Representations and Transformations

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Representations and Transformations
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Objectives

• Derive homogeneous coordinate transformation
  matrices
• Introduce standard transformations
• Rotations
• Translation
• Scaling
• Shear

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Scalars, Points, Vectors

• Three basic elements from geometry
• Scalars, Points, Vectors
• Scalars can be defined as members of a set which
  can be combined by the operations of addition and
  multiplication and obey the fundamental axioms
• associativity, commutivity, inversion
• Examples include the real and complex numbers
  under the rules we are all familiar with
• Scalars alone have no geometric properties

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Scalars, Points, Vectors

• A vector is a quantity with two attributes
  direction magnitude and its own rules as we saw
  last lecture
• The set defines a vector space

• But, vectors lack position
• Same length and magnitude -gt
• Vectors are insufficient to specify geometry
• We need points

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Scalars, Points, Vectors

• Points, we know, are locations in space
• Certain operations translate between points and
  vectors
• Point-point subtraction yields a vector
• - Leads to equivalent to point-vector addition

vP-Q
PvQ
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Vector Spaces

• Consider a basis v1, v2,., vn
• A vector is written va1v1 a2v2 .anvn
• The list of scalars a1, a2, . an then is the
  representation of v with respect to the given
  basis
• And we write the representation as a row or
  column array of scalars

aa1 a2 . anT
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Affine Spaces

• Vector spaces do not represent points
• Instead, we work in an affine space and add a
  special point, the origin, to the basis vectors,
  this is called a frame

v2
v1
P0
v3
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Affine Spaces

• An affine space is both point and vector space
• It allows operations from vectors, points and
  scalars
• Vector-vector addition
• Scalar-vector multiplication
• Point-vector addition
• All scalar-scalar operations
• Define our space with a coordinate Frame

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Frames

• A frame is determined by (P0, v1, v2, v3, ...)
• Within this frame
• Every vector can be written as
• va1v1 a2v2 .anvn
• And every point can be written as
• P P0 b1v1 b2v2 .bnvn

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Homogeneous Coordinates
Consider the point and the vector P P0 b1v1
b2v2 .bnvn va1v1 a2v2 .anvn They appear
to have the similar representations pb1
b2 b3 va1 a2 a3 which confuse the
point with the vector But, a vector has no
position
v
p
v
can be placed anywhere
fixed
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Homogeneous Coordinates

• An affine space combines point and vector
• with
• two new rules
• ( 1 )( P ) P
• ( 0 )( P ) 0 (zero vector)

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Homogeneous Coordinates
With these rules, we can keep track of the
difference va1v1 a2v2 a3v3 a1 a2 a3 0
v1 v2 v3 P0 T P P0 b1v1 b2v2 b3v3 b1
b2 b3 1 v1 v2 v3 P0 T Thus we obtain a
four-dimensional representation for both
v a1 a2 a3 0 T
p b1 b2 b3 1 T
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Homogeneous Coordinates

• Homogeneous coordinates are key to all computer
  graphics systems
• Hardware pipeline all work with 4 dimensional
  representations
• All standard transformations (rotation,
  translation, scaling) can be implemented by
  matrix multiplications with 4 x 4 matrices

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Homogeneous Coordinates

• Most generally, the form of homogeneous
  coordinates is
• px y z w T
• We can return to a simple three dimensional point
  by
• x'?x/w
• y'?y/w
• z'?z/w
• But if w0, the representation is that of a vector

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Transformations

• A transformation maps points to other points
  and/or vectors to other vectors

vT(u)
QT(P)
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Affine Transformations

• Line preserving
• Characteristic of many physically important
  transformations
• Rigid body transformations rotation, translation
• Non-rigid Scaling, shear
• Importance in graphics is that we need only
  transform vertices (points) of line segments and
  polygons, then system draws between the
  transformed points

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Translation

• Move (translate, displace) a point to a new
  location
• Displacement determined by a vector d
• Three degrees of freedom
• PPd

P
d
P
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Moving objects
When we move a point on an object to a new
location, to preserve the object, we must move
all other points on the object in the same way
object
translation every point displaced by the
same vector, d
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Translation Using Representations
Using the homogeneous coordinate representation
in some frame p x y z 1T px y
z 1T ddx dy dz 0T Hence p p d or
xxdx yydy zzdz
note that this expression is in four dimensions
and expresses that point vector point
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Translation Matrix
We can also express translation using a 4 x 4
matrix T in homogeneous coordinates pTp
where T T(dx, dy, dz) This form is better
for implementation because all affine
transformations can be expressed this way and
multiple transformations can be concatenated
together
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Rotation (2D)

• Consider rotation about the origin by q degrees
• radius stays the same, angle increases by q

What is this rotation about the z axis?
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Rotation (2D)

• Consider rotation about the origin by q degrees
• radius stays the same, angle increases by q

x r cos (f q) y r sin (f q)
xx cos q y sin q y x sin q y cos q
x r cos f y r sin f
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Rotation about the z axis

• Rotation about z axis in three dimensions leaves
  all points with the same z
• Equivalent to rotation in two dimensions in
  planes of constant z
• or in matrix notation (with p as a column)
• pRz(q)p

xx cos q y sin q y x sin q y cos q z z
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Rotation Matrix
Homogeneous Coordinates
R Rz(q)
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Rotation about x and y axes

• Same argument as for rotation about z axis
• For rotation about x axis, x is unchanged
• For rotation about y axis, y is unchanged

R Rx(q)
R Ry(q)
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Scaling
Expand or contract along each axis (fixed point
of origin)
xsxx ysyy zszz
pSp
S S(sx, sy, sz)
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Reflection
corresponds to negative scale factors
sx -1 sy 1
original
sx -1 sy -1
sx 1 sy -1
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Shear

• Helpful to add one more basic transformation
• Equivalent to pulling faces in opposite directions

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Shear Matrix
Consider simple shear along x axis
x x y shy y y z z
H(q)
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Inverses

• Although we could compute inverse matrices by
  general formulas, we can also use simple
  geometric observations, for example
• Translation T-1(dx, dy, dz) T(-dx, -dy, -dz)
• Rotation R -1(q) R(-q)
• Holds for any rotation matrix
• Note that since cos(-q) cos(q) and sin(-q)
  -sin(q)
• R -1(q) R T(q)
• Scaling S-1(sx, sy, sz) S(1/sx, 1/sy, 1/sz)

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Composite

• In modeling, we often start with a simple object
  centered at the origin, oriented with the axis,
  and at a standard size
• We apply an composite transformation to its
  vertices to
• Scale
• Orient
• Locate

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Composite Transformations

• Scaling about a fixed point
• - Simply applying the scale transformation also
  moves the object being scaled.

Q'
Q
P
P'
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Composite Transformations

• Scaling about origin does not move object
• Origin is a fixed point for the scale
  transformation
• We use composite transformations to create scale
• transformations with different fixed points

Q'
Q
P'
P
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Composite Transformations

• Fixed point scale transformation
• Move fixed point (px,py,pz) to origin
• Scale by desired amount
• Move fixed point back to original position

M T(px, py, pz) S(sx, sy, sz) T(-px, -py, -pz)
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Composite Transformations

• Rotating about a fixed point
• - basic rotation alone will rotate about origin
• but we want

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Composite Transformations

• Rotating about a fixed point
• Move fixed point (px,py,pz) to origin
• Rotate by desired amount
• Move fixed point back to original position

M T(px, py, pz) Rx(q) T(-px, -py, -pz)
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Composite Transformations