Operators: Translations, Rotations & Transformations

154. OPERATORS: TRANSLATIONS, ROTATIONS &TRANSFORMATIONSMotion, such as, translating points, rotating vectors or doing bothcan be described by certain matrix operators.A. TRANSLATION OPERATOR:figure Given frame {A}, we want to move point P1 to point P2. By theAP2=^P+4Q{A}P,P1AP2=AP,AP1QA2vector 4Q to AP1.In other words, moving point P1 to point P2 in equivalent to addingSince 16Translation can be accomplished by the matrix operator(7)Where is a homogeneous transformation, defined byqx9,9.=(8)000:110-m9xq^§- - The unit vector along ^Q/Q|- The length of 1QRemark:->Since P,P2= Q, moving a point from one location to another is thesame as the transform which describes a frame obtained bytranslating frame {A} by ^ Q (See 3)Assignment #1Chapters: 2,4,5,13,16 17B. ROTATION OPERATOR:Given {A}, we rotated a vector A P about a vector K by O degreesand call the resulting vector AP2. Denote this action as operatorROT(R,0), we writeK{A}P20AP2P10AP1It is expected that one can find a rotation matrix, denoted by R, suchthatAP2=R.^ P1Hence operator R 0(9) 18Question: How to find R?Special Cases100ROT(X,0)=cos@-sin Gsin Gcose(10 a)0001cos 00sin00ROT(1,0)=FIF-sin 9 0 cose(10 b)0cos 0-sin600ROT(9,0)=thesin Ocos 000(10 c)0 19To show (10 a), consider a vector AP1, we rotate AP about X-axisby 0 and denote the resulting vector AP2. Now if we also rotateframe {A} about X-axis by 0 and denote the resulting frame by{B}, then obviously we would haveAP,=" P2(11)ZA{B}{A}P20P1BP2P,YB4eYXThereforeP2=ARBP2=ARAP,AP2=ARAPThus ROT(X,0)= 1(12)Remark:We conclude that operator (9) which rotates a vector by R is thesame as the transform which describes a frame rotated by R relativeto reference frame {A} 20Example:given {A} and a vectorCompute the coordinates of vector P2 obtained by rotating AP aboutY, by -45°.Solution:where: 0RROT(91,-45°)=ZA{A}{B}-45°XB-45°-45°45P,-1/2P18X 21000n-182R=AR with B obtained by the same rotation, i.e., 22VECTOR: C.A GENERAL OPERATOR THAT ROTATES AND TRANSLATES AGiven {A} and a vector AP1. Let T be an operator that rotates thethen translates it (by vector Q) to get a new vector ^P2 i.e.vector about a rotation axis by 0 (through a rotation matrix R) and[AR]-1[R]the0 1 1Remarks:1. It follows from (a) and (b) that T is same as the transformationwhich describes a frame rotated by R and translated by Q relativeto the reference frame {A}2. The two actions does not commute sinceR:0:U#[: i