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 system? 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: http://vm.uconn.edu/~mcgavran/notes/wrapping.pdf http://www.math.msu.edu/~systeven/MTH103/T3.2.pdf 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: http://en.wikipedia.org/wiki/Circular_polarization 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: http://jwilson.coe.uga.edu/EMAT6680Fa05/Parveen/Assignment%2010/parametric_equations.htm 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 http://mathforum.org/dr.math/
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. Thanks!!!
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