Can you make sine waves from different shapes and mess around with those shapes?

[questionposter]

So if you can draw a sine-wave as a circle, what about other types of sine-waves? Can you draw them as other shapes too? Or what about secant? Can you draw that as a shape? Is there some way I could add two circles and get a bigger circle?

Edited December 20, 2011 by questionposter

[John Cuthber]

Do you mean this sort of thing?

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

[DrRocket]

So if you can draw a sine-wave as a circle, what about other types of sine-waves? Can you draw them as other shapes too? Or what about secant? Can you draw that as a shape? Is there some way I could add two circles and get a bigger circle?

 

A sine is not a circle.

 

The sine is a well-defined function. There is no such thing as "other types of sine-waves".

 

You can represent functions in terms of functions other than sine waves. This is something that you can study when you study Hilbert spaces or "orthogonal functions". One set of functions that are sometimes used are Walsh functions which are trains of square waves.

 

The secant can be graphed as the secant. It too is a well-defined function. It has the shape of a secant curve, just as the siine has the shape of a sine curve.

 

 

What in the world does it mean to add circles ? If you want a bigger circle just open up your compass and draw one.

[Cap'n Refsmmat]

I think questionposter is referring to the common way of representing sine and cosine through the unit circle, as in this animation:

 

http://js.2x.io/trig-paperjs.html

 

One could easily concoct other geometric shapes which produce the graphs of trigonometric functions when you animate them in some way.

[DrRocket]

I think questionposter is referring to the common way of representing sine and cosine through the unit circle, as in this animation:

 

http://js.2x.io/trig-paperjs.html

 

One could easily concoct other geometric shapes which produce the graphs of trigonometric functions when you animate them in some way.

 

The sine and cosine are not really "represented" through the unit circle so much as defined in terms of the unit circle. So, the basic answer to his question is that, no, you cannot replace the unit circle with something else in any natural way to produce the sine or cosine.

 

You can, of course, utilize computational power and graphics to kluge up some unnatural representation, but to do that would be to miss the point. The graphics in your link do pretty much that. I am quite sure that whoever put together that animation understands the sine and cosine quite well, and the exercise of putting it together would have been an good learning experience. But anyone who does not already understand the sine and cosine will only be baffled by the animation which generates sine and cosing traveling waves.

 

Just because you can concoct some animation based on other geometric shapes does not mean that you should do so. There is no natural way to do it. If one is going to force the issue using some ad hoc approach and lots of computing power, then I suggest that the geometric object be Salma Hayek -- there would be essentially no loss in mathematical content, but the geometric object would be at least be interesting.

[questionposter]

Do you mean this sort of thing?

http://en.wikipedia....Lissajous_curve

 

That's actually pretty close, but that seems to deal with more parametric and polar equations, I was thinking more of actual functions.

[TonyMcC]

I think mention of Lissajous figures is something of a red herring. They represent the action of TWO sine waves and never just ONE, in my experience and use of them. For example two sine waves of equal amplitude and 90 degrees of phase shift will produce a circle. http://en.wikibooks.org/wiki/Trigonometry/For_Enthusiasts/Lissajous_Figures

Edited December 21, 2011 by TonyMcC

[John Cuthber]

I think mention of Lissajous figures is something of a red herring.

The OP didn't.

Incidentally, if you take a fast sine wave and a long bit of cable you can make a circular Lissajous figure.

Also, if you combine more than 2 together you can get other pretty patterns.

[TonyMcC]

The OP didn't.

Incidentally, if you take a fast sine wave and a long bit of cable you can make a circular Lissajous figure.

Also, if you combine more than 2 together you can get other pretty patterns.

 

No, the OP didn't, but a few people trying to help did. As for the long cable, I assume you are using it as a delay line. Your two sine waves being the wave at the start of the cable and the wave after delay in going through the cable? Although you can make pretty patterns by combining more than two sine wave, as I understand it Lissajous figures are generally understood to be a combination of only two. These figures to be displayed on a cathode ray oscilloscope which only has two inputs, X and Y deflection. That is the only way I have used them for practical purposes.

[John Cuthber]

Well, I guess he was mistaken when he said it was "pretty close".

Anyway, there are indeed, generally 2 inputs on a scope (3 if it has a modulation input too).

The point I was making was that you can add two signals together say a sin(x) and b sin (3x) then plot that against , for example c cos (x).

 

If a and c are equal and b is zero you get a circle.

If b is non-zero then you get another pattern. That's the equivalent of looking at a distorted sine wave vs a cosine.

I'm not certain how much use it would ever be- but the patterns can be pretty.