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CMPE 466 COMPUTER GRAPHICS Chapter 9 3D Geometric Transformations Instructor: D. Arifler

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3D translation

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3D rotation

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3D z-axis rotation

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Rotations To obtain rotations about other two axes x  y  z  x E.g. x-axis rotation E.g. y-axis rotation

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General 3D rotations

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Arbitrary rotations

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Arbitrary rotations

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Rotations

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Rotations

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Rotations Two steps for putting the rotation axis onto the z-axis Rotate about the x-axis Rotate about the y-axis

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Rotations Projection of u in the yz plane Cosine of the rotation angle where Similarly, sine of rotation angle can be determined from the cross-product

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Rotations Equating the right sides where |u’|=d Then,

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Rotations Next, swing the unit vector in the xz plane counter-clockwise around the y-axis onto the positive z-axis

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Rotations

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Rotations

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In general

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Quaternions Scalar part and vector part Think of it as a higher-order complex number Rotation about any axis passing through the coordinate origin is accomplished by first setting up a unit quaternion where u is a unit vector along the selected rotation axis and θ is the specified rotation angle Any point P in quaternion notation is P=(0, p) where p=(x, y, z)

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Quaternions The rotation of the point P is carried out with quaternion operation where This produces P’=(0, p’) where Many computer graphics systems use efﬁcient hardware implementations of these vector calculations to perform rapid three-dimensional object rotations. Noting that v=(a, b, c), we obtain the elements for the composite rotation matrix. We then have

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Quaternions Using With u=(ux, uy, uz), we finally have About an arbitrarily placed rotation axis: Quaternions require less storage space than 4 × 4 matrices, and it is simpler to write quaternion procedures for transformation sequences. This is particularly important in animations, which often require complicated motion sequences and motion interpolations between two given positions of an object.