Search results
21 sty 2022 · This transformation is called arc length reparameterization or parameterization. And the most useful application of the arc length parameterization is that a vector function \(\vec{r}(t)\) gives the position of a point in terms of the parameter \(t\).
29 gru 2020 · This parameter \(s\) is very useful, and is called the arc length parameter. How do we find the arc length parameter? Start with any parametrization of \(\vecs r\). We can compute the arc length of the graph of \(\vecs r\) on the interval \([0,t]\) with \[\text{arc length } = \int_0^t\norm{\vecs r\,'(u)} du.\]
17 sie 2024 · A useful application of this theorem is to find an alternative parameterization of a given curve, called an arc-length parameterization. Recall that any vector-valued function can be reparameterized via a change of variables.
One simple example is $$ x (t) = \cos (t) \quad ; \quad y (t) = \sin (t) \quad (0 \le t \le 2\pi) $$ This a parameterization of the unit circle, and the arclength from the start of the curve to the point $ (x (t), y (t))$ is $t$.
Arc length parameterization ensures that the speed of movement along the curve is constant, making it useful in physics and engineering applications. When working with curves that have sharp turns or varying speeds, reparameterizing in terms of arc length allows for smoother and more manageable representations.
Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length).
The arc length of an epicycle ~r(t) = [t+ sin(t);cos(t)] parame-terized by 0 t 2ˇ. We have j~r0(t)j = p 2 + 2cos(t). so that L = R 2ˇ 0 p 2 + 2cos(t) dt. A substitution t = 2u gives L = R ˇ 0 p 2 + 2cos(2u) 2du = ˇ 0 p 2 + 2cos2(u) 2sin2(u) 2du = R ˇ 0 p 4cos2(u) 2du= 4 R ˇ 0 jcos(u)jdu= 8. Find the arc length of the catenary ~r(t) = [t ...
