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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
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$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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The circumference of a circle (the length of the curve that is the circle) as discussed in geometry.
Depending on which circle constant you use, given a radius $r$ of a circle $\mathcal{C}$ in the Euclidean plane $\mathbb{R}^2$, the circumference of a circle is either $C(r) = \tau r$ or $C(r) = 2 \pi r$.
In this proof, we are using the circle constant $\tau = 2 \pi$.
Given any Euclidean plane $\mathbb{R}^2$, one could select an orthonormal basis on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two orthonormal vectors $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as
Then the circumference of $\mathcal{C}$ is given by the following integral:
which evaluates to
In this proof, we are using the circle constant $\tau = 2 \pi$.
The perimeter of a regular polygon $\mathcal{P}_n$ (strictly speaking, a regular polygonal line) with $n$ sides and circumradius $r$ is given by the sequence of functions $P_\mathcal{P}:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$
which embeds in the $\mathbb{R}_+$-action $P_\mathcal{P}^\prime:\mathbb{R}_+ \to (\mathbb{R} \to \mathbb{R})$, defined as
The limit of $P^\prime$ as $n$ goes to infinity is the circumference of a circle with radius $r$:
Last revised on May 17, 2022 at 01:02:20. See the history of this page for a list of all contributions to it.