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Math · Advanced Math
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It is basic, in computer graphics, to be able to move points about, in3-space, by combinations of translations and rotations. Translation is easily accomplished by the operator F(X)-X+ AX, where X-Ix, y, z]r is the position vector to the point and ΔΧ-[Ax, Δι, Δ2P is the transla- tion. However, whereasitis convenient, inthe computer software, to express all translations and rotations as matrix transformations, the operator Let F: R3R3 isTtot linear, and hence not expressible as a matrix transformation. To circumvent this difficulty we define X = [x,y, z, 1 nstead, where the fourth component, unity, 1s îneluded for convenience. Ihen we canexpress 4, which, if we pay attention to onlythe first three components, effects the translation by means of multiplication by T, where T is the 4*4 matrix in (2) (b(25 points) Show that a rotation about the z axis, followed by a translation, is effected by the composite transformation 1] LI Does the order of the operations matter? That is, is TRz RT? NOTE Composite Transformation: If F and G are transformations for V into W, then we define the linear combination of F and G, (aF + 6G: V-W, by for allxinV.Giventransform ationsF: U→V and G.V→Wwedefinethe composition of F and G, (GF): U-W , by (GF)(x) GFx)) For all x in U

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