Can we find each curve/plane a corresponding function

[whybeing]

can we find each curve we see or we draw a corresponding function?
I mean, when I randomly draw a curve on the paperm, or look at a plane, say, the top of a hat, is there a definite function for it?

I learned from Steward's Calculus that a function for the population growth is formulated by modeling the data. so is somehow my question above connected with modeling things?

quite confused. any help?

 

[tiny-tim]

hi whybeing! welcome to pf!

can we find each curve we see or we draw a corresponding function?
I mean, when I randomly draw a curve on the paperm, or look at a plane, say, the top of a hat, is there a definite function for it?

yes, if you can draw (or make) the whole thing, then that is a function

I learned from Steward's Calculus that a function for the population growth is formulated by modeling the data. so is somehow my question above connected with modeling things?

ah, that's different …

the data aren't the whole thing, the data are just a small number of points, from which you have to guess the whole thing …

there's infinitely many different "guess" functions that will work for your data, but they won't necessarily be correct for filling in the gaps (ie for anyone else's data)

 

[hilbert2]

can we find each curve we see or we draw a corresponding function?
I mean, when I randomly draw a curve on the paperm, or look at a plane, say, the top of a hat, is there a definite function for it?

I learned from Steward's Calculus that a function for the population growth is formulated by modeling the data. so is somehow my question above connected with modeling things?

quite confused. any help?

Not always. You can definitely draw the curve $x^{2}+y^{2}=1$ on paper, but there's no single-valued function $y(x)$ that corresponds to it. If you want to represent the curve as a function, you have to use a parametric representation, where $x(t)$ and $y(t)$ are both functions of some parameter $t$.

 

[HallsofIvy]

hilbert2 is being very careful about the word "function". Certainly a specific "coordinate system" can be placed around any geometric object and the object then is described by some "relation" (not necessarily a "function" technically) in that coordinate system. The relation depends upon both the object and the coordinate system.

 

[whybeing]

hi whybeing! welcome to pf!


yes, if you can draw (or make) the whole thing, then that is a function


ah, that's different …

the data aren't the whole thing, the data are just a small number of points, from which you have to guess the whole thing …

there's infinitely many different "guess" functions that will work for your data, but they won't necessarily be correct for filling in the gaps (ie for anyone else's data)

thank you. that really helps. so can I assume that, among these "guess" functions, the one that goes smoothly through every point is a better guess? You know, there got to be one that best fits the reality, I think. and Is this somehow related to line integral, or Lagrangian in physics?

 

[whybeing]

Not always. You can definitely draw the curve $x^{2}+y^{2}=1$ on paper, but there's no single-valued function $y(x)$ that corresponds to it. If you want to represent the curve as a function, you have to use a parametric representation, where $x(t)$ and $y(t)$ are both functions of some parameter $t$.

thanks, that helps. so can I assume that any continuous curve, no mater how complicated, can be represented by a parametric representation?

 

[whybeing]

hilbert2 is being very careful about the word "function". Certainly a specific "coordinate system" can be placed around any geometric object and the object then is described by some "relation" (not necessarily a "function" technically) in that coordinate system. The relation depends upon both the object and the coordinate system.

thank you. now I think my attempt to find every curve a function(even for life) is like a deterministic perspective to look at the universe, and that mathematics or physics is so pure that they seem to be unable to describe reality where all things seem to be unpredictable.

"One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike -- and yet it is the most precious thing we have." --Albert Einstein

 

[Chronos]

It's pretty damn tricky to derive a function to plot a series of data points. No matter how 'smooth' it looks, I don't know any simple way to accurately derive a function to plot even a simple bell shaped curve. The least squares method is probably best, but, it is calculation intensive and not at bit simple.

 

[jbriggs444]

thank you. that really helps. so can I assume that, among these "guess" functions, the one that goes smoothly through every point is a better guess? You know, there got to be one that best fits the reality, I think.

For any finite set of data points, there is an [uncountable] infinity of smooth functions that exactly fit those data points. There is no one single unique preferred notion of "best fit" that would allow one of those smooth functions to be selected.

and Is this somehow related to line integral, or Lagrangian in physics?

There is the notion of Lagrange interpolating polynomials. For any set of n data points the Lagrange interpolating polynomial is the unique polynomial of degree n-1 or less that intersects every point.

http://en.wikipedia.org/wiki/Lagrange_polynomial

One problem is that a polynomial function is not always an appropriate model. So even though your polynomial fits all the measured data points, it may not fit the unmeasured points. Another more general problem is that if your data has experimental error then any function that you find that exactly fits the measured data will be attempting to replicate your experimental error as well.

Least squares regression is one "best-fit" technique that attempts to filter out experimental error by finding an underlying function within a particular class of functions that minimizes the deviation, in a least squares sense, of the selected function from the measured data. But you still have to pick the right class of functions to start with.

http://en.wikipedia.org/wiki/Least_squares

 

[LCKurtz]

thanks, that helps. so can I assume that any continuous curve, no mater how complicated, can be represented by a parametric representation?

If the curve is sufficiently smooth that the notion of its arc length exists, then yes. You can always parameterize it in terms of arc length. At least in theory you can. But remember A.E.'s statement": In theory, theory and practice are the same, but in practice they aren't."