What are some examples of quaternions in rectangular and exponential form?












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I've been interested in quaternions for their ability to represent positions and motions in spacetime in application to game development; I have searched for articles regarding representations of quaternions online for a few days, and a bit of it helped, but my knowledge of higher mathematics is limited and I struggled to make sense of many of the equations shown in these articles. I realize the study of quaternions is a deep subject closely linked with trigonometry and linear algebra, but at the moment my only interest is in the polar/exponential representation of quaternions.



During my search for information on this notion, I found an article that illustrated a particular form of expressing quaternions,
'Quaternion polar representation with a complex modulus and
complex argument inspired by the Cayley-Dickson form' by Stephen J. Sangwine and Nicolas Le Bihan



In the introduction of this document, it says any quaternion of the form $q = a+bhat i+chat j+dhat k$ may be expressed as $q = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$.



This article, 'Hypercomplex Signals—A Novel Extension of the
Analytic Signal to the Multidimensional Case' by Thomas Bülow and Gerald Sommer, states the same idea in part five of the document.



I've read through these articles and others, taking what I can from them with my current insight on trigonometry, but I have yet to grasp how to correctly convert a quaternion from rectangular form $a+bhat i+chat j+dhat k$ to the portrayed exponential form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$. I feel that seeing some examples of quaternions in both forms would help me understand the conversion.



Thus, given that $q = a+bhat i+chat j+dhat k = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$, what would the following quaternions be in the form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$?




  • $3+4hat i+2hat j+3hat k$

  • $3hat i+4hat j$

  • $5hat i+3hat j+hat k$






quaternions







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    I've been interested in quaternions for their ability to represent positions and motions in spacetime in application to game development; I have searched for articles regarding representations of quaternions online for a few days, and a bit of it helped, but my knowledge of higher mathematics is limited and I struggled to make sense of many of the equations shown in these articles. I realize the study of quaternions is a deep subject closely linked with trigonometry and linear algebra, but at the moment my only interest is in the polar/exponential representation of quaternions.



    During my search for information on this notion, I found an article that illustrated a particular form of expressing quaternions,
    'Quaternion polar representation with a complex modulus and
    complex argument inspired by the Cayley-Dickson form' by Stephen J. Sangwine and Nicolas Le Bihan



    In the introduction of this document, it says any quaternion of the form $q = a+bhat i+chat j+dhat k$ may be expressed as $q = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$.



    This article, 'Hypercomplex Signals—A Novel Extension of the
    Analytic Signal to the Multidimensional Case' by Thomas Bülow and Gerald Sommer, states the same idea in part five of the document.



    I've read through these articles and others, taking what I can from them with my current insight on trigonometry, but I have yet to grasp how to correctly convert a quaternion from rectangular form $a+bhat i+chat j+dhat k$ to the portrayed exponential form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$. I feel that seeing some examples of quaternions in both forms would help me understand the conversion.



    Thus, given that $q = a+bhat i+chat j+dhat k = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$, what would the following quaternions be in the form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$?




    • $3+4hat i+2hat j+3hat k$

    • $3hat i+4hat j$

    • $5hat i+3hat j+hat k$






    quaternions







    share|cite|improve this question











    $endgroup$















      2












      2








      2


      1



      $begingroup$


      I've been interested in quaternions for their ability to represent positions and motions in spacetime in application to game development; I have searched for articles regarding representations of quaternions online for a few days, and a bit of it helped, but my knowledge of higher mathematics is limited and I struggled to make sense of many of the equations shown in these articles. I realize the study of quaternions is a deep subject closely linked with trigonometry and linear algebra, but at the moment my only interest is in the polar/exponential representation of quaternions.



      During my search for information on this notion, I found an article that illustrated a particular form of expressing quaternions,
      'Quaternion polar representation with a complex modulus and
      complex argument inspired by the Cayley-Dickson form' by Stephen J. Sangwine and Nicolas Le Bihan



      In the introduction of this document, it says any quaternion of the form $q = a+bhat i+chat j+dhat k$ may be expressed as $q = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$.



      This article, 'Hypercomplex Signals—A Novel Extension of the
      Analytic Signal to the Multidimensional Case' by Thomas Bülow and Gerald Sommer, states the same idea in part five of the document.



      I've read through these articles and others, taking what I can from them with my current insight on trigonometry, but I have yet to grasp how to correctly convert a quaternion from rectangular form $a+bhat i+chat j+dhat k$ to the portrayed exponential form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$. I feel that seeing some examples of quaternions in both forms would help me understand the conversion.



      Thus, given that $q = a+bhat i+chat j+dhat k = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$, what would the following quaternions be in the form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$?




      • $3+4hat i+2hat j+3hat k$

      • $3hat i+4hat j$

      • $5hat i+3hat j+hat k$






      quaternions







      share|cite|improve this question











      $endgroup$




      I've been interested in quaternions for their ability to represent positions and motions in spacetime in application to game development; I have searched for articles regarding representations of quaternions online for a few days, and a bit of it helped, but my knowledge of higher mathematics is limited and I struggled to make sense of many of the equations shown in these articles. I realize the study of quaternions is a deep subject closely linked with trigonometry and linear algebra, but at the moment my only interest is in the polar/exponential representation of quaternions.



      During my search for information on this notion, I found an article that illustrated a particular form of expressing quaternions,
      'Quaternion polar representation with a complex modulus and
      complex argument inspired by the Cayley-Dickson form' by Stephen J. Sangwine and Nicolas Le Bihan



      In the introduction of this document, it says any quaternion of the form $q = a+bhat i+chat j+dhat k$ may be expressed as $q = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$.



      This article, 'Hypercomplex Signals—A Novel Extension of the
      Analytic Signal to the Multidimensional Case' by Thomas Bülow and Gerald Sommer, states the same idea in part five of the document.



      I've read through these articles and others, taking what I can from them with my current insight on trigonometry, but I have yet to grasp how to correctly convert a quaternion from rectangular form $a+bhat i+chat j+dhat k$ to the portrayed exponential form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$. I feel that seeing some examples of quaternions in both forms would help me understand the conversion.



      Thus, given that $q = a+bhat i+chat j+dhat k = |q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$, what would the following quaternions be in the form $|q|e^{hat iphi}e^{hat kpsi}e^{hat jtheta}$?




      • $3+4hat i+2hat j+3hat k$

      • $3hat i+4hat j$

      • $5hat i+3hat j+hat k$






      quaternions




      quaternions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited Jan 4 at 3:47









      Andrei

      11.7k21026




      11.7k21026










      asked Jan 3 at 22:35









      UnknownUnknown

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          Since $phi$, $psi$ and $theta$ are just Euler angles you can easily convert back and forth from quaternions to Euler angles following the formulae showed in wikipedia:



          https://en.m.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles






          share|cite|improve this answer









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            $begingroup$

            Since $phi$, $psi$ and $theta$ are just Euler angles you can easily convert back and forth from quaternions to Euler angles following the formulae showed in wikipedia:



            https://en.m.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              Since $phi$, $psi$ and $theta$ are just Euler angles you can easily convert back and forth from quaternions to Euler angles following the formulae showed in wikipedia:



              https://en.m.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                Since $phi$, $psi$ and $theta$ are just Euler angles you can easily convert back and forth from quaternions to Euler angles following the formulae showed in wikipedia:



                https://en.m.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles






                share|cite|improve this answer









                $endgroup$



                Since $phi$, $psi$ and $theta$ are just Euler angles you can easily convert back and forth from quaternions to Euler angles following the formulae showed in wikipedia:



                https://en.m.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 9 at 16:50









                Mauricio Cele Lopez BelonMauricio Cele Lopez Belon

                57748




                57748






























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