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chapter 4.2

Objectives

• Introduce standard transformations
  • Rotations
  • Translation
  • Scaling
  • Shear
• Derive homogeneous coordinate transformation matrices
• Learn to build arbitrary transformation matrices from simple transformations

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General Transformations

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

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Affine Transformations

• Line preserving
• Characteristic of many physically important transformations
  • Rigid body transformations: rotation, translation
  • Scaling, shear
• Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints
  

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Pipeline Implementation

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Notation

• We will be working with both coordinate-free representations of transformations and representations within a particular frame
  • P, Q, R: points in an affine space
  • u, v, w: vectors in an affine space
  • a, b, g: scalars
  • P, Q, R: representations of points
    • array of 4 scalars in homogeneous coordinates
  • u, v, w: representations of vectors
    • array of 4 scalars in homogeneous coordinates
      

 

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Translation

• Move (translate, displace) a point to a new location

• Displacement determined by a vector d
  • Three degrees of freedom
  • P`=P+d

How many ways?

• Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way

Translation Using Representations

• Using the homogeneous coordinate representation in some frame

  P=[ x y z 1]^T

  d=[dx dy dz 0]^T

• Hence P`= P + d or

  x`=x + dx

  y`=y + dy

  z`=z + dz
  

Translation Matrix

• We can also express translation using a 4 x 4 matrix T in homogeneous coordinates
  P`=TP whereT = 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 degrees

• radius stays the same, angle increases by

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
  • x`=x cos - y sin
    y`= x sin + y cos
    z`=z

 

• or in homogeneous coordinates
  P`=R _z()P
  

Rotation Matrix

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

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Scaling

• Expand or contract along each axis (fixed point of origin)

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Reflection

• corresponds to negative scale factors (Stencil Activity)
  

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Inverses

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

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Concatenation

• We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices
• Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p
• The difficult part is how to form a desired transformation from the specifications in the application

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Order of Transformations

• Note that matrix on the right is the first applied
• Mathematically, the following are equivalent
  • P`= ABCP = A(B(CP))
• Note many references use row matrices to represent points. In terms of row matrices
  P`^T = p^TC^TB^TA^T
• OpenGL assumes the column matrix form.

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General Rotation About the Origin

• A rotation by about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes
• R() = R_z(_z) R_y(_y) R_x(_x)
• _x, _y, _z are called the Euler angles

• Note that rotations do not commute. We can use rotations in another order but with different angles

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Rotation About a Fixed Point other than the Origin

• Move fixed point to origin
• Rotate
• Move fixed point back
• M = T(p_f) R() T(-p_f)

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Instancing

• 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 instance transformation to its vertices to
  • Scale
  • Orient
  • Locate

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Shear

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

Shear Matrix

• Consider simple shear along x axis

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