3d_transforms - CAP4730: Computational Structures in...

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CAP4730: Computational Structures in Computer Graphics 3D Transformations
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Outline 3D World What we are trying to do Translate Scale Rotate
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Transformations in 3D! Remembering 2D transformations -> 3x3 matrices, take a wild guess what happens to 3D transformations. ( 29 ( 29 = + = = + = 1 0 0 0 1 0 0 0 1 0 0 0 1 , , 1 0 0 1 0 0 1 , z y x z y x z y x y x y x y x t t t t t t z y x t t t T t t t t y x t t T T=(t x , t y , t z )
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Scale, 3D Style ( 29 ( 29 = = = = 1 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0 0 0 0 0 , , 1 0 0 0 0 0 0 * 0 0 , z y x z y x z y x y x y x y x s s s z y x s s s s s s S s s y x s s s s S S=(s x , s y , s z )
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Rotations, in 3D no less! R=(r x , r y , r z , θ ) What does a rotation in 3D mean? Q: How do we specify a rotation? A: We give a vector to rotate about, and a theta that describes how much we rotate. θ Q: Since 2D is sort of like a special case of 3D, what is the vector we’ve been rotating about in 2D?
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Rotations about the Z axis R=(0,0,1, θ ) What do you think the rotation matrix is for rotations about the z axis? θ ( 29 - = - = - = 1 0 0 0 0 1 0 0 0 0 cos sin 0 0 sin cos ) , 1 , 0 , 0 ( 1 0 0 0 cos sin 0 sin cos cos sin sin cos θ R R
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Rotations about the X axis R=(1,0,0, θ ) Let’s look at the other axis rotations θ - = 1 0 0 0 0 cos sin 0 0 sin cos 0 0 0 0 1 ) , 0 , 0 , 1 ( θ R
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Rotations about the Y axis R=(0,1,0, θ ) θ - = 1 0 0 0 0 cos 0 sin 0 0 1 0 0 sin 0 cos ) , 0 , 1 , 0 ( θ R
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Rotations for an arbitrary axis - = 1 0 0 0 0 1 0 0 0 0 cos sin 0 0 sin cos ) , 1 , 0 , 0 ( θ R - = 1 0 0 0 0 cos 0 sin 0 0 1 0 0 sin 0 cos ) , 0 , 1 , 0 ( R - = 1 0 0 0 0 cos sin 0 0 sin cos 0 0 0 0 1 ) , 0 , 0 , 1 ( R
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( 29 ( 29 ( 29 ( 29 ( 29 ( 29 α β θ x y z y x R R R R R R = - - 1 1 Rotations for an arbitrary axis α β θ Steps: 1. Normalize vector u 2. Compute α 3. Compute β 4. Create rotation matrix u
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Given a vector v , we want to create a unit vector that has a magnitude of 1 and has the same direction as v . Let’s do an example. V
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3d_transforms - CAP4730: Computational Structures in...

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