Computer Graphics 3: 2D Transformations

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Computer Graphics 32D Transformations
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Contents

• In todays lecture well cover the following
• Why transformations
• Transformations
• Translation
• Scaling
• Rotation
• Homogeneous coordinates
• Matrix multiplications
• Combining transformations

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Why Transformations?

• In graphics, once we have an object described,
  transformations are used to move that object,
  scale it and rotate it

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Translation

• Simply moves an object from one position to
  another
• xnew xold dx ynew yold dy

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Translation Example
(2, 3)
(1, 1)
(3, 1)
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Scaling

• Scalar multiplies all coordinates
• WATCH OUT Objects grow and move!
• xnew Sx xold ynew Sy yold

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Scaling Example
(2, 3)
(1, 1)
(3, 1)
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Rotation

• Rotates all coordinates by a specified angle
• xnew xold cos? yold sin?
• ynew xold sin? yold cos?
• Points are always rotated about the origin

y
6
5
4
3
2
1
x
0
1
2
3
4
5
6
7
8
9
10
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Rotation Example
(4, 3)
(3, 1)
(5, 1)
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Homogeneous Coordinates

• A point (x, y) can be re-written in homogeneous
  coordinates as (xh, yh, h)
• The homogeneous parameter h is a non-zero value
  such that
• We can then write any point (x, y) as (hx, hy, h)
• We can conveniently choose h 1 so that (x, y)
  becomes (x, y, 1)

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Why Homogeneous Coordinates?

• Mathematicians commonly use homogeneous
  coordinates as they allow scaling factors to be
  removed from equations
• We will see in a moment that all of the
  transformations we discussed previously can be
  represented as 33 matrices
• Using homogeneous coordinates allows us use
  matrix multiplication to calculate
  transformations extremely efficient!

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Homogeneous Translation

• The translation of a point by (dx, dy) can be
  written in matrix form as
• Representing the point as a homogeneous column
  vector we perform the calculation as

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Remember Matrix Multiplication

• Recall how matrix multiplication takes place

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Homogenous Coordinates

• To make operations easier, 2-D points are written
  as homogenous coordinate column vectors

Translation
Scaling
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Homogenous Coordinates (cont)
Rotation
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Inverse Transformations

• Transformations can easily be reversed using
  inverse transformations

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Combining Transformations

• A number of transformations can be combined into
  one matrix to make things easy
• Allowed by the fact that we use homogenous
  coordinates
• Imagine rotating a polygon around a point other
  than the origin
• Transform to centre point to origin
• Rotate around origin
• Transform back to centre point

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Combining Transformations (cont)
2
1
3
4
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Combining Transformations (cont)

• The three transformation matrices are combined as
  follows

REMEMBER Matrix multiplication is not
commutative so order matters
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Summary

• In this lecture we have taken a look at
• 2D Transformations
• Translation
• Scaling
• Rotation
• Homogeneous coordinates
• Matrix multiplications
• Combining transformations
• Next time well start to look at how we take
  these abstract shapes etc and get them on-screen

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Exercises 1

• Translate the shape below by (7, 2)

(2, 3)
(1, 2)
(3, 2)
(2, 1)
x
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Exercises 2

• Scale the shape below by 3 in x and 2 in y

(2, 3)
(1, 2)
(3, 2)
(2, 1)
x
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Exercises 3

• Rotate the shape below by 30 about the origin

(7, 3)
(6, 2)
(8, 2)
(7, 1)
x
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Exercise 4

• Write out the homogeneous matrices for the
  previous three transformations

Translation
Scaling
Rotation
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Exercises 5

• Using matrix multiplication calculate the
  rotation of the shape below by 45 about its
  centre (5, 3)

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Scratch
x
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Equations

• Translation
• xnew xold dx ynew yold dy
• Scaling
• xnew Sx xold ynew Sy yold
• Rotation
• xnew xold cos? yold sin?
• ynew xold sin? yold cos?