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

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In summary, the conversation discusses finding a vector-valued function that parametrizes the curve (x-1)^2 + y^2 = 1 and whether there is a specific algorithm or method to solve parametrization problems. The solution involves visualizing the curve and using trigonometric functions to create a parametrization. There is no one-size-fits-all procedure for parametrization problems, but there are ways to check that a given parametrization is correct.
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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?
 
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Yes, that would be a correct parametrisation. There are of course ways to check that it is correct.
 
  • #3
CAF123 said:
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?
 
  • #4
cytochrome said:
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.
 

1. What is the purpose of parametrizing a curve?

Parametrizing a curve means representing a curve using a set of parameters. This allows us to describe the curve in terms of a simpler equation or set of equations, making it easier to analyze and manipulate mathematically.

2. How is a curve typically parametrized?

A curve can be parametrized in several ways, but the most common method is to use a parameter t to represent the position on the curve. This is usually done by expressing the x and y coordinates of the curve as functions of t.

3. What are some advantages of parametrizing a curve?

Parametrization allows us to simplify the representation of a curve and make it more amenable to mathematical analysis. It also allows us to easily calculate important properties of the curve, such as its length, curvature, and tangent vectors.

4. Can any curve be parametrized?

Yes, any curve can be parametrized. However, different curves may require different parametrizations, and some curves may be more difficult to parametrize than others. In some cases, a curve may not have a single, unique parametrization.

5. How do you choose a suitable parametrization for a given curve?

The choice of parametrization depends on the specific properties and characteristics of the curve. In general, a good parametrization should be simple, smooth, and cover the entire curve without any gaps or overlaps. It should also be easy to differentiate and integrate, allowing for easy calculation of important properties.

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