MHB General definition of parameter

AI Thread Summary
The discussion revolves around the definition of a parameter in the context of parametrized curves, specifically questioning why the variable $t$ in the curve $\alpha(t)=(\cos t,\sin t)$ is considered a parameter. While some argue that $t$ is merely a variable, others clarify that a parameter is indeed a variable that can describe how other variables change, regardless of its rate of change. The term "parameter" is commonly used in coordinate systems where additional variables express coordinates in terms of one another. The conversation highlights a misunderstanding of the definition, particularly regarding the relationship between parameters and their associated variables. The conclusion emphasizes the importance of context in understanding mathematical terminology.
Ravik Rocha
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I'm reading a very general definition of parameter on this site Parameter definition - Math Insight.

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.
 
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I think the meaning of "parameter" is different in this context. This meaning is described in the article about parametrized curves.
 
In general we use the word "parameter" when we have an xy- coordinate system (in two dimensions) or and xyz- coordinate system (in three dimensions) but have the coordinates written in terms of some additional variables, other than x, y, and z.

"For me [FONT=MathJax_Math]t in this case is a variable too". Yes, t is a variable but that has nothing to do with whether it is a "parameter" or not. A parameter is always a variable.

Yes, the site you link to says "while parameters typically either don't change or change more slowly". That makes no sense to me even with the word "typically"! A common use of a "parameter" is in motion problems where the coordinates, x, y, and z, change as t, the time, changes. The case of a very slow moving an object is an example in which the parameter, t, changes much faster than the coordinates.

Don't believe everything you read on the internet!
 
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