Ask A Scientist© Physics Archive

name         Ivan 
status       student
age          17

Question -   Hello,
My physics teacher mentionned a while ago that we could fit a circle 
to every part of the sin(x) function.
For every arc, there is a circle with a certain circonference that 
can describe it. If not completely then approximatively.

Well, I was very intrigued by this and decided to check it out. I made 
a program for my graphic calculator to find the coordinates of the 
circles corresponding to every 0.1 rad of the sinus func.

It works pretty simple. I put in a one dimentional array (list) the 
x values {0, .1, .2, .3, .4, .... until 6pi} = 3 periods. Then I 
evaluate the y values at those points { sin(0), sin(.1), sin(.2) ...} in 
a second list. Then in a third list I put the derivatives of those points 
{ cos(0) ......}.

Then I find the equation of the normal lines to every points and then 
I find the coordinates of the intersection of those two lines = center.

Then I plotted the scatter of those points and it gave me a 
very interesting graph... one that doesn't resemble anything I have ever 
seen before.

I have some screenshots that I could send to you because without them 
you pretty much don't know what I'm talking about.

Q1: Has anybody seen anything like this before? Is this a new function ? 
Can it have any utility whatsoever?
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Hello,

You have accomplished quite a bit already.  I can imagine the shape of the 
plot, and it should look interesting as the radii of the circles increase 
and then decrease, and their locations also flip sides on the 
curve.  Connecting the centers of these circles also gives another 
interesting curve.  You can use this second curve and fit circles to it and 
continue the process.  I have to think more to see where that will get us.

Fitting the best circle to a point on a curve is a recurring problem in 
engineering.   For example, when you need to make an elliptical mirror but 
do not want to make an ellipse (because of costs, for example), you fit a 
circle to the desired point on the ellipse and make and use a circular 
rather than elliptical mirror.

So, the answers to your questions are yes, no, and yes.
You can generate additional interesting curves by using square of a sine 
wave or other trigonometric functions.   By the way, you can get the entire 
picture within one period (2pi) and the shape repeats itself after that.
Good luck.


In fact, you can generate a lot of interesting
Dr. Ali Khounsary
Advanced Photon Source
Argonne National Laboratory
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