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Math · Advanced Math
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It is basic, in computer graphics, to be able to move points about, in 3-space, by combinations of translations and rotations. Translation is easily accomplished by the operator where [x, y, 2] T Is the position vector to the point and ΔΧ-[Ax, Δ3, Δ2] T IS the transla tion. However, whereas it is convenient,inthecomputersoftware,to express all translations and rotations as matrix transformations, the operator Let F: R3 R3 is not linear, and hence not expressible as a matrix transformation. To circumvent this difficulty we define X = [x, y, z, 1 instead, where the fourth component, unity, is included for convenience. Then we can express F(x)- TX 0 1 0 ΔΧΙ.lx2 Ιγ+Δγ (2 4, which, if we pay attention to only the first three components, effects the translation by means of multiplication by T, where T is the 4*4 matrix in (2) (a)(25 points) Show that 0 0 1 0Z 0 0 0 1] LI effects a rotation about thez axis through an angle 0, taken according to the right-hand rule, where cz, sz are shorthand for cos θ, sin 6, respectively HINT: Letting x = r cos θ, y = r sin θ, show that

through an angle By, are effected by And 0 0 01 LO 0 0 1]L1」 respectively, where cx, so cy, Sy denote cos θ, sin θο cos θ» sin θ) respectively.

Please explain every thing.
Please write in the paper and then take a photo.

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