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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