A curve, or not a curve, that is the question?

[ValeryOchkov]

Here a dispute arose.
1. Is this closed curve (a phase portrait) an ellipse? The ellipse has two foci. How to calculate the abscissa of the focus of this "ellipse"?

2. Can this object be called a curve? The curve has a length. How to calculate the length of this curve?

[Werner_E]

Of  course is what you define a parameter representation of an ellipse but "t" must run up to approx. 2.00641 s with an appropriate step size to make it really a "closed" one.

But how can you expect this curve having a LENGTH? You are operating in a plane where the horizontal direction is of dimension time^-1 while the vertical direction is dimensionless. How wold you define LENGTH or DIRECTION in this context?

To make it work you either have to make all dimensionless or make equal dimensions in any direction (preferably dimension length, otherwise talking about the "length" of the curve would not make much sense 😉
To make the point more clear: In the following plot we can say that the slope is  0,75 kV/ms, but what would you say is the "length" of that straight line segment? Which unit would you apply???

It seems to make no sense to ask for a "length" in this setting with units.

P.S.: You did not attach the worksheet and I am too lazy to retype - whats the error message. My translation tells me that it means something like "Are you a total fool?" but this would not be very polite for an official error message.

[ValeryOchkov]

Thanks Werner for your reply.
But the questions were.
If it's not an ellipse, then what is it?
If that's not a curve, then what is?

[AlanStevens]

Does this answer your question Val?

Alan

[ValeryOchkov]

Я дико извиняюсь, но we cannot add values of different dimensions.

[AlanStevens]

Where did I add values of different dimensions?   omega/a has the same dimension as theta/b  (namely dimensionless)!

Alan

[ValeryOchkov]

OK!

But can you calculate the foci positions?

[AlanStevens]

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@ValeryOchkov wrote:

OK!

But can you calculate the foci positions?

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This depends on how you decide to define "length" for this situation as Werner has noted.

Alan

[Werner_E]

You may answer the very same questions for my simpler example:

Is it a curve, is it a segment of a straight line?. Does it have a length.

Guess the answers depends on how exactly(!) you define "curve", "line", "length" .... 🙂

So give us your exact and precise definitions of "curve", "ellipse", "arc length", etc. first before asking

The pure geometric shape sure is an ellipse (or a straight line in my example) and as such they have a (unitless) "length" defined.

But as soon as you add (different) units at the axis values, we are in the area of (physical) interpretation of the graphs and I can't think of any useful sensible physical interpretation of the length 5 of my line in terms of kV and ms or the circumfence of your ellipse.

But then - as usual its all a matter of definitions which we agree on ...

And while we are at definitions:

When you ask "is it an ellipse", please state what you exactly mean by "it".

The set of blue pixels in your picture? No, thats not an ellipse

The equations you show? No, these are mathematical expressions with a physical meaning, not an ellipse.

[ValeryOchkov]

>So give us your exact and precise definitions of "curve", "ellipse", "arc length", etc...

So "Give me a place to stand, and I shall move the World!"