How to Measure the Length of your Earphone from a Pic?| Cheenta Probability Series

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Describing a Line (like seriously?).

I mean a "Random" Line.

Thus the Space of all Lines in \( \mathbf{R}^2 \) can be represented in "kinda" polar coordinates S = {\({(\theta, p) \mid 0 \leq \theta \leq 2 \pi, p \geq 0},\)} where \(p, \theta\) are as indicated.

Now, we are now uniformly selecting this \( (\theta, p) \) over S. This is how we select a "Random" Line.

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Understanding Crofton's Gibberish

If C is the curve, whose length we are interested to measure. Then, \( L(C)= \frac{1}{2} \times \iint n(p, \theta) dp d \theta \).

\( L(C) \) = The Length of the curve

\( n(\theta, p) \) = The Number of times "Random Line" corresponding to \( (\theta, p) \) intersects C.

Crofton meant the following on a pretty, differentiable, and cute (regular) curve C that:

\( L(C)= \frac{1}{2} \times E_{(p, \theta)} [n(p, \theta)] \)

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The First Truth

Here, we will find the length of a Line using "Random Lines". We will see, how counting points on the line, can help us find the length of a line.

Here, we choose C = Line.

A Line intersects another Line exactly once. \( n(p, \theta) = 1 \). Hence,

\( L(C)= \frac{1}{2} \times \iint n(p, \theta) dp d \theta = \frac{1}{2} \times \iint dp d\theta = \frac{1}{2} \times \int_{0}^{2 \pi}\left(\int_{0}^{|\cos \theta|(1 / 2)} d p\right) d \theta = \frac{1}{2} \times \int_{0}^{2 \pi} l / 2|\cos \theta| d \theta= l \)

The Second Truth

Approximate Curves by Lines.

Refer to Approximation of Curves by Line Segments by Henry Stone.

Just Limit!

Now, take apply the result on these lines. By linearity of expectation and the uniform convergence of the Lines to the Curve, we get the same for the curve.

Suppose that the curve c is parameterized on the interval [a,b]. Define a partition \( P \) to be a collection of points that subdivides the interval, say { \( a = a_0, a_1, a_2, ..., a_n = b\) }.

Define \( L_{r}(c)=\sum_{i=1}^{n} |c\left({a}_{i-1}\right)-c\left({a}_{i}\right)| \)

Then \( L(c)= sup_{r} L{r}(c) \)

Now, each of the \( |c\left({a}_{i-1}\right)-c\left({a}_{i}\right)|\) are approximated by the Crofton's Formula. Now, taking the limit gives ust eh required result. QED.

Measuring the Length of an Earphone

Create a Mesh of Lines (Coordinate System) - A Fixed Sample representing the Population of the Random Lines. Then count the number of intersections using those mesh of lines only and add it up.

The mathematics of the above argument is as follows.

\( L(c)= \frac{1}{2} \times \iint n(p, \theta) dp d \theta \cong \frac{1}{2} \times n r \times \frac{\pi}{3} \)

An example is show below. This is done in the app named Cauchy-Crofton App, which unfortunately is not working now and I have reported.

Outstanding Statistics Program with Applications

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