Finding a Vector-Valued Function to Parametrize the Curve (x-1)^2 + y^2 = 1

[cytochrome]

Homework Statement

Find a vector-valued function f that parametrizes the curve (x-1)^2 + y^2 = 1

Homework Equations

(x-1)^2 + y^2 = 1

The Attempt at a Solution

The equation is the graph of a circle that is 1 unit to the right of the origin, therefore a parametrization would be

x(t) = cos(t) + 1
y(t) = sin(t)

Therefore a vector-valued function that parametrizes this curve is given by

r(t) = (cos(t) + 1)i + sin(t)jI've been having trouble with parametrization lately so I was wondering if this is correct. Also, is there a better method to go about this sort of thing? Is it entirely visual and you just have to have a "feel" for it?

 

[CAF123]

Yes, that would be a correct parametrisation. There are of course ways to check that it is correct.

 

[cytochrome]

Yes, that would be a correct parametrisation. There are of course ways to check that it is correct.

But is there an algorithm or anything to go about it? I just had to visualize it. Will every parametrization problem be like that?

 

[LCKurtz]

But is there an algorithm or anything to go about it? I just had to visualize it. Will every parametrization problem be like that?

I would say pretty much yes to that. There are always many possible parameterizations and often some are more natural or "better" than others for a particular purpose. In your example, visualizing it as a circle and thinking in terms of sines and cosines is exactly the appropriate approach. There isn't a magic procedure that will always mindlessly work.