What are the utilizations of Inverse transformation? Provide the Inverse transformation for translation, shearing, reflection, scaling and rotation.
Solution: We have observed the basic matrix transformations for translation, reflection; shearing, scaling and rotation are the origin of the coordinate system. Via multiplying these fundamental matrix transformations, we can build complicated transformations, as rotation regarding to an arbitrary point, mirror reflection regarding a line etc. This process is termed as concatenation of matrices and the consequential matrix is frequently referred to as the composite transformation matrix. Inverse transformations play a significant role when you are dealing with a composite transformation. They appear to the rescue of fundamental transformations through making them applicable throughout the construction of composite transformation. You can seen that the Inverse transformations for shearing, reflection, scaling, rotation and translation have the subsequent relations, and v, θ, a, b, sx, sy, sz are each parameter concerned in the transformations.
1) T_{v} ^{-1} =T_{-v}
2) R_{θ} ^{-1} = R_{-θ}
3) (i) Sh_{x}^{-1}(a) =Sh_{x}(-a)
(ii) Sh_{y}^{-1}(b) = Sh_{x}(-b)
(iii) Sh_{xy}^{-1}(a,b) Sh_{x}(-a,-b)
4) S^{-1}_{sx,sy,sz} =S_{1/sx,1/sy,1/sz}
5) The transformation for mirror reflection regarding the principal axes does not change after inversion.
(i) Mx^{-1} =M_{-x}= M_{x}
(ii) My^{-1} =M_{-y}= M_{y}
(iii) Mz^{-1} =M_{-z}= M_{z}
6) The transformation for rotations made regarding to x,y,z axes have the subsequent inverse:
(i) R^{-1} _{x,θ} = R_{x,-θ} = R^{T}_{x,θ}
(ii) R^{-1}_{y,θ} = R_{y,-θ} = R^{T}_{y,θ}
(iii) R^{-1}_{z,θ} = R_{z,-θ} = R^{T}_{z,θ}