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

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.

> 
>> 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?

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.)

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.)

-- 
Darrell