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03-Xfmns_2Arbitrary rotations, nonlinear deformation

03-Xfmns_2Arbitrary rotations, nonlinear deformation -...

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Arbitrary rotations, nonlinear deformations.

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Rotation about an arbitrary axis Rotate vector a by around a unit axis r r r r
Rotation about an arbitrary axis Rotation is linear ) ( ) ( ) ( ) ( ) ( a rot a rot b rot a rot b a rot b a b a ) ( b rot ) ( a rot ) ( b a rot

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Rotation about an arbitrary axis Define a projection operator '\': b b b a b a b a \ b a \ b a
Rotation about an arbitrary axis Create an orthonormal basis: 1 3 2 1 3 \ \ e e e r a r a e r e

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Rotation about an arbitrary axis Decompose before rotating: r r a a \ r r a a r r a r e e r a r e rot r a r e r a rot r rot r a rot r r a rot a rot ) ( ) ( sin ) \ ( cos ) sin (cos \ ) ( \ ) \ ( ) ( ) \ ( ) \ ( ) ( 2 1 1 1
An alternative view We can view the rotation around an arbitrary axis as a set of simpler steps We know how to rotate and translate around the world coordinate system Can we use this knowledge to perform the rotation?

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03-Xfmns_2Arbitrary rotations, nonlinear deformation -...

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