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The Wrapping Function, Unravelled

Date: 06/29/2012 at 20:11:20
From: Zack
Subject: How do you graph the Wrapping Function? 

When graphing the wrapping function, what coordinates do you use?

Could you graph the wrapping function on the Cartesian coordinate system?
Or is it something to do with polar coordinates or some other coordinate

I never thought about it until now, but the wrapping function seems
similar to a sine wave, forming a circular pattern oscillating between
[-1, 1] -- and yet different.

Is there any history to the wrapping function? Is it simply a modern
pedagogical tool?

Also, what is the exact meaning of the graph "opening" or being "centered"
from (1, 0)? Is this strange or unusual? For instance, when x = 0, the
coordinates are (1, 0) in terms of the wrapping function.

To get a visual handle on this I'm trying to imagine the graph of this
function, but having trouble. It seems to fail the vertical line test; but
for each element in the domain, there is exactly one.

Date: 06/30/2012 at 11:03:19
From: Doctor Peterson
Subject: Re: How do you graph the Wrapping Function? 

Hi, Zack.

I'd never heard of this function by that name, but I suppose you are
referring to the "wrapping function" explained here: 

This is not a function from x to y, which is why your vertical line test
fails; it is a function from a real number t to a POINT (x, y) on the unit
circle. It is a function because for any real number t, one point (x, y)
is determined. To put it another way, both x and y are functions of t.

If you wanted to "graph" this in a way similar to functions of x, you
would need three dimensions, with axes t, x, and y. For each value of t,
the point (t, x, y) would be determined by the wrapping function, so that
x and y are functions of t. The graph would be a spiral; the equivalent of
the vertical line test would be to show that the PLANE through any point
at t on the t-axis passes through the curve only once, at (t, x, y).

But I doubt that it will be helpful to think of it this way unless you are
at a higher level than basic trig; if you have heard of circularly
polarized waves, wrapping functions are the same idea. The graphs here are
similar to what I just described, except that they involve three space
dimensions and move with time: 

A clearer way to think is that (x, y) is a function of TIME t. At time 
t = 0, the point P = w(t) is (1, 0). As t increases, the point moves
counterclockwise along the unit circle.

I think you're right that this is mostly a pedagogical tool;
mathematicians don't need to treat this function as anything special. It
just represents how we measure angles in radians, so that for any real
number t we can form a corresponding angle of t radians. Then cos(t) is
the x-coordinate of P(t), and sin(t) is the y-coordinate of P(t). To a
mathematician, your wrapping function would be considered the natural
parameterization of the unit circle -- that is, a way to associate every
real number with a point on the unit circle:

Another perspective on this function is to treat it as a "mapping" of the
real line to the unit circle. When we think of a function as a mapping, we
focus not on graphing points (x, f(x)), but on relating points in the
domain to points in the range, much as a map of the earth relates points
on the earth to points on the map.

- Doctor Peterson, The Math Forum 

Date: 06/30/2012 at 13:37:43
From: Zack
Subject: Thank you (How do you graph the Wrapping Function? )

Great explanation, and so prompt! 

I was confusing the input of the function for the x-coordinate in the
output, but now I understand.

Associated Topics:
High School Functions

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