Help in re-parameterizing the curve

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In summary, the conversation is about re-parameterizing a curve in order to make it parameterized by arc-length, even when the speed is given in terms of t. This involves finding the integral of ||ds/du||, finding the function h(t) and its inverse f(t), and then composing the original curve with f(t) to get a new curve that is parametrized by arc length.
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sarah7
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Hi,

can someone help in re-parameterizing the curve

δ(t)=(2/3(√(L^2+9))cos(t),1/3(√(L^2+9))sin(t),L)

I found dδ/dt then I got the speed to be 1/3√(L^2+9)√(1+3sin^2(t))

L is just a constant z=L

I know how to re-parameterize curves to make them parameterized by arc-length when I get a constant speed but here the speed is in terms of t!

Thanks
 
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  • #2
You should get the integral of ||ds/du|| from zero to t, then you find the function h(t). After this you should find the inverse of h(t), say it is the function f(t), then you find the composition of your original curve and f(t), the new curve is a curve that is parametrized by arc length.
 

What is re-parameterization of a curve?

Re-parameterization of a curve is the process of changing the parameterization of a curve, while keeping the shape and geometry of the curve intact. It involves changing the way the curve is described, such as changing the parameter from time to distance.

Why is re-parameterization of a curve important?

Re-parameterization of a curve is important because it allows for easier manipulation and analysis of curves. It can also help to simplify complex curves and make them more understandable.

What are the common methods used for re-parameterization of a curve?

The most common methods for re-parameterization of a curve include arc-length parameterization, chord-length parameterization, and normalized arc-length parameterization.

What are the challenges in re-parameterizing a curve?

One of the main challenges in re-parameterizing a curve is finding a suitable parameterization that accurately represents the curve while also being easy to work with. Another challenge is ensuring that the re-parameterized curve maintains its original shape and properties.

How is re-parameterization of a curve used in real-world applications?

Re-parameterization of a curve is used in various applications, such as computer graphics, animation, and computer-aided design. It is also important in fields such as physics and engineering, where accurate representation and manipulation of curves are necessary for analysis and design.

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