| Topic I | Introduction & Summary for this Appendix |
| Topic II | Setting up the Planar Mapping |
| Topic III | A First Analysis |
| Topic IV | Finding the Maximum |
| Topic V | When { Beta } is [ 90 degrees ] |
This Appendix attempts to find the maximum when a { unit circle } is mapped onto an { ellipse } in a 2-Dimensional ( 2-D ) { linear mapping } process.
Let us look at this Planar Mapping :
where :
& 
respectively, in { Reference Frame B } on-the-right ,
are mapped onto :
& 
, respectively, in { Reference Frame A } on-the-left .
Let the point, { 
} , be any point on the { unit circle } , in { Reference Frame B } on-the-right :
such that :
is given by the equation :
This { unit-vector } , in { Reference Frame B } on-the-right, is then mapped onto the vector { 
} in { Reference Frame A } on-the-left :
Let us now introduce a pair of orthogonal { unit-vectors } , 
& 
respectively, in { Reference Frame A } on-the-left , as per this diagram below :
such that :
is in the same direction as the vector .
Let us now introduce the scalar values { 
} , { 
} and { 
} respectively, such that :
}is the length of the vector ,
}is the length of the vector , and
} is the length of the vector .
We can then write :
Let us recall, from above, that :
Substituting therein then yields :
Further yielding :
We can then write the following equation for the value of 
:
Consolidating terms then yields :
Let us now try to determine the value of { 
} such that the value { } is maximized.
We then take the { 1st derivative } of with respect to { } :
yielding :
And finally :
We can then set this { 1st derivative } equal to [ zero ] to find the maximum , i.e. setting :
yielding :
And we come up with this relation to find the value of { } at the maximum :
We note here that when 
is exactly { 90 degrees } , or { 270 degrees } , then :
yielding :
And consequently the value of is either { 0 degrees } , { 90 degrees } , { 180 degrees } , or { 270 degrees } .
Otherwise, i.e. for values of not equal to { 90 degrees } , { 270 degrees } , or equivalent , the value of cannot be { 0 degrees } or { a multiple of 90 degrees } .