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Using Eq. (2.7), the vertices of the reflected objects about the XY, YZ, XZ planes are determined and illustrated in Figure 2.12b to d, respectively. Taking as an example determination of the reflected object about the XY plane,

where [T]_{RE − XY} is given in Eq. (2.7) for the reflected object about the XY plane.

The coordinate transformation matrix for an object with a shearing transformation is represented as (TutorialsPoint 2018)

(2.8)

where [T]_SH is the shearing transformation matrix, d, g, i, b, c, f are the shearing factors of y along x, z along x, z along y, x along y, x along z, and y along z, respectively. A shearing is a relative angle deformation between two axes. Therefore, [T]_SH has to be symmetric, i.e. b = d, g = c, and i = f. In other words, [T]_SH only has three independent variables.

For a point P in its homogeneous coordinate (x, y, z, 1), Eq. (2.8) is applied to define its new position P′ (x′, y′, z′, 1) as

(2.9)