Transformations

1
Transformations

• Aaron Bloomfield
• CS 445 Introduction to Graphics
• Fall 2006
• (Slide set originally by David Luebke)

2
Graphics coordinate systems

• X is red
• Y is green
• Z is blue

3
Graphics coordinate systems

• If you are on the z axis, and y is up, then x
  is to the right
• Math fields have x going to the left

4
Outline

• Scaling
• Rotations
• Composing Rotations
• Homogeneous Coordinates
• Translations
• Projections

5
Scaling

• Scaling a coordinate means multiplying each of
  its components by a scalar
• Uniform scaling means this scalar is the same for
  all components

6
Scaling

• Non-uniform scaling different scalars per
  component
• How can we represent this in matrix form?

7
Scaling

• Scaling operation
• Or, in matrix form

scaling matrix
8
Outline

• Scaling
• Rotations
• Composing Rotations
• Homogeneous Coordinates
• Translations
• Projections

9
2-D Rotation
x x cos(?) - y sin(?) y x sin(?) y cos(?)
10
2-D Rotation

• This is easy to capture in matrix form
• 3-D is more complicated
• Need to specify an axis of rotation
• Simple cases rotation about X, Y, Z axes

11
Rotation example airplane
12
3-D Rotation

• What does the 3-D rotation matrix look like for a
  rotation about the Z-axis?
• Build it coordinate-by-coordinate
• 2-D rotation from last slide

13
3-D Rotation

• What does the 3-D rotation matrix look like for a
  rotation about the Y-axis?
• Build it coordinate-by-coordinate

14
3-D Rotation

• What does the 3-D rotation matrix look like for a
  rotation about the X-axis?
• Build it coordinate-by-coordinate

15
Rotations about the axes
16
3-D Rotation

• General rotations in 3-D require rotating about
  an arbitrary axis of rotation
• Deriving the rotation matrix for such a rotation
  directly is a good exercise in linear algebra
• Another approach express general rotation as
  composition of canonical rotations
• Rotations about X, Y, Z

17
Outline

• Scaling
• Rotations
• Composing Rotations
• Homogeneous Coordinates
• Translations
• Projections

18
Composing Canonical Rotations

• Goal rotate about arbitrary vector A by ?
• Idea we know how to rotate about X,Y,Z
• So, rotate about Y by ? until A lies in the YZ
  plane
• Then rotate about X by ? until A coincides with
  Z
• Then rotate about Z by ?
• Then reverse the rotation about X (by -?)
• Then reverse the rotation about Y (by -?)
• Show video

19
Composing Canonical Rotations

• First rotating about Y by ? until A lies in YZ
• Draw it
• How exactly do we calculate ??
• Project A onto XZ plane
• Find angle ? to X
• ? -(90 - ?) ? - 90
• Second rotating about X by ? until A lies on Z
• How do we calculate ??

20
3-D Rotation Matrices

• So an arbitrary rotation about A composites
  several canonical rotations together
• We can express each rotation as a matrix
• Compositing transforms multiplying matrices
• Thus we can express the final rotation as the
  product of canonical rotation matrices
• Thus we can express the final rotation with a
  single matrix!

21
Compositing Matrices

• So we have the following matrices
• p The point to be rotated about A by ?
• Ry? Rotate about Y by ?
• Rx ? Rotate about X by ?
• Rz? Rotate about Z by ?
• Rx ? -1 Undo rotation about X by ?
• Ry?-1 Undo rotation about Y by ?
• In what order should we multiply them?

22
Compositing Matrices

• Remember the transformations, in order, are
  written from right to left
• In other words, the first matrix to affect the
  vector goes next to the vector, the second next
  to the first, etc.
• This is the rule with column vectors (OpenGL)
  row vectors would be the opposite
• So in our case
• p Ry?-1 Rx ? -1 Rz? Rx ? Ry? p

23
Rotation Matrices

• Notice these two matrices
• Rx ? Rotate about X by ?
• Rx ? -1 Undo rotation about X by ?
• How can we calculate Rx ? -1?
• Obvious answer calculate Rx (-?)
• Clever answer exploit fact that rotation
  matrices are orthonormal
• What is an orthonormal matrix?
• What property are we talking about?

24
Rotation Matrices

• Rotation matrix is orthogonal
• Columns/rows linearly independent
• Columns/rows sum to 1
• The inverse of an orthogonal matrix is just its
  transpose

25
Rotation Matrix for Any Axis

• Given glRotated (angle, x, y, z)
• Let c cos(angle)
• Let s sin(angle)
• And normalize the vector so that (x,y,z 1
• The produced matrix to rotate something by angle
  degrees around the axis (x,y,z) is

26
Outline

• Scaling
• Rotations
• Composing Rotations
• Homogeneous Coordinates
• Translations
• Projections

27
Translations

• For convenience we usually describe objects in
  relation to their own coordinate system
• We can translate or move points to a new position
  by adding offsets to their coordinates
• Note that this translates all points uniformly

28
Translation Matrices?

• We can composite scale matrices just as we did
  rotation matrices
• But how to represent translation as a matrix?
• Answer with homogeneous coordinates

29
Homogeneous Coordinates

• Homogeneous coordinates represent coordinates in
  3 dimensions with a 4-vector
• x, y, z, 0T represents a point at infinity (use
  for vectors)
• 0, 0, 0T is not allowed
• Note that typically w 1 in object coordinates

30
Homogeneous Coordinates

• Homogeneous coordinates seem unintuitive, but
  they make graphics operations much easier
• Our transformation matrices are now 4x4

31
Homogeneous Coordinates

• Homogeneous coordinates seem unintuitive, but
  they make graphics operations much easier
• Our transformation matrices are now 4x4

32
Homogeneous Coordinates

• Homogeneous coordinates seem unintuitive, but
  they make graphics operations much easier
• Our transformation matrices are now 4x4

33
Homogeneous Coordinates

• Homogeneous coordinates seem unintuitive, but
  they make graphics operations much easier
• Our transformation matrices are now 4x4
• Performing a scale

34
More On Homogeneous Coords

• What effect does the following matrix have?
• Conceptually, the fourth coordinate w is a bit
  like a scale factor

35
More On Homogeneous Coords

• Intuitively
• The w coordinate of a homogeneous point is
  typically 1
• Decreasing w makes the point bigger, meaning
  further from the origin
• Homogeneous points with w 0 are thus points at
  infinity, meaning infinitely far away in some
  direction. (What direction?)
• To help illustrate this, imagine subtracting two
  homogeneous points

36
Outline

• Scaling
• Rotations
• Composing Rotations
• Homogeneous Coordinates
• Translations
• Projections

37
Homogeneous Coordinates

• How can we represent translation as a 4x4 matrix?
• A Using the rightmost column
• Performing a translation

38
Translation Matrices

• Now that we can represent translation as a
  matrix, we can composite it with other
  transformations
• Ex rotate 90 about X, then 10 units down Z

39
Translation Matrices

• Now that we can represent translation as a
  matrix, we can composite it with other
  transformations
• Ex rotate 90 about X, then 10 units down Z

40
Translation Matrices

• Now that we can represent translation as a
  matrix, we can composite it with other
  transformations
• Ex rotate 90 about X, then 10 units down Z

41
Translation Matrices

• Now that we can represent translation as a
  matrix, we can composite it with other
  transformations
• Ex rotate 90 about X, then 10 units down Z

42
Transformation Commutativity

• Is matrix multiplication, in general,
  commutative? Does AB BA?
• What about rotation, scaling, and translation
  matrices?
• Does RxRy RyRx?
• Does RAS SRA ?
• Does RAT TRA ?

43
Outline

• Scaling
• Rotations
• Composing Rotations
• Homogeneous Coordinates
• Translations
• Projections

44
Perspective Projection

• In the real world, objects exhibit perspective
  foreshortening distant objects appear smaller
• The basic situation

45
Perspective Projection

• When we do 3-D graphics, we think of the screen
  as a 2-D window onto the 3-D world
• The view plane

46
Perspective Projection

• The geometry of the situation is that of similar
  triangles. View from above
• What is x?

47
Perspective Projection

• Desired result for a point x, y, z, 1T
  projected onto the view plane
• What could a matrix look like to do this?

48
A Perspective Projection Matrix

• Answer

49
A Perspective Projection Matrix

• Example
• Or, in 3-D coordinates

50
A Perspective Projection Matrix

• OpenGLs gluPerspective() command generates a
  slightly more complicated matrix
• Can you figure out what this matrix does?

51
Projection Matrices

• Now that we can express perspective
  foreshortening as a matrix, we can composite it
  onto our other matrices with the usual matrix
  multiplication
• End result can create a single matrix
  encapsulating modeling, viewing, and projection
  transforms
• Though you will recall that in practice OpenGL
  separates the modelview from projection matrix
  (why?)