Talk:Holditch's theorem

Latest comment: 7 years ago by Michael Hardy in topic Proof

Proof

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Does anyone have a proof for this theorem? N Shar 02:33, 25 December 2005 (UTC)Reply

Here is a proof. Let $x(t)$ denote a parametrization of the curve and let $v(t)$ a vector of constant length $v$ along the curve. Let $\lambda$ be a real number and consider the curve $x_\lambda (t):= x(t) + \lambda v(t)$. A well-known application of Green's theorem gives a formula for the area $A_\lambda$ enclosed by $\x_\lambda$. Namely, \[ A_\lambda = 1/2 \int x'_\lambda (t) \wedge x_\lambda (t) dt \] Assuming that $x_1(t)$ parametrizes the same curve as $x_0(t)$ we have \[ A_1 = A_0 \] which implies \[ \int x' \wedge v + v' \wedge x dt = -\int v' \wedge v dt\] and so \[ A_\lambda - A_0 = -\lambda \int v' \wedge v + \lambda^2 \int v' wedge v dt = (1/2) \lambda\cdot(\lambda -1) \int v' \wedge v \, dt and since $v(t)$ goes once around a circle of radious $v$ the last integral is $\pi v^2$. Thus \[ A_0 - A_\lambda = pq \pi \] where $p=\lambda . v$ and $q= \lambda . v $. QED — Preceding unsigned comment added by Holonomia (talkcontribs) 16:23, 24 April 2012 (UTC)Reply

Here is a trascription of "Holonomia" 's comments:

Here is a proof. Let x(t) denote a parametrization of the curve and let v(t) a vector of constant length v along the curve. Let λ be a real number and consider the curve xλ(t):= x(t) + λv(t). A well-known application of Green's theorem gives a formula for the area Aλ enclosed by xλ. Namely,

 

Assuming that x1(t) parametrizes the same curve as x0(t) we have

 

which implies

 

and so

 

and since v(t) goes once around a circle of radius v the last integral is πv2.

Thus

 

where p = λ · v and q = λ · v. QED

Michael Hardy (talk) 19:05, 17 May 2017 (UTC)Reply

Pat Ballew's blog

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It seems too bad that policy dictates immediate removal of an external link to Pat Ballew's blog entry on Hamnett Holditch:

In this instance, at least, the blog entry gives helpful information of a strictly factual nature not in the article. An alternative is to duplicate this information in the article, for example in a new section on Holditch's biography, but that would not give due credit to Ballew as source. — Preceding unsigned comment added by 129.180.1.224 (talk) 04:12, 22 September 2013 (UTC)Reply