nLab circumference of a circle

Contents

Context

Analysis

Geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

The circumference of a circle (the length of the curve that is the circle) as discussed in geometry.

Definition/proposition and proof

Proposition

Depending on which circle constant you use, given a radius rr of a circle 𝒞\mathcal{C} in the Euclidean plane 2\mathbb{R}^2, the circumference of a circle is either C(r)=τrC(r) = \tau r or C(r)=2πrC(r) = 2 \pi r.

Proof by integration

Proof

In this proof, we are using the circle constant τ=2π\tau = 2 \pi.

Given any Euclidean plane 2\mathbb{R}^2, one could select an orthonormal basis on 2\mathbb{R}^2 by postulating an origin 00 at the center of the circle 𝒞\mathcal{C} and two orthonormal vectors i^\hat{i} and j^\hat{j}. The circle 𝒞\mathcal{C} could be parameterized by a function r:[0,τ] 2\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2 defined as

r(θ)rcos(θ)i^+rsin(θ)j^\overrightarrow{r}(\theta) \coloneqq r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}

Then the circumference of 𝒞\mathcal{C} is given by the following integral:

C(r)= 0 τ|r(θ)|dθC(r) = \int_{0}^{\tau} \vert \overrightarrow{r}(\theta) \vert d \theta

which evaluates to

C(r)= 0 τ|rcos(θ)i^+rsin(θ)j^|dθ= 0 τr((cos(θ)) 2+(sin(θ)) 2)dθ= 0 τrdθ=τrC(r) = \int_{0}^{\tau} \vert r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j} \vert d \theta = \int_{0}^{\tau} r((\cos(\theta))^2 + (\sin(\theta))^2) d \theta = \int_{0}^{\tau} r d \theta = \tau r

Proof by limits of regular polygons

Proof

In this proof, we are using the circle constant τ=2π\tau = 2 \pi.

The perimeter of a regular polygon 𝒫 n\mathcal{P}_n (strictly speaking, a regular polygonal line) with nn sides and circumradius rr is given by the sequence of functions P 𝒫:()P_\mathcal{P}:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})

P 𝒫(n)(r)=r(2n)sin(τ2n)P_\mathcal{P}(n)(r) = r (2 n) \sin\left(\frac{\tau}{2n}\right)

which embeds in the +\mathbb{R}_+-action P 𝒫 : +()P_\mathcal{P}^\prime:\mathbb{R}_+ \to (\mathbb{R} \to \mathbb{R}), defined as

P 𝒫 (n)(r)=r(2n)sin(τ2n)P_\mathcal{P}^\prime(n)(r) = r (2 n) \sin\left(\frac{\tau}{2 n}\right)

The limit of P P^\prime as nn goes to infinity is the circumference of a circle with radius rr:

C(r)=lim nP 𝒫 (n)(r)=lim nr(2n)sin(τ2n)=rlim m0sin(τm)m=rlim m0 msin(τm) mm=rlim m0τcos(τm)1=τrC(r) = \lim_{n \to \infty} P_\mathcal{P}^\prime(n)(r) = \lim_{n \to \infty} r (2 n) \sin\left(\frac{\tau}{2 n}\right) = r \lim_{m \to 0} \frac{\sin(\tau m)}{m} = r \lim_{m \to 0} \frac{\partial_m \sin(\tau m)}{\partial_m m} = r \lim_{m \to 0} \frac{\tau \cos(\tau m)}{1} = \tau r

See also

Last revised on May 17, 2022 at 01:02:20. See the history of this page for a list of all contributions to it.