Transformation

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Transformation Projection
Proseminar Computer Graphics

• Feng Yu

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Transformations

• What is a transformation?
• What kind of transformations are there?
• How can we compute them?

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Transformation

• 2D Transformations
• Homogeneous Coordinates and Matrix Representation
  of 2D Transformations
• Matrix Representation of 3D Transformations
• Transformations as a Change in Coordinate System

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2D Translations.
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2D Scaling from the origin.
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2D Rotation about the origin.
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2D Rotation about the origin.
y
P(x,y)
P(x,y)
r
?
y
r
?
x
x
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2D Rotation about the origin.
Substituting for r
Given us
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2D Rotation about the origin.
Rewriting in matrix form gives us
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Transformations.

• Translation.
• P?T P
• Scale
• P?S ? P
• Rotation
• P?R ? P
• We would like all transformations to be
  multiplications so we can concatenate them ?
  express points in homogenous coordinates.

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Homogeneous coordinates

• Add an extra coordinate, W, to a point.
• P(x,y,W).
• Two sets of homogeneous coordinates represent the
  same point if they are a multiple of each other.
• (2,5,3) and (4,10,6) represent the same point.
• At least one component must be non-zero ? (0,0,0)
  is not allowed.
• If W? 0 , divide by it to get Cartesian
  coordinates of point (x/W,y/W,1).
• If W0, point is said to be at infinity.

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Homogeneous coordinates

• If we represent (x,y,W) in 3-space, all triples
  representing the same point describe a line
  passing through the origin.
• If we homogenize the point, we get a point of
  form (x,y,1)
• homogenised points form a plane at W1.

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Translations in homogenised coordinates

• Transformation matrices for 2D translation are
  now 3x3.

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Concatenation.

• We perform 2 translations on the same point

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Concatenation.
Matrix product is variously referred to as
compounding, concatenation, or composition. This
single matrix is called the Coordinate
Transformation Matrix or CTM.
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Homogeneous form of scale.
Recall the (x,y) form of Scale
In homogeneous coordinates
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Concatenation of scales.
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Homogeneous form of rotation.
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3D Transformations.

• Use homogeneous coordinates, just as in 2D case.
• Transformations are now 4x4 matrices.
• We will use a right-handed (world) coordinate
  system - ( z out of page ).

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Translation in 3D.
Simple extension to the 3D case
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Scale in 3D.
Simple extension to the 3D case
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Rotation in 3D

• Need to specify which axis the rotation is about.
• z-axis rotation is the same as the 2D case.

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Rotation in 3D

• For rotation about the x and y axes

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Transformations of coordinate systems.
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Transform Left-Right, Right-Left
Transforms between world coordinates and viewing
coordinates. That is between a right-handed set
and a left-handed set.
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Projections

• Perspective Projection
• Parallel Projection

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Planar Geometric Projections

• Standard projections project onto a plane
• Projectors are lines that either
• converge at a center of projection
• are parallel
• Such projections preserve lines
• but not necessarily angles
• Nonplanar projections are needed for applications
  such as map construction

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Perspective Projection
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Parallel Projection
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Taxonomy of Planar Geometric Projections
planar geometric projections
perspective

• parallel

2 point
1 point
3 point
multiview orthographic
axonometric
oblique
isometric
dimetric
trimetric
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Orthographic Projection

• Projectors are orthogonal to projection plane

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Multiview Orthographic Projection

• Projection plane parallel to principal face
• Usually form front, top, side views

isometric (not multiview orthographic view)
front
in CAD and architecture, we often display three
multiviews plus isometric
side
top
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Advantages and Disadvantages

• Preserves both distances and angles
• Shapes preserved
• Can be used for measurements
• Building plans
• Manuals
• Cannot see what object really looks like because
  many surfaces hidden from view
• Often we add the isometric

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Oblique Projection

• Arbitrary relationship between projectors and
  projection plane

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Advantages and Disadvantages

• Can pick the angles to emphasize a particular
  face
• Architecture plan oblique, elevation oblique
• Angles in faces parallel to projection plane are
  preserved while we can still see around side
• In physical world, cannot create with simple
  camera possible with bellows camera or special
  lens (architectural)

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Perspective Projection

• Projectors coverge at center of projection

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Vanishing Points

• Parallel lines (not parallel to the projection
  plan) on the object converge at a single point in
  the projection (the vanishing point)
• Drawing simple perspectives by hand uses these
  vanishing point(s)

vanishing point
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One-Point Perspective

• One principal face parallel to projection plane
• One vanishing point for cube

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Two-Point Perspective

• On principal direction parallel to projection
  plane
• Two vanishing points for cube

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Advantages and Disadvantages

• Objects further from viewer are projected smaller
  than the same sized objects closer to the viewer
  (diminution)
• Looks realistic
• Equal distances along a line are not projected
  into equal distances (nonuniform foreshortening)
• Angles preserved only in planes parallel to the
  projection plane
• More difficult to construct by hand than parallel
  projections (but not more difficult by computer)

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END

• Thank you for your attentions