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Geometrical Transformation

Tong-Yee Lee

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Modeling TransformSpecify transformation for objects

Allow definitions of objects in own coordinate systemsAllow use of object definition multiple times in a scene

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Overview

2D transformationsBasic 2-D transformationsMatrix representationMatrix composition

3D transformationsBasic 3-D transformationSame as 2-D

Transformation HierarchiesScene graphs

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2-D Transformations

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2-D Transformations

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2-D Transformations

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2-D Transformations

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2-D Transformations

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2-D Transformations

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Basic 2D Transformations

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Basic 2D Transformations

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Basic 2D Transformations

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Rotation around the origin (2-D)

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Rotation around the origin (2-D)

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Rotation around the origin (2-D)

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Rotation (3-D)

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Rotation (3-D)

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Basic 2D Transformations

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Basic 2D Transformations

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Basic 2D Transformations

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Matrix Representation

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Matrix Representation

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2x2 Matrix

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Scaling

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Scaling Around A Point

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2x2 Matrix

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Shear (2-D)

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Shear (3-D)

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2x2 Matrix

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2x2 Matrix

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2x2 Matrix

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2D Translation

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Basic 2D Transformations

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Homogeneous Coordinates

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Matrix Composition

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Matrix Composition

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Matrix Composition

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Matrix Composition

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3D Transformations

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Basic 3D Transformations

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Basic 3D Transformations

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T1

T2

T3

root

P is a local coordinate

T3*P

T2*(T3*P)

T1*(T2(T3*P)) Final global coord. of P

Transform # (T1){ Transform { # (T2)

Transform { # (T3) } } }

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Transformation Hierarchies

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OpenGL transformation Matrices

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OpenGL transformation Matrices

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OpenGL transformation Matrices

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OpenGL transformation Matrices

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Transformation Example 1

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Transformation Example 2

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Transformation Example 2