0
$\begingroup$

I'm reading a very general definition of parameter on this site.

I didn't understand why we call the variable $t$ of the curve $\alpha(t)=(\cos t,\sin t)$ a parameter.

For me $t$ in this case is a variable too according to the definition of the site I linked. If the definition of this site is right a parameter is either constant or change slower than the variable, which is not the case with this curve.

  • 0
    Check out this https://www.researchgate.net/post/What_is_the_difference_between_variable_and_parameter-da_capo – 2017-02-18
  • 0
    @JoshuaRuiter Thank you for the link, but it doesn't solve my question. For me $t$ in this case is a variable too according to the definition of the site I linked. If the definition of this site is right a parameter is either constant or change slower than the variable, which is not the case with this curve. – 2017-02-18
  • 0
    Why does it matter to you what you call it, between a parameter and a variable? I'm just curious. It seems like an unnecessary distinction to me. – 2017-02-18
  • 0
    @JoshuaRuiter Maybe it seems a silly question at first glance, but I think is important to know why we call this a parameter. Every notation in mathematics has a reason to exist. Knowing this distinction correctly can avoid misinterpretations in other scenarios. – 2017-02-18

2 Answers 2

3

In my opinion, the site you are linking to is a bit misleading, since it suggests that "parameter" is a concept that has a precise mathematical definition and could be formally distinguished from "variable". Calling a variable a parameter is usually just done to clarify its role in the current context. (You can also see that the "definition" you have linked to is very far from being a formal definition, since it contains a hughe number of notions whose meaning is at least as unclear as the one of "parameter".) Moreover, the "definition" given on the linked site fits into a different context, than the problem you are talking about.

In the context of differential geometry that you are talking about, the main issue is as follows. What one really wants to study are unparametrized curves, so in the example you have written down the unit circle in $\mathbb R^2$. Imagine this just as a set of points, without having seen how it was drawn. Now there is the need to define smoothness of such curves, and the simplest way is via the fact that they admit regular smooth parametrizations. This means that you can write your point set at the image of a smooth function defined on an interval whose derivative is nowhere vanishing. Calling this a parametrization, it is fairly natural to call the variable in this curve a parameter. This is mainly to emphasize that the parametrization is an artificial additional choice and geometric concepts should be independent of this choice.

The meaning of "paramter" suggested in the site you have linked to would be used for example in the following context. Consider the map $(r,t)\mapsto (r\cos t, r\sin t)$ defined for $r>0$. Then you can say that these are polar coordinates using the variables $r$ and $t$. However, you may also say that the curve $\alpha$ you have written down originally nice fits into a family of curves defined by $\alpha_r(t)=(r\cos t, r\sin t)$ depending on a parameter $r>0$. In this case, you would certainly view $t$ as a variable and $r$ as a parameter. This however has no formal meaning, it is just different wording.

  • 0
    Thank you very much. Do you mean that historically the terms _parametrization_ and _parametrized curve_ have come first than _parameter_ in this context? – 2017-02-18
  • 0
    I would not want to make claims about history, but I would say that in the geometric context described in the second paragraph of my answer, "parameter" is derived from "parametrization" rathen than the other way round. – 2017-02-18
0

For me the distinction has always been a question of notation. That is $f(x)=\alpha\cdot x$ is a function with $x$ being a variable and $\alpha$ a parameter. Using $f(\alpha,x)$ for the same expression makes both $x$ and $\alpha$ variables. And for this reason any definition making distinction between parameters and variables will be artificial or forced for me.

On the other hand the sole fact that these two things have different words for them suggests some distinction. Variables vary and parameter comes from para (subsidiary) and metron (measure) (according to this etymology site).