III- 1 III 3D Transformation Homogeneous Coordinates The three dimensional point (x, y, z) is represented by the homogeneous coordinate (x, y, z, 1) In general, the homogeneous coordinate (x, y, z, w) represents the three dimensional point (x/w, y/w, z/w) The generalized transformation matrix:

III- 2 Scaling In general, this is done with the equations: x n = s x * x y n = s y * y z n = s z * z This can also be done with the matrix multiplication:

III- 3 Scaling can be done relative to the object center with a composite transformation Scaling an object centered at (c x, c y, c z ) is done with the matrix multiplication:

III- 4 Shearing Equivalent to pulling faces in opposite directions

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III- 9 Rotation Rotation can be done around any line or vector Rotations are commonly specified around the x, y, or z axis A positive angle of rotation results in a counterclockwise movement when looked at from the positive axis direction The matrix form for rotation –x axis

III- 10 –y axis –z axis

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III- 14 Reflection Reflection through the xy-plane: Reflection through the yz-plane: Reflection through the xz-plane:

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III- 17 Translations The amount of the translation is added to or subtracted from the x, y, and z coordinates In general, this is done with the equations: x n = x + t x y n = y + t y z n = z + t z This can also be done with the matrix multiplication:

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III- 23 Combining Transformations Matrices can be multiplied together to accomplish multiple transformations with one matrix A matrix is built with successive transformations occurring from right to left A combination matrix is typically built from the identity matrix with each new transformation added by multiplying it on the left of the current combination

III- 24 Rotation about an Arbitrary Axis in Space Assume an arbitrary axis in space passing through the point with direction cosines and rotation about this axis by some angle

III- 25 Direction cosines:

III- 26 The complete transformation is:

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III- 32 Reflection through an Arbitrary Plane

III- 33 The general transformation is:

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