Kvaternion q a b i c j d k displaystyle mathbf q a bi cj dk mozhna predstaviti u viglyadi pari skalyara ta 3 vimirnogo vektora q s v s a v b c d displaystyle mathbf q s vec v quad s a quad vec v b c d mnozhennya kvaternioniv bude virazhatis cherez skalyarnij ta vektornij dobutki 3 vimirnih vektoriv q 1 q 2 s 1 v 1 s 2 v 2 s 1 s 2 v 1 v 2 s 1 v 2 s 2 v 1 v 1 v 2 displaystyle mathbf q 1 q 2 s 1 vec v 1 s 2 vec v 2 s 1 s 2 vec v 1 cdot vec v 2 s 1 vec v 2 s 2 vec v 1 vec v 1 times vec v 2 Virazimo vektornij dobutok cherez dobutok kvaternioniv v 1 v 2 q 1 q 2 q 2 q 1 2 displaystyle vec v 1 times vec v 2 mathbf q 1 q 2 q 2 q 1 over 2 Povorot tochki navkolo osi v 3 vimirnomu prostorired Pokazhemo sho rezultatom povorotu vektora v displaystyle vec v nbsp na kut a displaystyle alpha nbsp vidnosno osi u displaystyle vec u nbsp odinichnij vektor bude v q v q 1 displaystyle mathbf v mathbf qvq 1 nbsp de v 0 v displaystyle mathbf v 0 vec v nbsp chisto vektornij kvaternion u 0 u displaystyle mathbf u 0 vec u nbsp chisto vektornij kvaternion q cos a 2 u sin a 2 q 1 q 1 q displaystyle mathbf q left cos frac alpha 2 vec u sin frac alpha 2 right qquad mathbf q 1 quad mathbf q 1 mathbf q nbsp Perepishemo ostannij kvaternion v inshij formi q cos a 2 u sin a 2 displaystyle mathbf q cos frac alpha 2 mathbf u cdot sin frac alpha 2 nbsp q 1 cos a 2 u sin a 2 displaystyle mathbf q 1 cos frac alpha 2 mathbf u cdot sin frac alpha 2 nbsp Spershu obchislimo neobhidnij nam viraz vikoristali vlastivist podvijnogo vektornogo dobutku u v u u v u u v u v 2 u u v displaystyle mathbf uvu vec u times vec v times vec u vec u cdot vec v vec u vec v 2 vec u vec u cdot vec v nbsp Obchislimo dobutok v cos a 2 u sin a 2 v cos a 2 u sin a 2 displaystyle mathbf v left cos frac alpha 2 mathbf u sin frac alpha 2 right mathbf v left cos frac alpha 2 mathbf u sin frac alpha 2 right nbsp v u v u sin 2 a 2 v cos 2 a 2 u v v u sin a 2 cos a 2 2 u u v v sin 2 a 2 v cos 2 a 2 u v 2 sin a 2 cos a 2 u u v 2 sin 2 a 2 v cos 2 a 2 sin 2 a 2 u v sin a u u v 1 cos a v cos a u v sin a u u v v u u v cos a u v sin a v v cos a u v sin a displaystyle begin matrix mathbf v amp amp mathbf uvu sin 2 frac alpha 2 amp amp mathbf v cos 2 frac alpha 2 amp amp mathbf uv vu sin frac alpha 2 cos frac alpha 2 amp amp 2 vec u vec u cdot vec v vec v sin 2 frac alpha 2 amp amp vec v cos 2 frac alpha 2 amp amp vec u times vec v 2 sin frac alpha 2 cos frac alpha 2 amp amp vec u vec u cdot vec v 2 sin 2 frac alpha 2 amp amp vec v cos 2 frac alpha 2 sin 2 frac alpha 2 amp amp vec u times vec v sin alpha amp amp vec u vec u cdot vec v 1 cos alpha amp amp vec v cos alpha amp amp vec u times vec v sin alpha amp amp vec u vec u cdot vec v amp amp vec v vec u vec u cdot vec v cos alpha amp amp vec u times vec v sin alpha amp amp vec v amp amp vec v bot cos alpha amp amp vec u times vec v bot sin alpha end matrix nbsp de v displaystyle vec v nbsp ta v displaystyle vec v bot nbsp komponenti vektora v displaystyle vec v nbsp paralelni i perpendikulyarni do u displaystyle vec u nbsp vidpovidno v v v displaystyle vec v vec v vec v bot nbsp u v displaystyle vec u vec v nbsp u v displaystyle vec u bot vec v bot nbsp Kozhen z troh dodankiv ye ortogonalnim do dvoh inshih Kilkist operacijred Obchislennya rezultatu dvoh povorotiv Zberigannya Mnozhennya Dodavannya Matricya povorotu R R 2 R 1 displaystyle mathbf R R 2 R 1 nbsp 9 27 18 Kvaternion q q 2 q 1 displaystyle mathbf q q 2 q 1 nbsp 4 16 12 Obchislennya povorotu tochki Zberigannya Mnozhennya Dodavannya Matricya povorotu v R v displaystyle mathbf v Rv nbsp 9 9 6 Kvaternion v q v q 1 displaystyle mathbf v qvq 1 nbsp 4 15 12Matricya povorotured Dokladnishe Matricya povorotu Povorotovi za dopomogoyu odinichnogo kvaterniona q a b i c j d k displaystyle mathbf q a bi cj dk nbsp vidpovidaye nastupna matricya povorotu R 1 2 c 2 d 2 2 b c 2 a d 2 a c 2 b d 2 a d 2 b c 1 2 b 2 d 2 2 c d 2 a b 2 b d 2 a c 2 a b 2 c d 1 2 b 2 c 2 displaystyle mathbf R begin pmatrix 1 2 c 2 d 2 amp 2bc 2ad amp 2ac 2bd 2ad 2bc amp 1 2 b 2 d 2 amp 2cd 2ab 2bd 2ac amp 2ab 2cd amp 1 2 b 2 c 2 end pmatrix nbsp Yaksho predstavimo kvaternion u viglyadi q cos a 2 u sin a 2 u x y z displaystyle mathbf q left cos frac alpha 2 vec u sin frac alpha 2 right quad vec u x y z nbsp todi R u a u u T I u u T cos a u sin a displaystyle mathbf R vec u alpha mathbf uu T I mathbf uu T cos alpha big mathbf u big times sin alpha nbsp Dodanki identichni dodankam iz formuli otrimanoyi cherez kvaternioni Dlya sproshennya obchislen zvedemo podibni dodanki ta vernemos do vektornoyi formi formula povorotu Rodrigesa v u u v 1 cos a v cos a u v sin a displaystyle mathbf v mathbf u mathbf u cdot mathbf v 1 cos alpha mathbf v cos alpha mathbf u times mathbf v sin alpha nbsp Pershij ta drugij dodanki vzhe ne ye obov yazkovo ortogonalnimi Otrimano z https uk wikipedia org w index php title Kvaternioni i povoroti prostoru amp oldid 30174802