1. Math
  2. Advanced Math
  3. please explain every thing please write in the paper and...

Question: please explain every thing please write in the paper and...

Question details

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

NOTE: Similarly, rotations about the x axiz through an angle 0z, and about the y axiz through am angle 8y, are effected by 0 0|-|y 1 001] F(X) = Ry * X = | 0 And 0 0 01 F(x) = Rx*X-10 011 sx cx 0 0 1] respectively, where cx, sr, Cy, Sy denote cos r, sin 6, cos θ) sin θΥ, respectively (b)(25 points) Show that a rotation about the z axis, followed by a translation, is effected by the composite transformation Δ),l.ly 1] LI |sz 0 cz F(x) TRZX 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, (oF + ßG) : V→W, by rallxinV. Giventransformations F:U-Vand G: V-W,wedefinethecomposition of F and G, (GF): U-W, by for all x in U (GF)(x) Ξ G(E(x)

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

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution