SmtC: Show me the Code!

SmtC: Show me the Code

Ole Peter Smith

Instituto de Matemática e Estatística

Universidade Federal de Goiás

http://www.olesmith.com.br

Roulettes

O dia que bosta valesse ouro.
Os pobres todos nascerão sem c*.

Grafitti.

< Trochoids | Epicycloid | Epitrochoid >

Epicycloid: Rolling a Circle on the outside of a Circle

Formulas

$\underline{\mathbf{r}}(t)=r\left( \omega\underline{\mathbf{e}}(t) - \underline{\mathbf{e}}(\omega t) \right)$	$\underline{\mathbf{r}}'(t)=r\omega\left( \underline{\mathbf{f}}(t) - \underline{\mathbf{f}}(\omega t) \right)$
$\underline{\mathbf{r}}''(t)=r\omega\left( - \underline{\mathbf{e}}(t) + \omega\underline{\mathbf{e}}(\omega t) \right)$	$\widehat{\underline{\mathbf{r}}}'(t)=r\omega\left( - \underline{\mathbf{e}}(t) + \underline{\mathbf{e}}(\omega t) \right)$
$v(t)^2=2r^2\omega^2\left( 1- \cos{ (\omega-1)t} \right)$	$D(t)=r^2\omega^2(1+ \omega)(1 -\cos{ (\omega-1)t})$
$\frac{d}{dt} v(t)^2=2r^2\omega^2(\omega-1)\sin{ (\omega-1)t}$	$D'(t)=r^2\omega^2(\omega^2-1) \sin{(\omega-1)t}$
$\underline{\mathbf{t}}(t)=\frac{ \underline{\mathbf{f}}(t) - \underline{\mathbf{f}}(\omega t) }{ \sqrt{ 2(1- \cos{ (\omega-1)t}) }}$	$\underline{\mathbf{n}}(t)=-\frac{ \underline{\mathbf{e}}(t) - \underline{\mathbf{e}}(\omega t) }{ \sqrt{ 2(1- \cos{ (\omega-1)t}) }}$
$\psi(t)=\frac{\omega+1}=\frac{R+2r}{2r}$	$\varphi(t)=\frac{2}{\omega+1}=\frac{2r}{R+2r}$
$\kappa(t)=\frac{\omega+1}{2\sqrt{2}r\omega} \cdot \frac{1}{\sqrt{1 -\cos{ (\omega-1)t}}}$	$\rho(t)=\frac{2\sqrt{2}r\omega}{\omega+1} \cdot \sqrt{1-\cos{ (\omega-1)t}}$
$\underline{\mathbf{c}}(t)=\frac{Rr}{R+2r}\left( \omega\underline{\mathbf{e}}(t) + \underline{\mathbf{e}}(\omega t) \right)$
$\underline{\mathbf{c}}'(t)=\frac{Rr}{R+2r}\omega\left( \underline{\mathbf{f}}(t) + \underline{\mathbf{f}}(\omega t) \right)$
$\rho'(t)=\sqrt{2}r\omega\cdot \frac{\omega-1 }{\omega+1} \cdot\frac{\sin{(\omega-1)t}}{\sqrt{1 -\cos{ (\omega-1)t}}}=\pm 2r\omega\cdot \frac{\omega-1 }{\omega+1} \cdot\cos{\left( \frac{\omega-1}{2} t \right)}$
$\mathcal{S}=\left\{t \in \mathbb{R}~\left\|~~2p\pi\frac{R}{r},~p \in \mathbb{Z}\right.\right\}$
$\mathcal{E}=\left\{t \in \mathbb{R}~~~\left\|(2 p+1) \pi\frac{R}{r},~p \in \mathbb{Z}\right.\right\}$
$s(t)-s(0)=\frac{2r\omega}{\omega-1} \cdot \left(1-\cos{ \frac{\omega-1}{2} t}\right),~0 < t < \frac{2\pi}{\omega-1}$