How can I determine the direction of parametrization for a curve?

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In summary, there is no general algorithm for finding a vector valued function that parametrizes a curve. It usually requires geometric or physical insight into the problem. In this case, recognizing the ellipse as a "warped circle" and using standard parametric equations for a circle with different values for "R" can give the desired result. Additionally, a unit vector can be assigned to the x and y equations to create a vector valued function.
  • #1
cytochrome
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Is there a general way to find a vector valued function that parametrizes a curve? I'm reading through a textbook and it says nothing in depth about parametrization and suddenly there's a question...

Find a vector valued function f that parametrizes the curve in the direction indicated.

4x^2 + 9y^2 = 36





Can someone please help me with this example and shed some light on parametrization?
 
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  • #2
There is no general "algorithm" for finding a parameterization- it usually results from some geometric or physical insight into the problem. Here, for example, I would recognise the graph of [itex]4x^2+ 9y^2= 36[/itex] as an ellipse which, in "standard form", would be [itex]x^2/9+ y^2/4= 1[/itex]. In that form I can see that the ellipse has center (0, 0) and has vertices at (3, 0), (-3, 0), (0, 2), and (0, -2).

And that I can think of as a "warped circle". Standard parametric equations for a circle with center at (0, 0) and radius R are [itex]x= R cos(\theta)[/itex] and [itex]y= R sin(\theta)[/itex]. Then it should be easy to see that choosing different values for "R" will give [itex]x= 3 cos(\theta)[/itex] and [itex]y= 2sin(\theta)[/itex] will give the desired ellipse.
 
  • #3
HallsofIvy said:
There is no general "algorithm" for finding a parameterization- it usually results from some geometric or physical insight into the problem. Here, for example, I would recognise the graph of [itex]4x^2+ 9y^2= 36[/itex] as an ellipse which, in "standard form", would be [itex]x^2/9+ y^2/4= 1[/itex]. In that form I can see that the ellipse has center (0, 0) and has vertices at (3, 0), (-3, 0), (0, 2), and (0, -2).

And that I can think of as a "warped circle". Standard parametric equations for a circle with center at (0, 0) and radius R are [itex]x= R cos(\theta)[/itex] and [itex]y= R sin(\theta)[/itex]. Then it should be easy to see that choosing different values for "R" will give [itex]x= 3 cos(\theta)[/itex] and [itex]y= 2sin(\theta)[/itex] will give the desired ellipse.

That was extremely helpful, thank you so much.

One more thing - when looking for a vector valued function, can I just assign the unit vectors to the x and y equations so that f(t) = 3cos(t)i + 2sin(t)j, where i and j are both unit vectors and t is the parameter?
 
  • #4
Yes, of course, I should have said that: If x= f(t) and y= g(t) then the "vector form" is v(t)= xi+ yj= f(t)i+ g(t)j.
 
  • #5
By the way, in your the post it says "in the direction indicated" but there was no "direction indicated". My parameterization "goes around" the ellipse as [itex]\theta[/itex] increases- in the counterclockwise direction. If, instead we use [itex]x= 3cos(t)[/itex], [itex]y= -2sin(t)[/itex], or v(t)= 3cos(t)i-2 sin(t)j goes around the ellipse in the clockwise direction.
 

What is parametrization of a curve?

Parametrization of a curve is the process of representing a curve in terms of one or more variables, known as parameters. This allows for the curve to be described in a mathematical form, making it easier to analyze and manipulate.

Why is parametrization of a curve important?

Parametrization of a curve is important because it allows for a curve to be expressed in a more simplified form, making it easier to study and understand. It also allows for the curve to be manipulated and transformed using mathematical techniques.

How is a curve parametrized?

A curve can be parametrized in several ways, depending on the specific curve and its characteristics. One common method is to use a parametric equation, where the coordinates of points on the curve are expressed as functions of a parameter. Another method is to use a vector function, where the position of a point on the curve is described by a vector function of the parameter.

What are the advantages of parametrizing a curve?

Parametrization of a curve has several advantages, including the ability to easily calculate the slope, arc length, and curvature of the curve. It also allows for the curve to be expressed in a more general form, making it applicable to a wider range of problems and situations.

Can any curve be parametrized?

Yes, any curve can be parametrized, although some curves may require more complex parametrization methods. Parametrization is a powerful tool in mathematics and can be applied to a wide variety of curves, from simple lines and circles to more complex curves such as fractals and spirals.

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