What is the relation between a "Sine Wave" and a "Circle"...

Hi,
What is the relation between a "Sine Wave" and a "Circle".......both these
appear Very frequently in DSP,Motor Control etc....

Thanks & Regards,
Shakes_ck

"shakes_ck" <shekar.external@infineon.com> writes:

> Hi,
> What is the relation between a "Sine Wave" and a "Circle".......both these
> appear Very frequently in DSP,Motor Control etc....

Imagine a circle of radius 1 centered at the origin of a cartesian
coordinate system. Then imagine a point on the circle that starts out
at (1,0) at time t = 0 seconds and travels around the circle at a
constant velocity of f revolutions per second.

Then the x-coordinate of the dot at any time t is cos(2*pi*f*t) and
its y-coordinate is sin(2*pi*f*t). If you make a plot of sin(2*pi*f*t)
as a function of t, then you've plotted a sine wave.

That, in a nutshell, is basic trigonometry. Check to see how you
can sign up for it at your high school. It is one of the most
useful math courses you will ever take.
-- 
%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
%% Fuquay-Varina, NC            %                    and kiss her interface, 
%%% 919-577-9882                %            til then, I'll leave her alone."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://home.earthlink.net/~yatescr

>"shakes_ck" <shekar.external@infineon.com> writes:
>
>> Hi,
>> What is the relation between a "Sine Wave" and a "Circle".......both
these
>> appear Very frequently in DSP,Motor Control etc....
>
>Imagine a circle of radius 1 centered at the origin of a cartesian
>coordinate system. Then imagine a point on the circle that starts out
>at (1,0) at time t = 0 seconds and travels around the circle at a
>constant velocity of f revolutions per second.
>
>Then the x-coordinate of the dot at any time t is cos(2*pi*f*t) and
>its y-coordinate is sin(2*pi*f*t). If you make a plot of sin(2*pi*f*t)
>as a function of t, then you've plotted a sine wave.
>
>That, in a nutshell, is basic trigonometry. Check to see how you
>can sign up for it at your high school. It is one of the most
>useful math courses you will ever take.
>-- 
>%  Randy Yates                  % "Maybe one day I'll feel her cold
embrace,
>%% Fuquay-Varina, NC            %                    and kiss her
interface, 
>%%% 919-577-9882                %            til then, I'll leave her
alone."
>%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO
  
>http://home.earthlink.net/~yatescr
>

Thanks!!!....Surely it was a stupid question...!!

"shakes_ck" <shekar.external@infineon.com> writes:

> Thanks!!!....Surely it was a stupid question...!!

No, not at all! 
-- 
%  Randy Yates                  % "Rollin' and riding and slippin' and
%% Fuquay-Varina, NC            %  sliding, it's magic."
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Living' Thing', *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Randy Yates wrote:

   ...

> That, in a nutshell, is basic trigonometry. Check to see how you
> can sign up for it at your high school. It is one of the most
> useful math courses you will ever take.

The concepts are extraordinarily enlightening, but the actual working 
out of formulas is most easily done with complex algebra. Formulas such 
as sec^2(x) = 1 + sin^2(x) and sin(2x) = 2 sin(x)cos(x) simply fall out 
of the hat. The second makes for an interesting construction with trig 
and demonstrated cleverness. It is trivial with complex algebra.

I submit that a high-school fluent in algebra and de Moivre's theorem 
who accepts without proof that exp(i*pi) = -1* can learn the entire 
content of the standard trigonometry course in two weeks of classes. 
Shekar can catch up quickly.

Jerry
__________________________
* The straight-forward proof uses the Taylor series for exp(x), usually 
covered in calculus.
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

shakes_ck wrote:

>[snip]
> Thanks!!!....Surely it was a stupid question...!!

May I expand on Randy's reply.
The *ONLY* "stupid/dumb" question is the one you do not ask.

When a "student", but not a "pupil", asks a question this group enjoys 
helping him or her.

[PS to further your education, use a *LARGE* unabridged dictionary to to 
understand derivation of "student" ;]

[so I'm fascinated with linguistics]
[can no enginurs be literat ;] [pps an in joke]

There are soooooo many ways to look at it too :

1. Sin^2(x) + cos^2(x) = 1. This is the basic equation of a circle.

2. Randy's explanation is exactly how the Simple Harmonic Motion is
defined Y(t) = A*sin(w*t). So in the time domain it is a sinusoidal
motion. But in the "w" domain, the sine wave traverses a simple circle.


Sastry


shakes_ck wrote:
> Hi,
> What is the relation between a "Sine Wave" and a "Circle".......both these
> appear Very frequently in DSP,Motor Control etc....
> 
> Thanks & Regards,
> Shakes_ck

Jerry Avins wrote:
> Randy Yates wrote:
>
>    ...
>
> > That, in a nutshell, is basic trigonometry. Check to see how you
> > can sign up for it at your high school. It is one of the most
> > useful math courses you will ever take.
>
> The concepts are extraordinarily enlightening, but the actual working
> out of formulas is most easily done with complex algebra. Formulas such
> as sec^2(x) = 1 + sin^2(x) and sin(2x) = 2 sin(x)cos(x) simply fall out
> of the hat. The second makes for an interesting construction with trig
> and demonstrated cleverness. It is trivial with complex algebra.

Maybe, but as soon as you make the move to complex algebra all sense of
reality disappears,  especially for beginners, its a big price to pay
just to simplify the equations.

joep1000@yahoo.com wrote:
> Jerry Avins wrote:
>> Randy Yates wrote:
>>
>>    ...
>>
>>> That, in a nutshell, is basic trigonometry. Check to see how you
>>> can sign up for it at your high school. It is one of the most
>>> useful math courses you will ever take.
>> The concepts are extraordinarily enlightening, but the actual working
>> out of formulas is most easily done with complex algebra. Formulas such
>> as sec^2(x) = 1 + sin^2(x) and sin(2x) = 2 sin(x)cos(x) simply fall out
>> of the hat. The second makes for an interesting construction with trig
>> and demonstrated cleverness. It is trivial with complex algebra.
> 
> Maybe, but as soon as you make the move to complex algebra all sense of
> reality disappears,  especially for beginners, its a big price to pay
> just to simplify the equations.

Doesn't it make more to spend a few weeks getting comfortable with 
complex numbers that to spend a semester learning trig without them?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

joep1000@yahoo.com skrev:
> Jerry Avins wrote:
> > It is trivial with complex algebra.
>
> Maybe, but as soon as you make the move to complex algebra all sense of
> reality disappears,  especially for beginners, its a big price to pay
> just to simplify the equations.

Wring. Learning complex numbers have two main benefits:

- It simplifies algebra and provides a means to understand DSP
  literature beyond the very basic
- It prepares the mind for abstract maths.

Accepting the existense and usefulness of sqrt(-1) is *the* hurdle
in mathematics. If you can get past it, you can do anything in
maths. By accepting sqrt(-1) you accept that mathematical
abstractions need not have an obvious physical interpretation,
which in turn prepares the mind for useful analytical work that
does not get bogged down by mere drudgery of arithmetics.

Complex numbers take only a little bit of work to learn, but in
return a whole new world of maths and physics, including DSP,
is opened up.

Rune