第二章坐标变换

1、APxAyzpPpp,xyzp p p(cylindrical)(cylindrical) spherical)spherical) ,BBBxyz ABRcos(,)cos(,)cos(,)cos(,)cos(,)cos(,)cos(,)cos(,)cos(,BABABAAAAABBBBBABABABABABABABABABABABABABABAxxyxzxRxyzxyyyzyxzyzzzxxyxzxxyyyzyxzyzzz111213212223313233)rrrrrrrrr100( , )0cossin0sincosR xcos0sin( , )010sin0cosR ycossin0

2、( , )sincos0001R zPABzz、AxByABzzPP、BxPAxPByPBxAyAyPAPBPcossinsincosABBxxyABByxyABzzppppppppcs0sc0001ABxxAByyABzzpppppp( , )AABBPR zP1,1AATABBBRRR10AAABBBAAAAAABBBBBBxyzxyyzzx AXAYAZ 00100( , )0010000100ABXYZcscsRsccsscscc cc s cs cc s cs ss cs s sc cs s cc ssc sc c ,scsin,cosBZBYBX BBBB ( )( )( )001

3、000010000100AABBBZ Y XBBBZYXRR R RRRRcscssccsscscc cc s cs cc s cs ss cs s sc cs s cc ssc sc c AK (, )ABR K( )KRAK1 cosv ,ATxyzKk kk111213212223313233( )xxxyzxzyABKxyzyyyzxxzyyzxzzk k vck k vk sk k vk sRk k vk sk k vck k vk sk k vk sk k vk sk k vcrrrrrrrrr Example: A Frame B is described as initiall

4、y coincident with A, then be rotated about the vector (passing through the origin) by degrees. Give the frame description of B. The frame description of B is: 300.707,0.70,0ATK 0.9330.0670.3540.0670.9330.3540.3540.3540.866xxxyzxzyABxyzyyyzxxzyyzxzzk k vck k vk sk k vk sRk k vk sk k vck k vk sk k vk

5、sk k vk sk k vc ,AABBORGBRPBARABORGPABORGPABORGPBPAPABABORGPPPABORGPABORGPAPBPAABBPR PABRBAR1AAATBBBRRRABR Example: Frame B is rotated relative to frame A about Z by 30 degrees. Here Z is pointing out of the page. Writing the rotation matrix, we obtain: Given We calculate The original vector P is no

6、t changed in space, we compute a new description of the vector relative to another frame.cossin00.8660.5000.000sincos00.5000.8660.0000010.0000.0001.000ABR0.02.00.0BP1.0001.7320.000AABBPR P Example: Consider two rotations, one about Z by 30 degrees and one about X by 30 degrees: The fact that the ord

7、er of rotations is important should not be surprising; furthermore, it is captured in the fact that we use matrices to represent rotations, because multiplication of matrices is not commutative in general.0.8660.5000.000(30.0)0.5000.8660.0000.0000.0001.000ZR1.0000.0000.000(30.0)0.0000.8660.5000.0000

8、.5000.866XR0.870.430.250.870.500.00(30.0)(30.0)0.500.750.43(30.0)(30.0)0.430.750.50.000.500.870.250.430.87ZXXZRRRRAPAABABBORGPR PPBPAPABRABORGP5,9,0TBP=000012cos30sin3000.8660.5000.0006,sin30cos3000.5000.8660.00000010.0000.0001.000AABORGBPR轾轾轾-犏犏犏犏犏犏=犏犏犏犏犏犏犏臌臌臌0.8660.5000.000 51211.0980.5000.8660.00

9、0 9613.5620.0000.0001.000 000AABABBORGPR PP轾轾轾轾-犏犏犏犏犏犏犏犏=+=+=犏犏犏犏犏犏犏犏臌臌臌臌BBCBCCORGPR PP()AABAABCBABBORGBCCORGBORGPR PPRR PPPBPAPBPAABBPT P100011AAABBBPRPP3 30010101AAAABBORGBORGBRPIPR_1000100010001OOOABxAByABABzPPTP0000Rot( , )00100001ABcsscTz000100Rot( , )000001ABcsTysc100000Rot( , )000001ABcsTxscB

10、BCCPT PAABABCBBCPT PT T P01ABABAAABBCBCORGBORGCBCR RR PPTT T()AABAABCBABBORGBCCORGBORGPR PPRR PPP(4, 3,7)( ,90)( ,90)1004001001000014010301001000100300171000001001070001000100010001ABTTransRot yRot z0014761003340107210000111AABBPT P ( ,90)(4, 3,7)( ,90)0010100401000017010001031000100310000017001001040001000100010001ABTRot yTransRot z001779100334010421000111AABBPT P BCORGP1BAATABBRRRBBAAT AAORGABORGBBORGPR PRP 01ATAT ABBBBORGARRPTAZAXAY0.8660.5000.0004.00.5000.8660.0003.00.0000.0001.0000.00001ABT0.8660.5000.0004.9640.5000.8660.0000.5980.0000.0001.0000.00 0 010001ATAT ABBBBORGARRPTUUAUUB