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To determine the shearing matrix, six independent variables must be determined in Eq. (2.9). Since the positions of nodes 1 and 5 are known before and after shearing, Eq. (2.9) is applied to determine the conditions that these variables must satisfy:

(2.10)

Solving Eq. (2.10) gives b = d = c = g = f = i = 0.5.

Figure 2.13b shows the sheared object whose vertices after shearing are determined as

The elements a to j in the reflection transformation matrix of Eq. (2.7) can also be determined for a rotation transformation matrix along an axis in a 3D space. The corresponding matrix can be derived from the rotational matrices for the points in Table 2.1. Figure 2.14 shows the rotational transformation matrices when x, y, and z are chosen as a rotational axis for a rotational angle of θ_x, θ_y, and θ_z, respectively.

Figure 2.14 Rational coordinate transformation of an object. (a) Object in the original position. (b) The x‐axis rotation (θ_x). (c) The z‐axis rotation (θ_z). (d) The y‐axis rotation (θ_y).

A series of the coordinate transformations, such as translation, scaling, shearing, rotation, and reflection, can be combined as a comprehensive transformation of a point or object. In such a case, the combined transformation matrix can be determined as

(2.11)

where [T]_COM is the combined transformation matrix, i = 1, 2, …, N is the order of N transformation operations, and [T]_i is the transformation matrix for the ith operation.

Note that the order of the multiplications of the matrices is reversed with that of the operations.