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Nima Kalantari

CSCE 441 - Computer Graphics

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Transformations 2D

Some slides from Ren Ng and Ravi Ramamoorthi

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Why study transforms?

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Projecting from 3D to 2D

Model from Pradeep Sen

Camera

COP

Image Plane

3D

2D

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Modeling A Robot Army

  • Can describe position of object instances

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iRobot movie

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Posing a Character’s Skeleton

  • Can describe relative position of connected body parts

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iRobot movie

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What Are Transforms?

  • Just functions acting on points

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  • Linear transforms
    • Condition:

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Outline

  • Basic transformations
  • Composing transformations

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Scale Transform (Uniform)

S0.5

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Scale Matrix (Uniform)

S0.5

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Scale (Non-Uniform)

S0.5,1.0

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Is scale a linear transform?

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Reflection

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Reflection Matrix

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Shear Matrix

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Rotation Matrix

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2-D Rotation

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2-D Rotation

Trigonometric Identity

Substitute

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Rotation Matrix

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Rotation Matrix

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Rotation Matrix

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Rotation Matrix

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Linear Transforms = Matrices

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Translation?

T1,1

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Translation

  • E.g., move x by +5 units, leave y
  • We need appropriate matrix. What is it?

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Transformations game demo

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Add an auxiliary dimension

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Add an auxiliary dimension

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Homogeneous Coordinates

  • Represent an n-dimensional Euclidean point using an n+1 dimensional point
  • For points: set dimension n+1 (w) to 1
    • E.g., 2D Euclidean -> 3D Homogeneous

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Euclidean

Homogeneous

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Homogeneous Coordinates

  • For points: to go from homogeneous coordinate to Euclidean
    • Divide by the last coordinate
    • Remove the normalized last coordinate

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Euclidean

Homogeneous

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Homogeneous Coordinates

  • Multiplication by scalar has no effect

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Euclidean

Homogeneous

=

=

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Homogeneous coordinates: some intuition

x

y

w = 1

w

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Homogeneous coordinates: points vs. vectors

w

w = 1

x

y

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Homogeneous coordinates (vectors)

  • To go to homogeneous, set the last dimension to 0
  • To come back to Euclidean, remove the last dimension

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Euclidean

Homogeneous

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General Translation Matrix

  • Point

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  • Vector

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Homogeneous coordinates

  • They are useful for two types of transformations
    • Translation
    • Projection (discussed later)

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Advantages of Homogeneous Coords

  • Unified framework for translation, rotation, etc.
  • Can concatenate any set of transforms to 3x3 matrix (4x4 for 3D)
  • Simpler formulas, no special cases
  • Standard in graphics software, hardware

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2D Transformations

  • Scale
  • Rotation
  • Translation

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Outline

  • Basic transformations
  • Composing transformations

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Composite Transform

?

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Rotate Then Translate

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Translate Then Rotate?

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Ordering matters!

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Ordering matters!

  • Matrix multiplication is not commutative

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  • Recall the matrix math represented by these symbols:

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  • Note that matrices are applied right to left:

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≠

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Composing Transforms

  • Sequence of transforms A1, A2, A3, ...
    • Compose by matrix multiplication
      • Very important for performance!

Pre-multiply n matrices to obtain a �single matrix representing combined transform

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Decomposing Complex Transforms

  • How to rotate around a given point c?
    • Translate center to origin
    • Rotate
    • Translate back
  • Matrix representation?

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Affine Transformations

  • Affine map = linear map + translation
  • Using homogenous coordinates:

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Inverse Transform

  • is the inverse of in both a matrix and geometric sense