How Do You Reparameterize a Curve with Respect to Arc Length?

[brandtw]

I think that I'm just having problems with my derivitives and integrals. If someone one could show me what the steps of this problem look like it would be greatly appreciated. "Reparameterize the curve with respect to the arc length measured from the point where t=0 in the direction of increasing t. r(t)=e^t*sint i +e^t*cost j"

 

[HallsofIvy]

Looks like homework to me!

Step 1: If you want to use arc-length as a parameter, then you will need to know what the arc-length is! The formula for arclength, given x and y in terms of parameter t, taking t= 0 as starting point and t>0 as positive arclength, is:
$$s(t)= \int_0^t\sqrt{x'^2+y'^2}dt$$.

Step 2: That will give you arclength, s, as a function of t. Now solve for t as a function of s and substitute that formula into your parametric equations.

 

[M98Ranger]

Reparameterizing a curve means to change the parameter used to describe the curve. In this case, we are asked to reparameterize the curve with respect to the arc length, which is the distance measured along the curve.

To do this, we first need to find an expression for the arc length of the curve. This can be done using the arc length formula:

s = ∫√(x'(t)^2 + y'(t)^2) dt

where x'(t) and y'(t) are the derivatives of the x and y components of the curve with respect to t.

In this case, we have r(t) = e^t*sint i + e^t*cost j, so x(t) = e^t*sint and y(t) = e^t*cost. Taking the derivatives, we get x'(t) = e^t*sint + e^t*cost and y'(t) = e^t*cost - e^t*sint.

Plugging these into the arc length formula, we get:

s = ∫√(e^2t*sint^2 + e^2t*cost^2) dt

= ∫√(e^2t) dt

= ∫e^t dt

= e^t + C

Now, we can use this expression for the arc length to reparameterize the curve. We want to find a new parameter, say u, such that when we plug it into the original curve, we get the same points on the curve as when we use t. In other words, we want to find a function u(t) such that r(t) = r(u(t)).

To do this, we can use the inverse function theorem, which states that if a function f is invertible, then the inverse function f^-1 can be used to reparameterize the curve. In our case, we can use the inverse of the arc length function we found earlier, so u = e^t + C.

Plugging this into our original curve, we get:

r(u) = e^(e^t + C)*sint i + e^(e^t + C)*cost j

= e^u*sint i + e^u*cost j

This is the reparameterized curve with respect to the arc length measured from the point where t=0 in the direction of increasing t