CAP4730: Computational Structures in Computer Graphics

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CAP4730 Computational Structures in Computer
Graphics
2D Transformations
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2D Transformations

• World Coordinates
• Translate
• Rotate
• Scale
• Viewport Transforms
• Putting it all together

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Transformations

• Rigid Body Transformations - transformations that
  do not change the object.
• Translate
• If you translate a rectangle, it is still a
  rectangle
• Scale
• If you scale a rectangle, it is still a rectangle
• Rotate
• If you rotate a rectangle, it is still a rectangle

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Vertices

• We have always represented vertices as (x,y)
• An alternate method is
• Example

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Matrix Vector
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Matrix Matrix
Does AB BA? What does the identity do?
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Practice
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Translation

• Translation - repositioning an object along a
  straight-line path (the translation distances)
  from one coordinate location to another.

(x,y)
(tx,ty)
(x,y)
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Translation

• Given
• We want
• Matrix form

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Translation Examples

• P(2,4), T(-1,14), P(?,?)
• P(8.6,-1), T(0.4,-0.2), P(?,?)
• P(0,0), T(1,0), P(?,?)

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Which one is it?
(x,y)
(tx,ty)
(tx,ty)
(x,y)
(x,y)
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Recall

• A point is a position specified with coordinate
  values in some reference frame.
• We usually label a point in this reference point
  as the origin.
• All points in the reference frame are given with
  respect to the origin.

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Applying to Triangles
(tx,ty)
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What do we have here?

• You know how to

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Scale

• Scale - Alters the size of an object.
• Scales about a fixed point

(x,y)
(x,y)
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Scale

• Given
• We want
• Matrix form

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Non-Uniform/Differential Scalin
(x,y)
(x,y)
S(1,2)
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Rotation

• Rotation - repositions an object along a circular
  path.
• Rotation requires an ? and a pivot point

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Rotation
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Example

• P(4,4)
• ?45 degrees

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What is the difference? Revisited
V(-0.6,0) V(0,-0.6) V(0.6,0.6) Translate
(1.2,0.3) V(0,0.6) V(0.3,0.9) V(0,1.2)
Translate (1.2,0.3) V(0.6,0.3) V(1.2,-0.3)
V(1.8,0.9) V(0,0.6) V(0.3,0.9) V(0,1.2)
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Rotations
V(-0.6,0) V(0,-0.6) V(0.6,0.6) Rotate -30
degrees V(0,0.6) V(0.3,0.9) V(0,1.2)
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Combining Transformations
Q How do we specify each transformation?
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Specifying 2D Transformations

• Translation
• T(tx, ty)
• Translation distances
• Scale
• S(sx,sy)
• Scale factors
• Rotation
• R(?)
• Rotation angle

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Combining Transformations

• Using translate, rotation, and scale, how do we
  get

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Combining Transformations

• Note there are two ways to combine rotation and
  translation. Why?

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Lets look at the equations
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Combining them

• We must do each step in turn. First we rotate
  the points, then we translate, etc.
• Since we can represent the transformations by
  matrices, why dont we just combine them?

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2x2 -gt 3x3 Matrices

• We can combine transformations by expanding from
  2x2 to 3x3 matrices.

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Homogenous Coordinates

• We need to do something to the vertices
• By increasing the dimensionality of the problem
  we can transform the addition component of
  Translation into multiplication.

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Homogenous Coordinates

• Homogenous Coordinates - term used in mathematics
  to refer to the effect of this representation on
  Cartesian equations. Converting a pt(x,y) and
  f(x,y)0 -gt (xh,yh,h) then in homogenous
  equations mean (vxh,vyh,vh) can be factored
  out.
• What you should get By expressing the
  transformations with homogenous equations and
  coordinates, all transformations can be expressed
  as matrix multiplications.

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Final Transformations - Compare Equations
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Combining Transformations
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How would we get
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How would we get
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Coordinate Systems

• Object Coordinates
• World Coordinates
• Eye Coordinates

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Object Coordinates
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World Coordinates
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Screen Coordinates
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Coordinate Hierarchy
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Lets reexamine assignment 2b
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Transformation Hierarchies

• For example

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Transformation Hierarchies

• Lets examine

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Transformation Hierarchies

• What is a better way?

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Transformation Hierarchies

• What is a better way?

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Transformation Hierarchies

• We can have transformations be in relation to
  each other

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More Complex Models