CS 551 / 645: Introductory Computer Graphics

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CS 551 / 645 Introductory Computer Graphics

• Geometric Transforms

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Administrivia

• Return assignment 1
• Hand out assignment 2
• Todays reading material FvD, Chapter 5

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Recap Rendering Pipeline (3-D)
Scene graphObject geometry

• Result
• All vertices of scene in shared 3-D world
  coordinate system
• Vertices shaded according to lighting model
• Scene vertices in 3-D view or camera
  coordinate system
• Exactly those vertices portions of polygons in
  view frustum
• 2-D screen coordinates of clipped vertices

ModelingTransforms
LightingCalculations
ViewingTransform
Clipping
ProjectionTransform
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Recap Transformations

• Modeling transforms
• Size, place, scale, and rotate objects parts of
  the model w.r.t. each other
• Object coordinates ? world coordinates

Y
Z
X
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Rigid-Body Transforms

• Goal object coordinates?world coordinates
• Idea use only transformations that preserve the
  shape of the object
• Rigid-body or Euclidean transforms
• Includes rotation, translation, and scale
• To reiterate we will represent points as column
  vectors

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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

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Scaling

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

? 2
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Scaling

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

X ? 2,Y ? 0.5
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Scaling

• Scaling operation
• Or, in matrix form

scaling matrix
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2-D Rotation
(x, y)
(x, y)
x x cos(?) - y sin(?) y x sin(?) y cos(?)
?
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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

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3-D Rotation

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

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3-D Rotation

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

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3-D Rotation

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

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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
• Standard approach express general rotation as
  composition of canonical rotations
• Rotations about X, Y, Z

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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 -?)

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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 ??

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Composing Canonical Rotations

• Why are we slogging through all this tedium?
• A Because youll have to do it on the test

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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!

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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?

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Compositing Matrices

• Short answer 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.
• So in our case
• p Ry?-1 Rx ? -1 Rz? Rx ? Ry? p

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Rotation Matrices

• Notice these two matrices
• Rx ? Rotate about X by ?
• Rx ? -1 Undo rotation about X by ?
• How can we calculate Rx ? -1?

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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

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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?

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Rotation Matrices

• Orthonormal matrix
• orthogonal (columns/rows linearly independent)
• Columns/rows length of 1
• The inverse of an orthogonal matrix is just its
  transpose

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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

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Homogeneous Coordinates

• Homogeneous coordinates represent coordinates in
  3 dimensions with a 4-vector
• (Note that typically w 1 in object coordinates)

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Homogeneous Coordinates

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

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Homogeneous Coordinates

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

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Homogeneous Coordinates

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

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Homogeneous Coordinates

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

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Homogeneous Coordinates

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

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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

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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

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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

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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

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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 ?

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More On Homogeneous Coords

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

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Homogenous Coordinates

• In particular, increasing w makes things smaller
• We think of homogenous coordinates as defining a
  projective space
• Increasing w ? getting further away
• Points with w 0 are, in this sense, projected
  on the plane at infinity
• Will come in handy for projection matrices

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Perspective Projection

• We talked about geometric transforms, focusing on
  modeling transforms
• Ex translation, rotation, scale, gluLookAt()
• These are encapsulated in the OpenGL modelview
  matrix
• Can also express perspective projection (and
  other projections) as a matrix
• Next few slides representing perspective
  projection with the projection matrix

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Perspective Projection

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

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Perspective Projection

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

How tall shouldthis bunny be?
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Perspective Projection

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

Viewplane
P (x, y, z)
X
x ?
(0,0,0)
Z
d
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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?

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A Perspective Projection Matrix

• Answer

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A Perspective Projection Matrix

• Example
• Or, in 3-D coordinates

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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 a single matrix encapsulating
  modeling, viewing, and projection transforms

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Matrix Operations In OpenGL

• Certain commands affect the current matrix in
  OpenGL
• glMatrixMode() sets the current matrix
• glLoadIdentity() replaces the current matrix with
  an identity matrix
• glTranslate() postmultiplies the current matrix
  with a translation matrix
• gluPerspective() postmultiplies the current
  matrix with a perspective projection matrix
• It is important that you understand the order in
  which OpenGL concatenates matrices

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Matrix Operations In OpenGL

• In OpenGL
• Vertices are multiplied by the modelview matrix
• The resulting vertices are multiplied by the
  projection matrix
• Example
• Suppose you want to scale an object, translate
  it, apply a lookat transformation, and view it
  under perspective projection. What order should
  you make calls?

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Matrix Operations in OpenGL

• Problem scale an object, translate it, apply a
  lookat transformation, and view it under
  perspective
• A correct code fragment
• glMatrixMode(GL_PERSPECTIVE)
• glLoadIdentity()
• gluPerspective()
• glMatrixMode(GL_MODELVIEW)
• glLoadIdentity()
• gluLookAt()
• glTranslate()
• glScale()
• / Draw the object... /

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Matrix Operations in OpenGL

• Problem scale an object, translate it, apply a
  lookat transformation, and view it under
  perspective
• An incorrect code fragment
• glMatrixMode(GL_PERSPECTIVE)
• glLoadIdentity()
• glTranslate()
• glScale()
• gluPerspective()
• glMatrixMode(GL_MODELVIEW)
• glLoadIdentity()
• gluLookAt()
• / Draw the object... /