3DTransform - 3D Geometric Transformations Chapter 5 Intro....

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3D Geometric Transformations Chapter 5 Intro. to Computer Graphics Spring 2009, Y. G. Shin
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3D Transformation 1 1 0 0 0 ' ' ' 3 2 1 3 2 1 3 2 1 z y x t z z z t y y y t x x x h z y x z y x z x y Right-handed coordinate system
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3D Transformation 1 0 0 0 1 0 0 0 1 0 0 0 1 z y x t t t Translation 1 0 0 0 0 0 0 0 0 0 0 0 0 z y x s s s Scaling
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3D Rotation 1 0 0 0 0 1 0 0 0 0 cos sin 0 0 sin cos ) ( z R 3D rotations do NOT commute! 1 0 0 0 0 cos sin 0 0 sin cos 0 0 0 0 1 ) ( x R
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Rotation About Arbitrary Axis
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Rotation about an arbitrary axis 1. Translation : rotation axis passes through the origin 2. Make the rotation axis on the z-axis ) ( ) ( y x R R 3. Do rotation ) ( z R 4. Rotation & translation ) ( ) ( 1 1 1 x y R R T T R R R R R T R x y z y x ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 Rotation about an arbitrary axis ) , , ( 1 1 1 z y x 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 z y x T
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Rotation About Arbitrary Axis Rotate u onto the z-axis
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Rotation About Arbitrary Axis Rotate a unit vector u onto the z-axis u’ : Project u onto the yz-plane to compute angle u’’ : Rotate u about the x-axis by angle Rotate u’’ onto the z-asis
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Rotation About Arbitrary Axis Rotate u’ about the x-axis onto the z-axis Let u =(a,b,c) and thus u’ =(0,b,c) Let u z =(0,0,1) 2 2 cos c b c z z u u u u b sin x z x z u u u u u u 2 2 sin c b b b z u u x z y u ) 1 , 0 , 0 ( z u u’
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Rotation About Arbitrary Axis Rotate u’ about the x-axis onto the z-axis Since we know both cos and sin , the rotation matrix can be obtained Rotate u’’ onto the z-asis With the similar way, we can compute the angle 1 0 0 0 0 0 0 0 0 0 0 1 ) ( 2 2 2 2 2 2 2 2 c b c c b b c b b c b c x R x z y u u’ u’’
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Rotation about an arbitrary axis using orthogonal matrix Unit row vector of R rotates into the principle axes x, y, and z.
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3DTransform - 3D Geometric Transformations Chapter 5 Intro....

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