2D GEOMETRIC TRANSFORMATION

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• 2D GEOMETRIC TRANSFORMATION

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• 2D Transforms
• Translation
• Rotation
• Scaling
• Composition
• Homogeneous Coordinates

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2D Transforms

• In many applications it is necessary to modify
  the position, orientation or size of a 2D object.
• The most widely used transformations are
• Translation move one object to another position
• Rotation rotate the object around the origin
• Scaling change the size of the object

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2D Transforms

• We are going to see how to transform one point.
• In order to transform an object, we transform the
  points that define it. For instance
• Polygon its vertices
• Circle its center and, perhaps, its radius

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2D Transforms Translation

• Add a distance in x and y to the point.

V
T
V
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• Translation

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Scaling

• Changing the size of an object
• In the figure, sx sy 0.5.
• The size of the line is reduced and moved closer
  to the origin.

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• Scaling

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Rotation

• A point is rotated around the origin.
• Rotation is through an angle q.

x, y
q
x, y
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• Rotation

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Translation in Homogeneous Coordinates (HC)
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Scaling in HC
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Rotation in HC
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Composition of transformationsImages taken from
Hearn Baker. Computer Graphics with OpenGL.
3rd Ed.

• Example Rotate an object around an arbitrary
  point.
• Translate P so that the axis is at the origin
• Rotate the object
• Translate the object so that P returns to its
  position

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Composition of Transformations

• The final transform becomes a multiplication of
  several matrices.

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Composition of transformationsImages taken from
Hearn Baker. Computer Graphics with OpenGL.
3rd Ed.

• Example Scale an object in its place

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Composition of transformations

• Multiplication of matrices

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Inverse transformations

• The inverse of the transformation is the inverse
  of its matrix
• Translation

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Inverse transformations

• Rotation

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Inverse transformations

• Scaling

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Transformation composition properties

• Multiplication of matrices is associative

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Transformation Composition properties

• Multiplication of matrices is not commutative

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Transformation composition propertiesImages
taken from Hearn Baker. Computer Graphics with
OpenGL. 3rd Ed.
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Other two-dimensional transformations

• Reflection around the x axis (y 0).

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Other two-dimensional transformations

• Reflection around the y axis (x 0).

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Other two-dimensional transformations

• Reflection relative to the coordinate origin.

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Shear

• A transformation that distorts the shape of an
  object such that the transformed shape appears as
  if the object were composed of internal layers
  that had been caused to slide over each other is
  called a shear.

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Shear

• The transformed positions are