 z2)
unit vector u
p1?(x1? y1? z1?)
p1 (x1 y1 z1)
24
3D Rotation Step2-1

• In Step2-1, how much angle to rotate? It is
  convenient if we consider the projection of u
  onto yz-plane, u?.
• Rotation about x is represented by
• Note that a itself is not important but we need
  sina cosa.
• How can we compute sina cosa?

u(a b c)
u? (0 b c)
uz (0 0 1)
25
3D Rotation Step2-1 (contd)

• Recall the definitions for dot product and cross
  product.

26
3D Rotation Step2-2

• Rotation about y
• Rotation about y is represented by
• Note that ? itself is not important but we need
  sin? cos?.
• How to compute sin? cos?? Same as in Step2-1!

? Rotation about x-axis leaves us x-component
unchanged, i.e. a. ? u?? is on xz-plane, and so
y-component is 0. ? u?? should be a unit vector,
and so z-component is .
?
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3D Rotation Step2-2 (contd)

• Recall the definitions for dot product and cross
  product.
• u??? uz (a 0 d) ? (0 0 1) d
• u??uzcos? cos? ? d cos?
• u?? x uz (a 0 d) x (0 0 1) (0 -a 0)
• uy u??uzsin? (0 sin? 0)
  ? -a sin?

?
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3D Rotation Steps 3, 4 and 5

• After Step1 and Step2, we are in the following
  situation.
• So, Step3 is simply
• Step4 an inverse of the matrix computed at Step2
• Step5 an inverse of the matrix computed at Step1
• R(?) is a single 4x4 matrix. All done!!!!

29
Orthogonal Matrix

• A lot of matrix multiplications in R(?) lead to
• Inefficiency
• error accumulation
• There is a less intuitive but better method -
  Use orthogonal matrix.
• 2 columns vi vj are orthonormal if
• i.e. ? each column is a unit vector, and
• ? any two different columns are
  orthogonal.
• Orthogonal matrix a matrix with orthonormal
  columns.
• e.g.

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Orthogonal Matrix (contd)

• A matrix Q is orthogonal ? QTQ I (identity
  matrix)
• In other words, QT Q-1, i.e. for orthogonal
  matrices, the transpose is the inverse.
• Then, QQT I
• ? (QT)TQT I
• ? QT is an orthogonal matrix if Q is an
  orthogonal matrix.

0
...
0
0
1
0
...
0
1
0
...

0
...
1
0
0
...
...
...
...
...
...
1
0
0
0
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Orthogonal Matrix (contd)

• We can decompose QTQ I in the following way.
• An orthogonal matrix transforms its rows into
  coordinate axes.
• Rotation is a special orthogonal matrix. Lets
  see.

0
...
0
0
1
orthogonal matrix
0
...
0
1
0
...

0
...
1
0
0
...
...
...
...
...
...
1
0
0
0
nxn nxn
nxn
0
1
0
0
0
1




0
0
0
...
...
...
...
...
...
1
0
0
nxn nx1 nx1 nxn
nx1 nx1 nxn
nx1 nx1
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Orthogonal Matrix Rotation

• Suppose that you have the following two
  orthonormal vectors.
• Make an orthogonal matrix M with v1 and v2 as the
  1st and 2nd rows respectively.
• Then, M transforms v1 and v2 into ux and uy,
  respectively.
• Rotation is a special orthogonal matrix whose
  determinant is 1. For example, an orthogonal
  matrix with v2 and v1 as the 1st and 2nd rows is
  not a rotation. Every rotation is an orthogonal
  matrix.
• In summary, we found that
• a rotation can be generated as an orthogonal
  matrix, and
• for a rotation, its inverse is the transpose.

v1
v2
45º
v2
v1
M
M
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3D Orthogonal Matrix
uy

• Same in 3D
• R transforms v1, v2 and v3 into ux, uy and uz,
  respectively.

v1
v2
an orthogonal matrix
v3
ux
34
Application of Orthogonal Matrix

• Recall that, after Step1, our goal was to rotate
  u into uz.
• For this purpose, lets make an orthogonal matrix
  whose 3rd row is u.

z
35
Application of Orthogonal Matrix (contd)
y
u v3
uy
v2
x
ux
v1
uz
z

• Recall that R(?)T-1Rx-1(?)Ry-1(?)Rz(?)Ry(?)Rx(?)T
  
• The orthogonal matrix R replaces Ry(?)Rx(?)!
• Maybe more importantly, RT replaces
  Rx-1(?)Ry-1(?)?
• This alleviates error accumulation makes the
  overall transformation efficient.

36
Structure-deforming Transformations

• At the opposite end of rigid-body transformations
  are structure-deforming transformations such as
  tapering.

• Note that all y- z-coordinates of the object
  are unchanged.
• However, x-coordinates are scaled(reduced) along
  the z-axis.