steveupson Posted February 27, 2017 Share Posted February 27, 2017 To recap: The arc length of a velocity vs time (or position vs time) graph is a line (minimum length) if the underlying data is a constant function and will not be a line if the underlying data is not a constant function. If the variation in the function is cyclic then the arc length is proportional to frequency and amplitude of the variation. The amplitude is simply a scale factor, so the frequency component must be relevant part. For a cyclic (sinusoidal) function, the frequency is simply the angular velocity. Angular velocity is the change in angular position over time. Angular position is direction. This quantity (arc length of a velocity vs time graph) has a base quantity (or dimension) of direction. (Yes, I do understand that this is not in the wiki for this subject, but let's try and be scientists here and look at the actual issues. The questions that are raised in the OP will still exist whatever my behavior is and no matter what I have to say about it. If the math that is being presented is incorrect, please try and focus on fixing the math instead of dwelling on trying to fix the author of this post.) We know that direction is always expressed as a ratio between two lengths (xyz vectors or diameter to circumference) so it’s natural to assume that when we say that the base quantity is direction that we mean that the dimension is length. But this is wrong because base quantities (or loosely, a dimension) must be in units that are numbers, and not a ratio. What is missing here is the number or quantity that would express direction as a number and not as a ratio. There is way to do this but it does involve doing some math in order to understand how it is done. There isn’t any intuitive way that I know of that would allow a person to quantify direction as a base unit without doing math. Fortunately, the math is simply trigonometry and involves solving some triangles. The model can be viewed as an interesting math problem. The truly competent folks seem to view it this way, having made an honest effort to find a solution before expressing an opinion. Others, who also seem to be extremely competent but not quite as careful, complain that the problem is so simple and trivial that it isn’t worth the time or effort to try and solve. The solution that we’ve come up with gives two equations that are a parameterization of a multivariable function. At least that is what I’ve been told. The two equations are: [latex] \cot\alpha = \cos\upsilon\tan\frac{\phi}{2 } [/latex] [latex] \sin\frac{\lambda}{2 } = \sin\frac{\phi}{2 }\sin{\upsilon} [/latex] The function [latex] \alpha=f(\lambda) [/latex] turns out to be: [latex] \alpha={\cot}^{-1 }(\cos\upsilon\tan{\sin}^{-1}(\frac{\sin\frac{\lambda}{ 2}}{ \sin\upsilon})) [/latex] This smooth function approaches a sine curve when [latex] \upsilon\to0 [/latex], and that approaches a hyperbola when [latex] \upsilon\to\frac{\pi}{2} [/latex] , and allows a way to assign a meaningful value to direction, rather that simply keep considering it to be a simple ratio (which it is - but it's also more, much more, than that.) More about the interesting math problem is here: https://forum.cosmoquest.org/showthread.php?161772-New-Math-Function-Redux (the solution attempts begin at post #164) Link to comment Share on other sites More sharing options...
imatfaal Posted February 27, 2017 Share Posted February 27, 2017 Stop hijacking and stop overcomplicating due to lack of understanding. The arc-length is well defined for a smooth continuous fnction and is simply the integral of the norm of the differential or more usefully as [latex] L_{ab}=\int_{a}^{b} \sqrt{1+\left( \frac {dy}{dx} \right) ^2} \cdot dx [/latex] Link to comment Share on other sites More sharing options...
steveupson Posted February 27, 2017 Author Share Posted February 27, 2017 (edited) Stop hijacking and stop overcomplicating due to lack of understanding. I'm pretty sure that I haven't overcomplicated anything, and I'm pretty sure that I haven't hijacked anything, and if I do have the lack of understanding that you are so sure that I have, well, then that is specifically why I am here asking questions. The image shows what I've been saying. How is the math over complicating things? If there's a simpler explanation of either the math that has been presented or of the question that was asked in the OP then share it with the rest of us. If I've said anything that contradicts your calculus methodology then please correct me because I don't know what it is that I said that you believe to be wrong. Sure, I disagree with the way we've always done things, that's correct. But can't we put on our grown-up pants and discuss why the math that I've presented is simply dismissed in favor of the old "we do it this way because we've always done it this way" rationale? Have you even looked at the math before commenting on it? I think not. We generally have a low opinion of people who ignore the math in favor of pablum. Don't be one of those people. What, specifically, have I said - EVER - that you take issue with? If members do believe this to be a thread hijack then vote this post down. If you're interested in this discussion then please vote it up. Edited February 27, 2017 by steveupson -4 Link to comment Share on other sites More sharing options...
swansont Posted February 28, 2017 Share Posted February 28, 2017 I'm pretty sure that I haven't overcomplicated anything, You haven't shown a calculation of the arc length with Fourier analysis. and I'm pretty sure that I haven't hijacked anything, and if I do have the lack of understanding that you are so sure that I have, well, then that is specifically why I am here asking questions. ... If members do believe this to be a thread hijack then vote this post down. If you're interested in this discussion then please vote it up. (Asking questions not directed at the OP is generally considered hijacking) Link to comment Share on other sites More sharing options...
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