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Geometric Transformations. Geometric Transformations. Translate X’ = X + dx Y’ = Y + dy X’ = X + (-6) Y’ = Y + (-4). dy. dx. Geometric Transformations. Scale X’ = X * S x Y’ = Y * S y X’ = X * 2 Y’ = Y * 0.5 Only origin is stable. Geometric Transformations. Scale X’ = X * S x

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Geometric transformations1
Geometric Transformations

  • Translate

    • X’ = X + dx

    • Y’ = Y + dy

    • X’ = X + (-6)

    • Y’ = Y + (-4)

dy

dx


Geometric transformations2
Geometric Transformations

  • Scale

    • X’ = X * Sx

    • Y’ = Y * Sy

    • X’ = X * 2

    • Y’ = Y * 0.5

  • Only origin is stable


Geometric transformations3
Geometric Transformations

  • Scale

    • X’ = X * Sx

    • Y’ = Y * Sy

    • X’ = X * -1

    • Y’ = Y * 1

  • Only origin is stable


Geometric transformations4
Geometric Transformations

  • Rotate

    • X’ = cos(a)X – sin(a)Y

    • Y’ = sin(a)X + cos(a)Y

    • X’ = cos(45)X – sin(45) Y

    • Y’ = sin(45)X + cos(45)Y

  • Only origin is stable


General form
General Form

  • Translate

    • X’ = 0 * X + 0 * Y + dx

    • Y’ = 0 * X + 0 * Y + dy

  • Scale

    • X’ = Sx * X + 0 * Y + 0

    • Y’ = 0 * X + Sy * Y + 0

  • Rotate

    • X’ = cos(a)*X – sin(a)*Y + 0

    • Y’ = sin(a)*X + cos(a)*Y + 0


Homogenous coordinates
Homogenous Coordinates

  • Translate

    • X’ = 1 * X + 0 * Y + dx = [1, 0, dx] * [ X, Y, 1]

    • Y’ = 0 * X + 1 * Y + dy = [0, 1, dy] * [ X, Y, 1]

  • Scale

    • X’ = Sx * X + 0 * Y + 0 = [Sx, 0, 0] * [ X, Y, 1]

    • Y’ = 0 * X + Sy * Y + 0 = [0, Sy, 0] * [ X, Y, 1]

  • Rotate

    • X’ = cos(a)*X – sin(a)*Y + 0

      = [cos(a), -sin(a), 0] * [ X, Y, 1]

    • Y’ = sin(a)*X + cos(a)*Y + 0

      = [sin(a), cos(a), 0] * [ X, Y, 1]


Matrix form
Matrix form

[ X Y 1] * a d 0 = [X’ Y’ 1]

b e 0

c f 1

a b c * X = X’

d e f Y Y’

0 0 1 1 1


Matrix form1
Matrix form

1 0 dx * X = X’

0 1 dy Y Y’

0 0 1 1 1

T(dx,dy)

S(Sx, Sy)

R(a)

R(sin(a),cos(a))

Sx 0 0 * X = X’

0 Sy 0 Y Y’

0 0 1 1 1

cos(a) -sin(a) 0 * X = X’

sin(a) cos(a) 0 Y Y’

0 0 1 1 1


T(2, 2) S(3/4, 1/3) T(-2,-2)

1 0 2

0 1 2

0 0 1

3/4 0 0

0 1/3 0

0 0 1

X

Y

1

X’

Y’

1

1 0 -2

0 1 -2

0 0 1

=


T(2, 2) S(3/4, 1/3) T(-2,-2)

1 0 2

0 1 2

0 0 1

3/4 0 0

0 1/3 0

0 0 1

X

Y

1

X’

Y’

1

1 0 -2

0 1 -2

0 0 1

=


R(45) S(2,1)

S(2,1) R(45)


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