User:Tohline/Appendix/Ramblings/ConcentricEllipsodalDaringAttack
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Daring Attack
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Background
Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of the socalled T6 (concentric elliptic) coordinate system, here we take a somewhat daring attack on this problem, mixing our approach to identifying the expression for the third curvilinear coordinate. Broadly speaking, this entire study is motivated by our desire to construct a fully analytically prescribable model of a nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids.
Direction Cosine Components for T6 Coordinates  
          

As before, let's adopt the firstcoordinate expression,



but for the thirdcoordinate expression we will abandon the trigonometric expression and instead simply use,



This modified thirdcoordinate expression means that the last row of the above table changes, as follows.
Daring Attack  
          
Notice that the direction cosine functions for the (as yet, unknown) secondcoordinate function remain the same. This is because the directioncosine functions associated with both and remain unchanged, so it must be true that the cross product of the first and third unit vectors leads to the same components for the second unit vector.
New Approach
Setup
The surface of an ellipsoid with semimajor axes (a, b, c) is defined by the expression,



This is identical to our expression for if we make the associations,



Now, given that does not functionally depend on , let's consider that the choice of is tightly associated with the specification of the second coordinate, . Specifically, let's adopt the definition,



in which case, we see that,



and,






[Note that in the case of spherical coordinates (q^{2} = p^{2} = 1), , and this "second" coordinate, , becomes .] Combining this last expression with the relationship that is provided by the definition of , gives,



In general, the exponent of that appears in the first term on the righthand side of this expression prevents us from being able to analytically prescribe the function, . But a solution is obtainable for selected values of .
Examine the Case: q^{2} = 2
If we set , then this last combined expression becomes a quadratic equation for . Specifically, we find,









(Note that, for reasons of simplicity for the time being, in this last expression we have retained only the "positive" solution.) Again, calling upon the relationship that is provided through the definition of , we find (when q^{2} = 2),









Summary (q^{2} = 2)

For convenience, we have defined,






Test Example 



Do we get the correct values of ?
 
Evaluate a few partial derivatives …
This matches the numerical value for as determined below, but it does not match the numerical value obtained previously (0.730058) for . The most likely piece that needs adjustment is the partial of "z" with respect to λ_{1}. It needs to be … .
Alternatively,

Next, let's examine all nine partial derivatives, noting at the start that,









We have,






























What about the derived scalefactors?












Written in terms of Cartesian coordinates, this becomes,












Note that,






Hence, the scale factor becomes,












Compare this expression with the one derived earlier, namely,



Well … first we recognize that, when q^{2} = 2,



Hence,









which means,



Think Again
Firm Relations
In addition to the functions that are specified in our above Daring Attack Table, we appreciate that,









Check …


(Yes!) 
Also,









Check …


(Yes!) 
And, last …









Speculation
First
From the directioncosine expressions for that have been summarized in our above Daring Attack Table, it seems reasonable to suggest that,



in which case,









and,









Second
Alternatively, after examining the directioncosine expressions for that we have just provided, one might suggest that,



in which case, the expressions provided for and must be swapped relative to our First speculation.
Third
Noticing that is proportional to and that is inversely proportional to , let's consider both as possible behaviors for the 2^{nd} scale factor. Let's try the first of these behaviors. Specifically, what if we assume …









Then,






Primary implication:
Secondary implication:

Now, what specifically is the function, ? Start by rewriting the three partial derivatives as,









Suppose that,



Then we have,



and, 


Great! 
But this cannot be the correct expression for because,



which does not match the desired partial derivative with respect to .
Fourth
Alternatively, if we assume …









then,


















Let's check for consistency with one of the directioncosines.












This does not match the term in the expression for — namely, — that is expected from the original tabulation.
Better Organized
From our above Daring Attack Table, we appreciate that the three direction cosines associated with the (as yet unknown) second curvilinear coordinate are,









It is easy to see that the desired orthogonality relationship,



is satisfied because,



Now, as we attempt to determine the functional form of the second curvilinear coordinate, , a seemingly useful intermediate step is to determine the functional form of each of the three partial derivatives of this key coordinate function, namely, , for i = 1, 3. Here, we will accomplish this intermediate step by guessing the functional form of the second scale factor, , then applying the relation,



Notice that, without violating the abovestate orthogonality relationship, we can adopt virtually any functional form for and deduce that,









as long as,



This key, leading coefficient function is unity — and, hence, is independent of position — if, as in our First speculation above, we guess that . If, as in our Second speculation above, we guess that , we find that, . Our above Third speculation is replicated if we guess that ; we immediately see that, in this Third case,









Study the Functional Forms
We know the functional forms of two of the desired curvilinear coordinates, namely,






but we do not yet have a valid expression for the 2^{nd} coordinate, . Nevertheless, let's see if we can guess the functional forms for , by inverting the two known curvilinearcoordinate functions. As a starting point, let's impose the following mappings:









This means, for example, that,






Derivatives of x












Struggling
I have noticed that, in this last set of expressions, there are recurring terms of the form, and . So, while keeping the same definition of the ccordinate, , let's replace and with a pair of coordinates defined as follows:

and, 

This means that,









Is this a set of orthogonal coordinates? Well … No!
New Insight
Following the development of our above, Better Organized discussion, we reverted to several hours of pen & paper derivations, primarily investigating whether it will help us to rewrite various expressions using the [MF53] DirectionCosine Relations. We discovered that if we set,



then,









This seems to be a promising method of attack because — in all three cases, i = 1,3 — the derivative of with respect to does not depend on . Perhaps this simplification will help us identify the function that defines . This proposed prescription for and some of its implications are reflected in the following "New Insight" table. (Keep in mind that, although the expressions for remain correct, the tabulated expression is a guess for and, hence, the tabulated expressions for all three are pure speculation.)
New Insight  
  
See Also
© 2014  2021 by Joel E. Tohline 