Consider a unit cube centered at The origin It is deformed a

Consider a unit cube centered at The origin. It is deformed according to The following expressions: X1 = 0.882 times 1 + 0.295 times 2 - 0.589 times 3 X2 = -0.72 times 1 + 0.72 times 2 X3 = 0.882 times 1 + 0.295 times 2 + 0.589 times 3 (5) Calculate F, B, and C for The deformation. (5) Does this deformation preserve volume? (5) Based on your calculation of C, what are The stretch matrix U and The rotation matrix R such that RU = F? (5) Describe in words The stretch associated with U. (5) About what axis is The rotation associated with R? (5) Based on your calculation of B, what are The stretch matrix V and The rotation matrix R such that VR = F? (5) The stretch associated with V for this problem is less obvious than that for U. Based on your understanding of how U, V, R, and F are related, discuss what The stretch for V is and why it is different from The stretch in U.

Solution

FOR SIMPLICITY & TO AVOID CONFUSION USING .. MATRIX X FOR X1,X2,X3 AND MATRIX Y FOR Y1=x1 , Y2=x2 , Y3=x3 ….. WE HAVE .. Y = A*X WHERE …. Y= X = A= Y1 X1 0.882 0.295 -0.589 Y2 X2 -0.72 0.72 0 Y3 X3 0.882 0.295 0.589 a)…. FIND F , B , C ….. PLEASE EXPLAIN YOUR SYMBOLS ..I SHALL RESPOND ON KNOWING WHAT YOU MEAN BY F , B , C … b)..LET US TAKE A CUBE OF 2 UNITS SIZE & CHECK … CUBE IS CENTERED AT ORIGIN SO IF WE CONSIDER ITS 4 VERTICES AS A,B,C,D CORRESPONDING TO ITS 3 COTERMINOUS EDGES AB ,AC & AD THEN THEY ARE GIVEN BY ... ORIGINAL DEFORMED X1 X2 X3 Y1 Y2 Y3 A -1 -1 -1 -1.177 0 -1.177 B 1 -1 -1 0.587 -1.44 0.587 C -1 -1 1 -1.766 0 -0.588 D -1 1 -1 -0.587 1.44 -0.587 AB= 2 0 0 1.764 -1.44 1.764 AC= 0 0 2 -0.589 0 0.589 AD= 0 2 0 0.59 1.44 0.59 VOLUME = [AB , AC , AD ] = 8             V \' = [A\'B\' , A\'C\' , A\'D\' ] = 3.993137 VOLUME IS NOT PRESERVED ………………ANSWER c , d , e , f , g … PLEASE EXPLAIN YOUR SYMBOLS ..I SHALL RESPOND ON KNOWING WHAT YOU MEAN BY F , B , C …