Review:%20Transformations

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ReviewTransformations
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Transformations

• Modeling transformations
• build complex models by positioning
  (transforming) simple components relative to each
  other
• Viewing transformations
• placing virtual camera in the world
• transformation from world coordinates to camera
  coordinates
• Perspective projection of 3D coordinates to 2D
• Animation
• vary transformations over time to create motion

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Transformations - Modeling
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Transformations - Viewing
CAMERA
OBJECT
WORLD
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Modeling Transformations

• Transform objects/points

Transform coordinate system
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Affine Transformations

• Transform P (x, y, z) to Q (x, y, z) by M.
• Affine transformation

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Translation

• Translation by (tx, ty, tz)

(tx, ty, tz)
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Scaling

• Scaling by (sx, sy, sz)

Uniform v. non-uniform scaling
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Rotation

• Rotation counter-clockwise by angle ? around the
  z-axis
• x x cos(?) y sin(?)
• y x sin(?) y cos(?)
• z z

Proof x r cos(?) y r sin(?) x r cos(?
?) r cos(?) cos(?) r sin(?) sin(?) x cos(?)
y sin(?) y r sin(? ?) r cos(?) sin(?)
r sin(?) cos(?) x sin(?) y cos(?)
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Rotation around x-axis

• Rotation counter-clockwise by angle ? around the
  x-axis
• x x
• y y cos(?) z sin(?)
• z y sin(?) z cos(?)

z
?
y
x
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Rotation around y-axis

• Rotation counter-clockwise by angle ? around the
  y-axis
• y y
• z z cos(?) x sin(?)
• x z sin(?) x cos(?)
• Or
• x x cos(?) z sin(?)
• y y
• z x sin(?) z cos(?)

x
?
z
y
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Matrix Multiplication
Scaling by (sx, sy, sz) x sx x y sy y z
sz z

• Rotation counter-clockwise by angle ? around the
  z-axis
• x x cos(?) y sin(?)
• y x sin(?) y cos(?)
• z z

Translation by (tx, ty, tz) x x tx y y
ty z z tz
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Homogeneous Coordinates
Represent P by (x, y, z, 1) and Q by (x, y, z,
1). (Homogeneous coordinates.)
Translation by (tx, ty, tz) x x tx y y
ty z z tz

• Scaling by (sx, sy, sz)
• x sx x
• y sy y
• z sz z

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Rotation Matrices

• Rotation counter-clockwise by angle ? around the
  x-axis
• x x
• y y cos(?) z sin(?)
• z y sin(?) z cos(?)

Rotation counter-clockwise by angle ? around the
y-axis x x cos(?) z sin(?) y y z x
sin(?) z cos(?)
Rotation counter-clockwise by angle ? around the
z-axis x x cos(?) y sin(?) y x sin(?)
y cos(?) z z
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Affine Transformation Matrix

• Affine transformation
• x m11 x m12 y m13 z m14
• y m21 x m22 y m23 z m24
• z m31 x m32 y m33 z m34
• Transformation Matrix

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Shearing

• Shear along the x-axis
• x x hy
• y y
• z z

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Reflection

• Reflection across the x-axis
• x (-1) x
• y y
• z z

Reflection is a special case of scaling!
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Elementary Transformations

• Translation
• Rotation
• Scaling
• Shear

What about inverses?
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Inverse Transformations
Translation
Scale
Rotation
Shear
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Compose Transformations
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Compose Transformation Matrices
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Compose Transformation Matrices

• Scale by (2, 1, 1)

• Translate by (20, 5, 0)

• Rotate by 30 counter-clockwise around the
  z-axis

• Translate by (0, -50, 0).

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Order Matters!

• Transformations are not necessarily commutative!

• Translate by (20, 0, 0).
• Rotate by 30?.

• Rotate by 30?.
• Translate by (20, 0, 0).

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Order of Transformation Matrices

• Apply transformation matrices from right to left.

• Translate by (20, 0, 0).
• Rotate by 30?.

• Rotate by 30?.
• Translate by (20, 0, 0).

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Which transformations commute?

• Translation and translation?
• Scaling and scaling?
• Rotation and rotation?
• Translation and scaling?
• Translation and rotation?
• Scaling and rotation?

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Elementary Transformations

• Elementary transformations
• Translation
• Rotation
• Scaling
• Shear.

• Theorem Every affine transformation can be
  decomposed into elementary operations.

Eulers Theorem Every rotation around the origin
can be decomposed into a rotation around the
x-axis followed by a rotation around the y-axis
followed by a rotation around the z-axis. (or a
single rotation about an arbitrary axis).
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Vectors

• Vector V (Vx, Vy, Vz).
• Homogeneous coordinates (Vx, Vy, Vz, 0).
• Affine transformation

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Vectors

• Rotation
• Scaling
• Translation
• (Note No change.)

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Transform Parametric Line
V
P(u)
u
P0
M
M
u
W
Q0
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Line
transformed point on a line point on
transformed line
Theorem Affine transformations transform lines
to lines.
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Affine Transformation of Lines
y
y
x
x
z
z
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Affine Combinations
Q
b
R
a
P
M
b
M
Q
a
P
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Affine Combinations

• Theorem Affine transformations preserve affine
  combinations.

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Properties of Affine Transformations

• Affine transformations map lines to lines
• Affine transformations preserve affine
  combinations
• Affine transformations preserve parallelism
• Affine transformations change volume by Det(M)
  
• Any affine transformation can be decomposed into
  elementary transformations.
• Affine transformations does/does not preserve
  angles?
• Affine transformations does/does not preserve
  the intersection of two lines?
• Affine transformations does/does not preserve
  distances?

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Properties of Transformation Matrices
First column is how (1,0,0,0) transforms

• Second column is how (0,1,0,0) transforms

Third column is how (0,0,1,0) transforms
Matrix holds how coordinate axes transform and
how origin transforms
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Properties of Pure Rotation Matrices

• Rows (columns) are orthogonal to each other
• A row dot product any other row 0
• Each row (column) dot product times itself 1
• RT R-1

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Coordinate Frame
y
j
?
i
k
x
z

• Coordinate frame is given by origin ? and three
  mutually orthogonal unit vectors, i, j, k -
  defined in (x,y,z) space
• Mutually orthogonal (dot products) ij ? ik
  ? jk ?.
• Unit vectors (dot products) ii ? jj ?
  kk ?.

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Orientation
Right handed coordinate system
Left handed coordinate system
y
(Into page)
z
x
(Out of page)
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Orientation
Right handed coordinate system
Left handed coordinate system
k
j
i
?
Cross product i x j ?
Cross product i x j ?
How do you test whether (i,j,k) is left handed or
right handed?
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Coordinate Transformations

• Given object data points defined in the (i, j,
  k, ?) coordinate frame,
• Given the definition of (i, j, k, ?) in (x, y, z)
  coordinates,
• How do you determine the coordinates of the
  object data points in the (x, y, z, 0) frame?

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Coordinate change (Translation)
b
(?x, ?y, ?z)
a
(0,0,0)
c
Change from (a,b,c,?) coordinates to (x,y,z,0)
coordinates

1. Move (a,b,c, ?) to (x,y,z,0) and invert
2. Move data relative from (0,0,0) out to ? position
   by adding ?

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Coordinate change (Rotation)
x
y
?
z-axis rotation by ?
(0,0,0)
z
Change from (a,b,c,?) coordinates to (x,y,z,0)
coordinates

1. Rotate (a,b,c) by ? and invert
2. Rotate data by -?

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Object Transformations
Given (i,j,k,p) defined in the (x,y,z,0)
coordinate frame, Transform points defined in the
(i, j, k, ?) coordinate frame to the (x,y,z,0)
coordinate frame. Rotate object to get x to line
up with i, then translate to ?
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Object Transformations
Affine transformation matrix
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Coordinate Transformations
Given (i,j,k,f) defined in the (x,y,z,0)
coordinate frame, Transform points in (i,j,k,0)
coordinate frame to (x,y,z, ?) coordinate
frame. translate by - ?, then rotate to align i
with x then invert
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Coordinate Transformations
T
R
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Coordinate Transformations
Apply RT to coordinate system Apply (RT)-1 to
data T-1R-1
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Composition of coordinate change
y
y
M1
y
?
z
M2
x
x
?
z
0
z
x

• M1 changes from coordinate frame (x,y,z,?) to
  (x,y,z,?).
• M2 changes from coordinate frame (x,y,z,?) to
  (x,y,z,?).
• Change from coordinate frame (x,y,z,?) to
  (x,y,z,0) ?

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Composition of transformations - example
B
B
y
q
A
x
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Transformations of normal vectors
n
?

• n is a unit normal vector to plane ?.
• M is an affine transformation matrix.
• How is n transformed, to keep it perpendicular to
  the plane, under
• translation?
• rotation by ??
• uniform scaling by s?
• shearing or non-uniform scaling?

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Non-uniform scale transformation
n

• n is a unit normal vector to plane ? (-nx,0)
• M is a non-uniform scale transformation matrix
  that only modifies x-coordinates
• If we just transform the n as a vector, then M nT
  n
• But we want n that has a non-zero
  y-coordinate value

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Transformations of normal vectors

• Planar equation a x b y c z d 0.
• Let N (a, b, c, d). (Note (a, b, c) is normal
  vector)
• Let P (x, y, z, 1) be a point in plane ?.
• Planar equation N P 0.
• M is an affine transformation matrix. (MT is M
  transpose.)
• Let P M PT.
• Find N such that N P 0 (transformed planar
  equation)
• N P 0
• N PT 0
• N (M-1 M) PT 0
• (N M-1) (M PT) 0
• (N M-1) PT 0
• So N NM-1
• To put in column-vector form, (N)T (NM-1)T
  (M-1)TNT
• So N ((M-1)T NT)T and (M-1)T is the
  transformation matrix to take NT into NT
• Note If M is a rotation matrix, (M-1)T M.

n
?
P