![]() ![]() To $\dllp(t_1)$ is simply $\dllp(t_1) - \dllp(t_0)$, so the length of the $\dllp(t_1)$, the second line segment goes from the point $\dllp(t_1)$ ![]() Line segment goes from the point $\dllp(t_0)$ to the point Segments, we could define $t_0, t_1, \ldots, t_n$ so that the first What's the length of each line segment? If there are $n$ line Consequently, the total length $L(\Delta t)$ of the line segments Increases, and the line segments become closer and closer to the Length of each line segment), the total length of the line segmentsīecomes a better estimate of the slinky arc length.Īs $\Delta t$ approaches zero, the length ofĮach line segment shrinks toward zero, the number of line segments However, if you increase the number of line segments (decreasing the The length of the line segments underestimate the arc length of the slinky. Of the line segments, and denote the total length of the line Let $\Delta t$ specify the discretization interval Lengths of all the line segments, you'll get an estimate of the length The length of the line segments is easy to measure. As $\Delta t \to 0$, the length $L(\Delta t)$ of the line segment approximation approaches the arc length of the helix from below. The discretization size of line segments $\Delta t$ can be changed by moving the cyan point on the slider. The green lines are line segments that approximate the helix. The vector-valued function $\dllp(t)=(\cos t, \sin t, t)$ parametrizes a helix, shown in blue. The slinky, rather than being a curved wire, was really composed of aīunch of short straight wires. You do to estimate its length? One thing you could do is pretend that If you can't stretch the slinky into one straight line, what could That stretching it into a straight line is virtually impossible. Tried to do that with a slinky (or a strong spring), you'd discover Would be to stretch it out into one straight line. Probably the easiest way to calculate the length of the slinky It's much easier to calculate the length of straight pieces of Unfortunately, it's difficult to calculate the length of a curved piece Which we call the arc length of the parametrized curve. Imagine we wanted to estimate the length of the slinky, As you change $t$ by moving the cyan point along the interval $$, the red point traces out the helix. For each value of $t$, the red point represents the vector $\dllp(t)$. ![]() This helix is the image of the interval $$ (shown in cyan) under the mapping of $\dllp$. ![]()
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