Re: parametric: x = cos(t), y = sin(t)

In article <cMedna3sY9adN8HanZ2dnUVZ_vyinZ2d@[EMAIL PROTECTED]
>,
Darrell wrote:

> Daniel C. Bastos wrote:
>> In article <EIKdnQETnvkdF8banZ2dnUVZ_ruqnZ2d@[EMAIL PROTECTED]
>,
>> Darrell wrote:
>> 
>>> Daniel C. Bastos wrote:
>>>
>>>> Okay. I was having trouble with your notation, so I wrote x + 3 = t
and
>>>> then y = 3(x + 3) - 11 = 3x + 9 - 11 = 3x - 2. So these two are
>>>> equivalent. They must also have the same graph. I still follow you.
>>> The parametric equations are obviously not equivalent.
>>>
>>> The graphs, although they may look the same after completion of a 
>>> sketch, are not the same.  How were they traced?  Or to ask it more 
>>> properly, what are the orientations of the graphs?
>>>
>>> The graph of a set of parametric equations is defined as the set of 
>>> points (x,y) obtained as the parameter varies over some interval.
>>>
>>> Consider the same parametric equations and the interval [0,1].
>>>
>>> a) x = t
>>>     y = 3t - 2
>>>
>>> For t = 0, x = 0
>>>             y = 3(0) - 2 = -2  ...the point (0,-2)
>>> For t = 1, x = 1
>>>             y = 3(1) - 2 = 3 - 2 = 1   ...the point (1,1)
>>>
>>> b) x = t - 3
>>>     y = 3t - 11
>>>
>>> For t = 0, x = 0 - 3 = -3
>>>             y = 3(0) - 11 = -11  ...the point (-3,-11)
>>> For t = 1, x = 1 - 3 = -2
>>>             y = 3(1) - 11 = 3 - 11 = -8   ...the point (-2,-8)
>> 
>> Hm, I see. They are traced differently.
>> 
>>> Note that all four points satisfy the rectangular equation y = 3x - 2.

>> 
>> What do you mean? All the points above satisfy y = 3x - 2. I don't know
>> what a rectangular equation is, though; what is it?
>
> i.e. a "Cartesian" equation, with x and y (no parameter.)  The 
> rectangular equation y=3x-2 is obtained from eliminating the parameter 
> of the aforementioned parametric equations.

Okay.

>>> Now reconsider what were were discussing about the equivalence
relation 
>>> between 360deg, and -360deg.  Specifically, think of how this applies
to 
>>> a circle traced clockwise from a certain point vs. the same circle 
>>> traced counterclockwise from the same point.
>> 
>> They'd be traced differently. Is this a problem to have 360, -360, 0
all
>> be the same angle?
>
> Yes, if you mean angles of measure 360deg, -360deg, and 0deg.  As a 
> practical application, do you get the same sound if you play a tape 
> forwards as you do backwards?  Or if the tape doesn't move at all?

In usual cases, the sound would change; but it might be possible to have
a tape that plays the same either way; no? I don't know though. But this
is one case. Here's another: do you have more candies if you end up with
2 after giving 5 to your friend or 2 candies after earning 1 when had
just another --- or if you had 2 and nothing happened at all?

> BTW the parameter can be anything according to the application involved.

>    t is often used for "time" and theta is often used as "angle."  In 
> applications, such as motion problems, you will see the im****tance more 
> clearly than my feeble explanation. For example, in certain motion 
> problems the rectangular equation (and graph) tells you where an object 
> has been, but it fails to tell you _when_ it was at a particular point. 
>   The parametric equations complete that information (since t=time in 
> these cases.)

Yeah. I see that.

> My point was simply to get you to recognize that a rectangular 
> (Cartesian) graph does not have unique parametric equations, thus when 
> needed we must consider the entire "plane curve" (the graph _and_ its 
> parametric equations.)

I see it now. Thanks much, Darrell.