9-Transformations

9-Transformations - Modeling Transformations Jason Lawrence...

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Unformatted text preview: Modeling Transformations Jason Lawrence CS 4810: Graphics Acknowledgment: slides by Misha Kazhdan, Allison Klein, Tom Funkhouser, Adam Finkelstein and David Dobkin Overview • 2D Transformations Basic 2D transformations Matrix representation Matrix composition • 3D Transformations Basic 3D transformations Same as 2D Simple 2D Transformations Translation x y P P’ Simple 2D Transformations Translation x y Simple 2D Transformations Scale x z y Simple 2D Transformation Rotation x z y 2D Modeling Transformations Scale Rotate Translate Scale Translate x y World Coordinates Modeling Coordinates 2D Modeling Transformations x y World Coordinates Modeling Coordinates Let ʼ s look at this in detail… 2D Modeling Transformations x y Modeling Coordinates 2D Modeling Transformations x y Modeling Coordinates Scale .3, .3 Rotate -90 Translate 5, 3 2D Modeling Transformations x y Modeling Coordinates Scale .3, .3 Rotate -90 Translate 5, 3 2D Modeling Transformations x y Modeling Coordinates Scale .3, .3 Rotate -90 Translate 3, 5 World Coordinates Basic 2D Transformations • Translation: x ʼ = x + tx y ʼ = y + ty • Scale: x ʼ = x * sx y ʼ = y * sy • Rotation: x ʼ = x*cos Θ- y*sin Θ y ʼ = x*sin Θ + y*cos Θ Transformations can be combined (with simple algebra) Basic 2D Transformations • Translation: x ʼ = x + tx y ʼ = y + ty • Scale: x ʼ = x * sx y ʼ = y * sy • Rotation: x ʼ = x*cos Θ- y*sin Θ y ʼ = x*sin Θ + y*cos Θ Basic 2D Transformations • Translation: x ʼ = x + tx y ʼ = y + ty • Scale: x ʼ = x * sx y ʼ = y * sy • Rotation: x ʼ = x*cos Θ- y*sin Θ y ʼ = x*sin Θ + y*cos Θ x ʼ = x *sx y ʼ = y *sy (x,y) (x ʼ ,y ʼ ) Basic 2D Transformations • Translation: x ʼ = x + tx y ʼ = y + ty • Scale: x ʼ = x * sx y ʼ = y * sy • Rotation: x ʼ = x *cos Θ- y *sin Θ y ʼ = x *sin Θ + y *cos Θ (x ʼ ,y ʼ ) x ʼ = (x*sx) *cos Θ − (y*sy) *sin Θ y ʼ = (x*sx) *sin Θ + (y*sy) *cos Θ Basic 2D Transformations • Translation: x ʼ = x + tx y ʼ = y + ty • Scale: x ʼ...
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9-Transformations - Modeling Transformations Jason Lawrence...

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