Talk:Holditch's theorem

Proof

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

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 (talk • contribs) 16:23, 24 April 2012 (UTC)

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 x_λ(t):= x(t) + λv(t). A well-known application of Green's theorem gives a formula for the area A_λ enclosed by x_λ. Namely,

$A_{\lambda }={\frac {1}{2}}\int x'_{\lambda }(t)\wedge x''_{\lambda }(t)\,dt$

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

$A_{1}=A_{0}\,$

which implies

$\int x'\wedge v''+v'\wedge x''\,dt=-\int v'\wedge v''\,dt$

and so

$A_{\lambda }-A_{0}=-\lambda \int v'\wedge v''+\lambda ^{2}\int v'\wedge v''\,dt={\frac {1}{2}}\lambda \cdot (\lambda -1)\int v'\wedge v''\,dt$

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

Thus

$A_{0}-A_{\lambda }=pq\pi \,$

where p = λ · v and q = λ · v. QED

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

Pat Ballew's blog

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

• Pat's blog.

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)