Yes, this is true. But we have already determined that only one point in common exists between all of these planes/angles. In Mathematica the guts of the program can be told to reference each of the other algebraic equations to this point, somehow. How does it do that? Because there is no existing algebraic equation that can do that. That's why I'm asking for help.
If you have written the necessary equations in Excel why can't you tell us what those equations are? And if it is so well understood, why are you still asking for help.
Not everything that can be described by a set of equations fits the mathematical definition of a function.Can you explain this? I don't understand how this function would be exceptional. Everyone here claims that it isn't, and it seems that you are claiming that it might be.
Ok: What I am saying is clear enough.
- There is no evidence that your model exists anywhere in the world except in that Mathematica model. There also exists an (unrelated?) Excel spreadsheet somewhere and maybe other ((unrelated?) documents.
- That Mathematica model and other documents have been seen by only a few people as far as we know.
- None of those people have bothered to express the angle between G and B as a function of angle E.
- Thus no one has bothered to express the angle between G and B as a function of angle E because few people in the world have seen that model.
There will be other reasons, e.g. this is a geometry question that is seen as boring or trivial; you have not been able to describe the problem clearly enough; inappropriate audiences - an appropriate audience would be a math forum or mathematics professionals. Google "math forum". Google "mathematicians" at your local university. Maybe a university math club?
The evidence is that you are wrong about some of the mathematics you have written about because you do not know enough mathematics.
The separate evidence is that your example is a relatively trivial to a mathematician example of a series of rotations.
Last edited by Reality Check; 2016-Jul-07 at 10:29 PM.
A position, mathematically speaking, does not really exist. There are position vectors sometimes shortened to "position". You do not seem to be talking about them otherwise that post would not assert "In 2D there is no method for splitting length and direction from one another". A 2D position vector is defined as a length and direction from an origin!
Given that you cannot even be bothered to try to rephrase your input into something that would make life easier for someone solving this I don't see why I should waste my time translating your pictures and vague descriptions into something practical. Especially since, without more input from you, the odds are good you won't agree that I have done it right because you have it in your mind that this is some fiendishly complex piece of highly significant new mathematics.
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I am sorry you are talking utter ...
Of course there are algebraic equations that describe the stuff you want, you only don't seem to know them, just as you are clueless about what was in the Mathematica program.
This discussion is pointless, until you ask your former collaborator to give us the Mathematica program or give the actual input of your excell sheet.
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It would be more interesting (to me) to deal with the mathematical problem than continue a back-and-forth series of recriminations. Let someone peer review this general plan:
The circle has radius and the coordinates of the center are .
To express the point P as function of angle E, it is convenient to find two orthogonal unit vectors that are parallel to the plane that passes through the circle.
We can normalize the vector defined by the segment from the center of the circle to the "north pole" (0,0,1) to get as the "up" unit vector in that plane. For the "sideways unit vector in that plane we can use since (intuitively) the plane of the circle is parallel to the x-axis.
From the usual application of trigonometry in plane, the coordinates of the point satisify:
Lets find two orthogonal unit vectors that are parallel to the G plane. Since G is perpendicular to the x,y plane, one vector is . For the other vector we can take the line segment from the (0,0,0) to P, project it on the x,y plane and normalize it. The projection is The length of the projection is . For the time being, I'll denote this length as .
So
Next let's consider finding two orthogonal unit vectors that are parallel to the B plane.
In the 2-D trigonometry of the unit circle a tangent to point with coordinates is parallel to a radius drawn from to . So I think the vector (in 3D) is parallel to a tangent to the circle at the point P and thus to the plane since contains that tangent.
So, let . If necessary, we can evaluate that expression to give the three components of . Conveniently, is a unit vector.
It's natural to consider the vector from to . Unfortunately, this vector is not necessarily orthogonal to . However, given two independent vectors, we can construct two orthogonal unit vectors by applying the "Gram-Schmidt" process.
One plan for completing the solution is the following.
1. Apply the Gram-Schmidt process to vectors and to find a unit vector orthogonal to and in the plane .
2. We can take the cross product of two vectors, each parallel to a plane, to find a vector that is normal to that plane. So we can find a unit normal to the plane from . We can find a unit normal to the plane from .
3. The angle between the planes and can be found from the formula for the inner product of the unit normals and . This inner product involves a term with the cosine of the angle between the planes. The formula can be solved for the angle between the planes, giving that angle as the arccosine of other things which depend only on angle .
I don't want to carry out that plan till the previous work has been checked for mistakes.
The attachments are our last revision on this. The method for determining the slope (page 3) doesn't work due to spherical excess. In our Excel model, we replaced that method with another method using the rotations shown in the attachments in #55. (Also, we found an indispensable article, Virtues of the Haversine by R.W. Sinnott in a 1984 edition of Sky and Telescope which provided a code snippet that we incorporated into the model in order to avoid using cosine.)
An algebraic expression has never been composed. Let me make this a little more clear. I need help writing an algebraic expression of this equation.And if it is so well understood, why are you still asking for help.
What is it that you don't understand about this request for assistance? It seems specific enough to me. Nothing at all vague or misleading that I am aware of.
Can't we look at the graph of the set of equations and tell one way or the other? I thought we went over this many times.Not everything that can be described by a set of equations fits the mathematical definition of a function.
I think you are making a semantic argument, and a losing one at that.
Once again, from wiki:
"There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation."
https://en.wikipedia.org/wiki/Functi...ing_a_function
Odd to me that this is the exact same link that I have been told to study, over, and over again, even in this thread!
There has to be some type of user interface in Mathematica. The author told me during our collaboration that he did not have to calculate the outputs from the inputs because Mathematica "automatically" does that.
So yes, I do understand the calculations were done internally in the bowels of Mathematica somewhere, sure, but no one, afaict, knows exacly how it did it.
And sure, the rotations are functions. I never meant to imply otherwise.
Let me look this over. It'll take me an afternoon (or two).
on edit>>>>
More than two, I imagine. I made it through the description of and it became a blur.
The only suggestion I have is to switch this up so that we express the angle between B and G as a function of the change in position of G (instead of angle E.) This will allow other functions (for small circle sizes other than used here) to be graphed as a set. In other words, if we proceed down this path (using angle E), the domain will vary with the size of the small circle. If done the other way, the domain will always be 90 degrees (relative to the pole) while the range remains unchanged (the tangent sweeps 180 degrees relative to the circle, and 90 degrees relative to the sphere.)
Last edited by steveupson; 2016-Jul-08 at 09:15 PM.
Originally Posted by HornblowerAre you using "flip side" in an explicitly defined mathematical sense, or is it merely a figure of speech?Originally Posted by steveupson
Yes indeed, as I see it, direction is a positional attribute that is commonly quantified by angular displacement, the measure of which is commonly quantified by a ratio of two lengths. I do not see a problem with that. Could you explain why you find it to be a problem, should that be the case?It does seem self-evident that direction can be readily described quantitatively, except that when you delve much, much deeper into it you can see that we don’t really have a quantitative way of expressing it, by itself, without length.
To be brutally honest, it is my sincere hunch that you are overthinking something, to the point of having an idea about "what direction is", and that your idea defies attempts to articulate it to others either verbally or mathematically.The fact that we always tether direction to some relationship between lengths leads to a lot of misunderstanding (or more precisely, lack of understanding) about what direction, as a quantifiable property, is.
Please try to explain, in appropriate mathematical detail, what you mean by separating direction from length, and why doing so should be necessary to make it obvious that direction is a commutative attribute.Once we can separate direction from length, it becomes obvious (at least to me) that direction is the property that commutes (mathematically) between two positions in spacetime, rather than length or distance.
That isn’t a mathematical term. Mathematically it would be “direction is the multiplier or multiplicand of position, with length being the other multiplier or multiplicand of position.”
It really isn’t that much different than the way we’ve always done things. The only difference is that we can express direction without a metric. Mathematically, this should allow relativistic transformations to be done differently.To be brutally honest, it is my sincere hunch that you are overthinking something, to the point of having an idea about "what direction is", and that your idea defies attempts to articulate it to others either verbally or mathematically.
You seem to understand how the math is normally accomplished. The mathematical detail is very simple. We quantify direction as a ratio between directions.Yes indeed, as I see it, direction is a positional attribute that is commonly quantified by angular displacement, the measure of which is commonly quantified by a ratio of two lengths. I do not see a problem with that. Could you explain why you find it to be a problem, should that be the case?
…..
Please try to explain, in appropriate mathematical detail, what you mean by separating direction from length, and why doing so should be necessary to make it obvious that direction is a commutative attribute.
The obviousness has to do with the ladder paradox or Lorentz–FitzGerald contraction. There is another way to express the phenomenon using relationships that occur in 3-D. This will require creation of another geometry.
The y-axis lies in G when G is at its initial position. Then G revolves around the segment from (0,0,1) to (0,0,0). The angle made between its initial position and the plane defined by (0,0,1)(0,0,0) and P is what we are looking for, and it will differ from the angle made by the intersection of G with the xy plane.
Last edited by steveupson; 2016-Jul-09 at 11:45 AM. Reason: for clairity
Please show us, in appropriate mathematical detail, how you would express direction without a metric. Then show us, in appropriate mathematical detail, how you would apply it to relativistic transformations.Originally Posted by steveupson
Please show us, in appropriate mathematical detail, how you would quantify direction as a ratio between directions. How do you define those original directions? Perhaps it would help if you give us an explicit definition of direction in your line of thought.The mathematical detail is very simple. We quantify direction as a ratio between directions.
Please walk us through the ladder paradox and the Lorentz-Fitzgerald contraction, in appropriate mathematical detail, and show us how you would apply your concept of direction.The obviousness has to do with the ladder paradox or Lorentz–FitzGerald contraction. There is another way to express the phenomenon using relationships that occur in 3-D. This will require creation of another geometry.
It is impossible for me to ascertain your apparent alternative concept of "direction" if you don't or won't do the math and demonstrate it for us. The animations of the resultants of compound rotations of linked planes don't show us anything that cannot be derived from conventional analytic techniques. Your responses are reinforcing my hunch that you have created in your mind an articulation-resistant idea as a result of overthinking the concept of direction.
Suggestion to all: stop discusssing the against-the-mathematical-mainstreamhypothesisdefinitions and return to solving this high-school-level trigonometry problem.
If you can.
If you still care...
Last edited by 01101001; 2016-Jul-09 at 03:06 PM. Reason: caring
0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan
What do you mean by "what we are looking for" ? As I understand it you want a function. For example, if we have a function alpha = f( phi) then what does "what we are looking for" refer to? Does it refer to alpha? or to phi? I'm trying to determine what you want the argument phi of the function to be.