steveupson

You have to click on the link: "This function is the one that I have a lot of trouble with, for some reason or other. How is this function expressed?" The function doesn't use vectors. So did I. The green is the longitude plane, correct, and the tangent plane is the one that's moving in a conical orbit around the small circle, dark blue. I gave it a lot of effort and was not successful. It should be straightforward (for any math genius).

One of what others of what? I explained exactly, precisely, what I am looking for in my very first post: http://community.wolfram.com//c/portal/getImageAttachment?filename=NSTF.gif&userId=93385 There is a function that expresses the relationship between the small numbers that are changing in the top of the image and the numbers that are changing in the rectangular box. I want to express this function in algebraic fashion. Also, an additional request has been made for help in deriving the function of the group of functions that will reveal their relationship to one another. Yes, that's one of the things that I noticed when I started playing around with the function and doing some thought experiments with it. I found the function in some of my notes from ten years ago, so I don't recall exactly how it came about, but I do know that I was working with small circles on a sphere, trying to develop some geometry to map their slope as a function. Some months ago I found it in my notes and couldn't stop thinking about it, and thinking about how it might be useful for something. There are a lot of peculiarities about it, that much I know. So far, no one I've discussed it with has ever seen a function that is even close to this one, or even in the same genre. I do know for a fact that there is no existing trig that will deal with small circles on as sphere. Why would direction have any importance at all for deriving the function? If you tell me why it's necessary then I'll figure out how the express it. It seems to me as though we have all the necessary information to proceed.

Let's try it on as a semantic argument. We are used to saying that a vector has magnitude and direction, and it works fine, because everyone does it that way, because we have quantified length as a property which allows us to assign it to the scalar value in the vector. The only real difference here is, now we are saying something we are not used to saying, which is that a vector has direction and magnitude, which should also work just fine, because we have quantified direction as a property which allows us to assign it the scalar value in a vector. But honestly, that isn't what I'm trying to do right now. I'm not trying to apply the function to anything at all. I'm just trying to derive the function. Again, for the gazillionth time (I should go back and count just for fun) what I am trying to do, what I would like someone to help me with, is first, composing the function in generic algebraic lingo. I was hoping that we wouldn't be necessary to actually express the function as a formula, just express the nature of how it's to be structured, using algebra. The model was built in Mathematica. The function should give the same inputs and outputs as the model. I wish everyone would at least admit that they understand what I'm trying to do, and I would be very grateful for anyone's help. I don't think anyone is. I am simply trying to derive a function. I am simply trying to explain to everyone that I am trying to derive a function.

I'm not sure how this same argument would be made for the [math]/ cos[/math] function. I really believe that if someone here would help derive the function, or at least compose it, then everyone could at least be talking about the same thing. There might be some misunderstanding of what we're talking about. The statements that I made about not needing a coordinate system were in regards to composing and deriving the function. I'm not sure how it's done, I've never had to figure it out for myself, but I would be a little bit surprise to find that a coordinate system needs to be decided on before it can be done. If we just had something that we could all look at together, so that we knew we were all talking about the same thing, maybe in some form of symbolic language....

Thanks for this. I think that in the new system there won't be a requirement to translate the lines. instead there will be a requirement for a projection. Thanks for this explanation. I didn't understand that that was what they are doing. I no longer feel a bit inadequate about my algebra skills. No one else seems to be able to do any better at composing the function.

We are using the new function to replace [math]\cos [/math]. The difference is that we don't need a scalar length any longer. While this stuff does nothing to alter existing vector analysis, the two geometries cannot be readily mixed and matched. That's what threw me off track the other day. ​Every time a knowledgeable individual tells me "it won't work because of this," I try to look at everything in order to translate it into the new system. While trying to do this with circulation and swirl as studiot suggested, I sort of lost my footing there for a while, if you know what I mean. I'm still not sure if that stuff is translatable or not, it's way above my pay grade at this point. The problem that everyone seems to be having is that they are used to looking at things a certain way. I certainly used to... on edit> That isn't quite right, what I said about not needing the scalar length. what I should have said is that we don't use the length as the scalar or the metric. Length is the dependent part of the angle expressed by the inner product. Direction becomes the scalar.

In the same way that having a length does not necessitate a coordinate system is my guess. Representing a line without a coordinate system would seem awkward. We aren't really trying to represent an angle (analogous to a line), we're trying to represent the magnitude of the angle (analogous to a length). I think that by using two projections it would be possible to measure the difference in direction between two lines that don't intersect in three dimensions. I'm not sure how one would go about it, exactly, but I don't see why it couldn't be doable, using math, of course.

They aren't really unit vectors because those things relate to either a point or a surface. Unit vectors can't be the metric but direction, once quantified, can be. what the function returns 2.doc what the function returns 2.doc

I'm with you so far, carry on. (and no, just like what happened here, I haven't been able to convince anyone over there that this is a function, either) After reexamining things, I do believe that there is a tether to orient things in the frame. It is the plane in which the two directions lie. The difference between this function and an angle is that angles don't commute and this function should commute. We've come a long way since the days when I used to argue that this was a sphere. Thank you for dragging me out of the weeds on that.

Because it's a function. Look at the animation: http://community.wolfram.com//c/portal/getImageAttachment?filename=NSTF.gif&userId=93385 The little numbers that are changing at the top of the image relate to the little numbers that are changing in the rectangular box. The relationship is establish through a function. Also, there is a function that is different, but similar to this for every direction. Also, there is a function that will relate these other functions to one another. <---- This is what I want.

I'm not sure how useful it is either, but I think it's pretty cool. This is a pictorial view of what I have been saying: file attached below This is my way of saying what you have also been saying. A vector must have a reference. The reference is always in a plane. All you are doing is triangulating by arranging three of these operations orthogonality. When you look at the basic relationships that are determined by the geometry, it is clear that all of these relationships are rooted in plane geometry. And I'm sorry, the rest of what you said is nothing like what I've been asking about, at all. I have a thread in the math forum that explains a new function. I've asked innumerable times for help expressing this function in algebraic terms. I cannot make any more clear than that. It confounds me to no end that you still cannot hear what I am saying, at all. Clue: whatever I'm trying to say algebraically should have a f() in it somewhere! what the function returns.doc

I've always done the math myself. That's how I roll. I only look to see how other people do it when I either don't have time to figure it out for myself, or when it's too difficult for me to learn. Composing functions and rearranging algebra expressions is not anything that I've ever used. When you figure it out for yourself there's no real reason to write it down anywhere, is there? Other than to keep track of things during complicated operations. In those cases it's simple enough to invent the book keeping on the fly.

It doesn't seem to be useful for what I initially intended, although if it was paired with a reflection of itself it might do what I was thinking. Or not. Needs more work. Because my understanding of it is geometric and not algebraic. That's my best explanation. I can see where everything is pointing and it's all pointing differently than other objects that I've looked at. I hope you understand that the directions (what I've been calling cardinal and ordinal) have a remote relationship with the function itself. They are not related to the function at all in the plane in which the two directions lie. This is what made me think the angle between them could be quantified into a magnitude.

Please see my edit above. The idea was to create a sort of protractor in order to handle direction directly (or independently). The length would then have become the vector dependent property. I honestly don't know how I was expecting it to work, looking at it now. Although, if there were some sort of backdrop to set this whole thing on (which there isn't, is there?) then it might look more like what I thought it looked like until now.

<see edit at bottom of post> I'm sorry, I do understand what you're saying, and I do understand how vectors, tensors, and coordinate systems work. I don't have the level of sophistication that others may have, but I do have a very solid understanding of the basics. I know when dot products and cross products should be used (not that I could do a dot product from memory). I know a little bit about polar coordinate systems and I'm pretty good with spherical trig (or at least I used to be). My problem is that I only know the things that I've used because that's the only things that I've bothered to learn. I have no formal training. This isn't about that. What we're doing doesn't change any of that stuff at all. I'm comfortable in knowing that those things can be always be used in the conventional manner, only we won't be using any of them here because we are not doing any of those operations. What we are trying to accomplish in this discussion is something that is in addition to those tools. What I need, what I cannot do without help, is compose a function. I just don't know how to do it. Your guess as to what I'm trying to do is only half right, and it also isn't precise enough. If we can get the basic form correct, then we shouldn't have any problem polishing it up since we'll both be looking at the same thing. We are not going to be using any coordinate systems. We'll have to figure out how to plug this into a frame later (or we may not need to). What we have are two angles. Angle alpha will always range between 90 to 0. Angle E will have a range that will vary according the magnitude of the direction. (Yes, I know that it's annoying for me to say it that way, I mean the size of the difference between the reference and the direction. If you have a preferred way to express that, please speak up and we'll use that term instead of magnitude.) When we are at 45 degrees, E will range between 0 to 90. When we are at 30 degrees, E will range between 0 to 60. It will always be twice the magnitude of the direction. The primary function is that for every angle E there will be a unique angle alpha. Let me stop here and see if we see eye to eye so far. On edit> I've taken a closer look at the circulation of vector fields that studiot wanted me to look at and I think I finally see what I have been missing. They do have a similar structure to what I was talking about, with one major difference. For almost a year I have been looking at this and assuming that there was a boundary there that would orient the thing in the frame and now it's finally dawning on me that there isn't one, it just continues to wrap around, so to speak. This is a problem. I fear it's a fatal one. I think that at this point I should retire from this thread unless there are any further developments or questions, which I don't expect that there will be. I don't know quite what to say to everyone. Will a thank you suffice?

And what did i say that led you to believe that I have a problem with any of that? I'm curious, because I didn't intentionally say anything that was even remotely critical of any of that. What I am asking is whether or not it is possible to put my words into mathematical symbols. There is rigor that is required in order to do that. I think that no one is hearing what I am actually saying because they won't help me make the effort to express it algebraically. If the result is drivel then y'all have my permission to mock me mercilessly until I get to 1000 posts. Once we have the expression for you to evaluate, then we can go over all of the reasons why it won't work. But we can't (legitimately) evaluate the the expression until we know precisely what it is, can we? Of course not. What do you think the function is that I'm going on and on about? Can you tell me what you think I'm claiming?

I'll reword the post to say whatever you want it to say if we can answer at least the first of my three questions. You seem to think that we have some sort of disagreement or something over vectors or tensors or something. I assure that is not the case, and if I said anything at all to give that impression then I completely misspoke. Yes, it is worthwhile for me to learn much more about how these things relate to each other, but that is a side issue. I know that the stuff works. We went to the moon using these same techniques. I get that. I am not arguing with you about it. What this discussion is about is direction, with a specific focus on a method for quantifying it. Let me ask (politely) whether or not it is possible for us to use the proper notation to show what I am looking for? Math is very precise. I want that precision. I just don't know how to do it in a manner that anyone else would understand. What I'm after is something along the lines of: E is a function of alpha, and for every different direction the function changes.

That isn't even close to any of the questions that I asked. There was a point at which I asked you to show me what you think I'm talking about, using the model, and you refused. I was hoping that by watching you work it through I could stop you at the point where we are not communicating and then try and correct that problem. I do appreciate that you want to help, and without a doubt this discussion has been fruitful to some degree, but unless we listen to each other we won't be able to solve this, imho. The questions that I have are basically three, and they are math questions. First, is there some way to algebraically express that the thing we are searching for is a function of a function? Next, is it possible to derive the function? Finally, if it isn't practical to define the function then is it possible to graph it?

Unfortunately, I don't really need any help with translation on a graph. Too bad for me, I guess, because you're so willing to help and all. What I need is some help with the actual questions that I asked. studiot, I've made an effort to look this over. You know what, though, I have to ask a very huge favor from you. Can you try and tell me exactly what you think I am claiming and then I'll try and clarify all of the misunderstandings. Everyone seems to be hearing me say things that are much different than what I've actually been saying. Can you please, in as much detail as you can, explain the model to me?

This shouldn't be this painful for the two of us. Maybe we should give it a rest for a while and let some things sink in. We seem to keep repeating ourselves. Thanks for helping with this. I think we've made progress, but I'm sure that you don't really agree. cheers, steve

Look, it's math, it's either correct or it's incorrect, right? The model presents a description of a function that defines the relationship between a reference and some other direction. Call it what you will. On edit> A while back you mentioned rotational symmetry. Think about what "rotation" is. In this case there is no rotation because the function defines the direction relationship in all of the positions that would be encountered under a rotation, simultaneously. It is a smooth function.

Yes, that is exactly the point. It's what I've been trying to tell you. In one environment (plane geometry) the length is the scalar while in another environment ( _______ geometry) the direction is the scalar.

It isn't changing anything. That's what is difficult to understand. Rather than expressing direction on a two-dimensional surface using vectors, it is expressed as a quantity in three dimensions. Once you actually see what's going on, your entire concept of direction will be stood on it's head. If you don't want that to happen to you, look away now. Is there any physical law that says whether position or direction is the preferred "reference frame?" Again, if you haven't looked at Daniel Cross' paper on "The physical origin of torque and of the rotational second law" then you really should consider doing so in order to better understand the question.

I think you're beginning to understand the issue. This has now been discussed for a while, with others, before the discussion moved here. We need more horses. We need to find someone who can either graph the function or derive it through some magical method. As I said before, my attempt is nonsensical. More likely than not, I used sin instead of sin2 somewhere, or some such.