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I am reading an article by Vernon Chi on quaternions and rotations in 3-space. The title of the article is as follows:View attachment 3979
I am concerned that I do not follow the proof of one of the properties of unit quaternions in Section 3.1.3 of the article.
Section 3.1.3 reads as follows:
View attachment 3980
In the above text we read the following (regarding an important property of unit quaternions):
" ... ... A less obvious, but very useful one (property) is,\(\displaystyle Q_u = R_u cos \phi + P_u sin \phi = cos \phi + P_u sin \phi ... ... ... (1)
\)where \(\displaystyle R_u = (1,0,0,0)\) is a real quaternion and \(\displaystyle P_u = (0, i p_2 , j p_3 , k p_4 )\) is a vector unit quaternion ... ..."
Chi then presents what is meant to be a proof of the property (1) above ... indeed he shows that if (1) is true then \(\displaystyle | Q_u |^2 = 1 \) ... ... ... (2)follows.... ... BUT ... does he not have to show that for the property (1) to be true then we also have to show that if (2) is true then (1) follows ... ... Can someone please clarify this for me ...Further, if it is indeed the case that we need to show (2) implies (1) ... then can someone indicate how we would prove this ...Hope someone can help ...
Peter
To give readers an idea of the notation, definition and theory preceding the above section of Chi's article I am providing Sections 2 to 3.1.2 as follows:View attachment 3981
View attachment 3982
View attachment 3983
I am concerned that I do not follow the proof of one of the properties of unit quaternions in Section 3.1.3 of the article.
Section 3.1.3 reads as follows:
View attachment 3980
In the above text we read the following (regarding an important property of unit quaternions):
" ... ... A less obvious, but very useful one (property) is,\(\displaystyle Q_u = R_u cos \phi + P_u sin \phi = cos \phi + P_u sin \phi ... ... ... (1)
\)where \(\displaystyle R_u = (1,0,0,0)\) is a real quaternion and \(\displaystyle P_u = (0, i p_2 , j p_3 , k p_4 )\) is a vector unit quaternion ... ..."
Chi then presents what is meant to be a proof of the property (1) above ... indeed he shows that if (1) is true then \(\displaystyle | Q_u |^2 = 1 \) ... ... ... (2)follows.... ... BUT ... does he not have to show that for the property (1) to be true then we also have to show that if (2) is true then (1) follows ... ... Can someone please clarify this for me ...Further, if it is indeed the case that we need to show (2) implies (1) ... then can someone indicate how we would prove this ...Hope someone can help ...
Peter
To give readers an idea of the notation, definition and theory preceding the above section of Chi's article I am providing Sections 2 to 3.1.2 as follows:View attachment 3981
View attachment 3982
View attachment 3983