** Twisted Functions**

by

Patty Wagner

In this write-up I will explore parametric equations and consider the concept behind them. Parametric equations are functions with a fun twist. Consider the functions:

x = cos tandy = sin tIf we allow these two functions to define our points

(x,y)at a particular value,t, then we see what occurs when we change the value oft. Since we are working with cosine and sine, it makes sense to allow our values fortto range froms0 ≤ t ≤ 2π.

You may recognize this as the unit circle. You can graph it by putting in specific values for

tand connecting the resulting points. For example, when

t = 0,x = 1andy = 0and the resulting point is(1,0).Here is a unit circle with well know values for

tplotted:One of the fun things that can be done with parametric equations is the graphing of three dimensional curves. In reality we live in 3-dimensional space. So it's helpful to consider the graph above as

x = cos t, y = sin t,andz = 0.What happens if we assign a value to

zdifferent from zero? You may immediately realize that you just end up with your two-dimensional circle "raised" or "lowered" to a different plane perpendicular to z. The interesting stuff happens when you letzvary, such as whenz = t.

These graphs show what happens when

tranges from about0to50. We know that we get a unit circle astmoves from0to2π. The circle also moves linearly up the z-axis as the circle is created. We can tell thattranges from zero to about 50 since there are approximately 8 circles and

8(2π) ≈ 50

Going back to 2-dimensional space (to keep things simple), let's consider what happens when we place coefficients into our functions:

x = a cos tandy = b sin tTo the right we see

x = 2 cos tandy = sin tIt appears that the unit circle has been "stretched" along the x-axis and the range of

xis now-2 ≤ x ≤ 2. This makes sense since we know that the range of

cos twhentvaries from0to2πis-1to1.Obviously,

2 cos twill give us a range that is-2to2.Using what we know about the range of sine and cosine, we can predict that if we let

b = 2in the above equation fory, we will get a unit circle "stretched" along the y-axis, with y values ranging from -2to2.If we leta = 1, then we'll get an elliptical figure like the one above except stretched along the y-axis:

This leads us to conjecture that we will get a circle when

a = b. Here is a graph of

x = 5 cos tandy = 5 sin t

It appears that we've exhausted all that could be of interest in this case. However, there's one thing we haven't considered yet. What if

a = b = t? We would have

x = t cos tandy = t sin t

This is a special case of

a = bwhere the graphdoesn'tproduce a circle. We can see what is happening by considering a series of circles of the form

x = a cos tandy = a cos twhere

aincreases linearly. As,(x,y)moves around the circle, the circles gradually get larger in diameter.

We can see that the graph crosses the x-axis for the first time at

-3.14and crosses for the second time atx = 2π ≈ 6.28which is what we would expect.This elegant sprial is called "Archimedes's Spiral" and you can learn more about it here:

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