text:
\section{\protect\smallskip \textbf{INTRODUCTION}}

Focal surfaces are known as line congruences. The concept of line
congruences is defined for the first time in visualization in 1991 by Hagen
and Pottman \cite{HPD}.

Let $M:X(u,v)$ be a surface defined as a real-valued function and $N\left(
u,v\right) $ be a unit normal vector on the surface. The line congruence is
defined a
\begin{equation}
C\left( u,v,z\right) =X\left( u,v\right) +zE\left( u,v\right) ,  \label{1.1}
\end{equation
where $E\left( u,v\right) $ is the set of unit vectors. For each $\left(
u,v\right) $, the equation (\ref{1.1}) indicates a line congruence and
called generatrix. Here, the parameter $z$ is a marked distance. In
addition, there exist two special points (real, imaginary or unit) on the
generatrix of $C.$ These points are called as focal points which are the
osculator points with generatrix. Therefore, focal surfaces are defined as a
geometric locus of focal points. In general, there exist two focal surfaces.
If $E\left( u,v\right) =N\left( u,v\right) $, then $C=C_{N}$ is normal
congruence. Thus, the focal surface $C_{N}$ has the following parametric
representatio
\begin{equation*}
X^{\ast }_{i}\left( u,v\right) =X\left( u,v\right) +\kappa _{i}^{-1}\left(
u,v\right) N\left( u,v\right) ,
\end{equation*
where $\kappa _{1}$ and $\kappa _{2}$ are the principle curvature functions
of the surface $M:X\left( u,v\right) $ \cite{HH}. The center of curvature of
the normal section curve corresponds to a certain level of the normal vector
at a point $X\left( u_{0},v_{0}\right) $ on $M.$ The extreme values are the
center of the curvature of two principle directions. These points correspond
to the focal points. For this reason, the congruence of lines is considered
as a set of lines which are tangent to two surfaces. Also, these two
surfaces are focal surfaces of congruence of the lines. Thus, focal points
of the normal congruence are the centers of curvature of two principle
directions. Some studies can be found about focal curves and focal surfaces
in Euclidean spaces \cite{HH, O, OA2, OA}.

Canal surface with a significant place in geometry provides benefits in
showing human internal organs, long thin objects, surface modelling, CG/ CAD
and graphics. A canal surface $X\left( u,v\right) $ obtained by spin curve
\gamma \left( u\right) $ is the combination of the spheres which are
determined by the radius function $r(u)$ and center $\gamma \left( u\right)$. If the radius function is constant, then this surface is called tubular surface. In \cite{S}, the authors studied tubular surface in Euclidean $3-
space. Recently, in \cite{KO, KO2, KOA}, the authors have attended to
tubular surfaces in Euclidean $4-$space $\mathbb{E}^{4}.$ Thanks to
geometric structure of these types of surfaces, they are also used in
reshaping and planning the movement lines of the robots (see, \cite{MPSY
).

In differential geometry, frame fields are important tools for analyzing
curves and surfaces. Frenet frame is the most familier frame field but there
is also the other frame field such as Darboux frame.
Besides Frenet frame is constructed on a curve with its velocity and the
acceleration vectors, Darboux frame is constructed on a surface with the
curves' velocity vector and the surface' normal vector. There have been so many studies about Darboux frame such as \cite{DDKO, KA, KT}.

In the current work, we study focal surfaces of a tubular surface which are
constructed by the Frenet frame and Darboux frame. We
calculate the mean and Gaussian curvatures of the focal
surfaces and get the necessary and sufficient conditions of these surfaces
to become flat and minimal.

\section{\protect\smallskip \textbf{BASIC CONCEPTS}}

Let $\gamma =\gamma (s):I\subseteq\mathbb{R}\rightarrow \mathbb{E}^{3}$ be a unit speed
curve in the Euclidean space
\mathbb{E}^{3}$. Then the derivatives
of the Frenet frame $\left\{ T,N_{1},N_{2}\right\} $ of $\gamma $
(Frenet-Serret formula)
\begin{equation*}
\left[
\begin{array}{c}
T^{^{\prime }} \\
N_{1}^{^{\prime }} \\
N_{2}^{^{\prime }
\end{array
\right] =\left[
\begin{array}{ccc}
0 & \kappa & 0 \\
-\kappa & 0 & \tau \\
0 & -\tau &
\end{array
\right] \left[
\begin{array}{c}
T \\
N_{1} \\
N_{2
\end{array
\right] ,
\end{equation*
where $\tau $, $\kappa $ are the torsion and curvature of the curve $\gamma $,
respectively \cite{C}.

Let $\gamma :$ $I\rightarrow M$ be a unit speed curve on the surface $M$.
Then, $\left\{ T,Y=T\times N,N\right\}$ is called as Darboux frame which is defined along
the curve $\gamma$. Here $T$ is the tangent vector of $\gamma$ and $N$ is the
unit normal vector of $M$. Darboux frame formulas are expressed as
\begin{equation*}
\left[
\begin{array}{c}
T~^{\prime } \\
Y^{\prime } \\
N^{\prime
\end{array
\right] =\left[
\begin{array}{ccc}
0 & k_{g} & k_{n} \\
-k_{g} & 0 & \tau _{g} \\
-k_{n} & -\tau _{g} &
\end{array
\right] \left[
\begin{array}{c}
T \\
Y \\

\end{array
\right] ,
\end{equation*
where $k_{g}$ is the geodesic curvature, $k_{n}$ is the normal curvature and
$\tau _{g}$ is the geodesic torsion of the curve $\gamma $.

The relation between the Darboux frame and the Frenet frame is given as
follows
\begin{equation*}
\left[
\begin{array}{c}
T \\
Y \\

\end{array
\right] =\left[
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos \theta  & \sin \theta  \\
0 & -\sin \theta  & \cos \theta
\end{array
\right] \left[
\begin{array}{c}
T \\
N_{1} \\
N_{2
\end{array
\right] ,
\end{equation*
where $\theta $ is the angle between the vectors $Y$ and $N_{1}$. Here
Darboux curvatures are defined by $k_{g}=\kappa \cos \theta ,$ $k_{n}=\kappa
\sin \theta $ and $\tau _{g}=\tau -\theta ^{\prime }$.

Let $M$:$X(u,v)$ be a regular surface in $\mathbb{E}^{3}$. The tangent space of
M$ is spanned by the vectors $X_{u}$ and
X_{v}$ at a point $p=X(u,v)$. The coefficients of the first fundamental form of $M$ are defined as
\begin{equation}
E=\langle X_{u},X_{u}\rangle ,\text{ \ \ }F=\left\langle
X_{u},X_{v}\right\rangle ,\text{ \ \ }G=\left\langle
X_{v},X_{v}\right\rangle ,  \label{A1}
\end{equation
where $\left\langle ,\right\rangle $ is the Euclidean inner product. We
set $W^{2}=EG-F^{2}\neq 0$.

Then the unit normal vector field of $M$ is defined a
\begin{equation}
N=\frac{X_{u}\times X_{v}}{\left\Vert X_{u}\times X_{v}\right\Vert }.
\label{A2}
\end{equation
The coeffiicients of the second fundamental form are as follow:
\begin{equation}
l=\left\langle X_{uu},N\right\rangle ,\text{ \ \ }m=\left\langle
X_{uv},N\right\rangle ,\text{ \ \ }n=\left\langle X_{vv},N\right\rangle,
\label{A3}
\end{equation
\cite{C}.

The shape operator matrix of a surface is defined as

\begin{equation*}
A_{N}=\left[
\begin{array}{cc}
\frac{l}{E} & \frac{1}{W}\left( m-\frac{F}{E}l\right) \\
\frac{1}{W}\left( m-\frac{F}{E}l\right) & \frac{1}{W^{2}}\left( En-2Fm+\frac
	F^{2}}{E}l\right
\end{array
\right] ,
\end{equation*
\cite{Be}.

The Gaussian and the mean curvatures of $M$ are given as

\begin{equation}
K=\frac{ln-m^{2}}{EG-F^{2}},  \label{ilim18}
\end{equation
an
\begin{equation}
H=\frac{En+Gl-2Fm}{2\left( EG-F^{2}\right) },  \label{ilim19}
\end{equation
respectively \cite{C, IAO}.

Let $\gamma :$ $I\rightarrow M$ be a unit speed curve on the surface $M$.
Then $\gamma $ is an asymptotic curve for which the normal curvature
vanishes in the direction $\gamma ^{\prime }$. Recall that $\gamma $ is
asymptotic if and only if $\gamma ^{\prime \prime }$ is perpendicular to the normal
vector $N$ of the surface $M$. Furthermore, $\gamma $ is a geodesic curve on
$M$ if the tangential component $(\gamma ^{\prime \prime })^{T}$ of the
acceleration of $\gamma $ vanishes \cite{C}.

\section{\protect\smallskip \textbf{FOCAL SURFACE OF A TUBULAR SURFACE WITH FRENET FRAME}}

Tubular surface with Frenet frame was studied by \"{O}zt\"{u}rk et al.,\ in
\cite{OBBA}. In this section, we handle the tubular surface with Frenet
frame and give the focal surface of this surface in $\mathbb{E}^{3}$.

Let $\gamma \left( u\right) =\left( \gamma _{1}\left( u\right) ,\gamma
_{2}\left( u\right) ,0\right) \subset \mathbb{E}^{3}$ be a curve
parametrized by arclength. Then the famous Frenet formulas becom
\begin{equation*}
\begin{array}{c}
\gamma ^{\prime }=T, \\
T^{\prime }=\kappa N_{1}, \\
N_{1}^{\prime }=-\kappa T, \\
N_{2}^{\prime }=0
\end{array
\end{equation*}

The tubular surface with respect to Frenet frame has the parametrization:
\begin{equation}
M:X\left( u,v\right) =\gamma \left( u\right) +r\left( \cos vN_{1}\left(
u\right) +\sin vN_{2}\right) ,  \label{e1}
\end{equation
where $r=const.$ is the radius of the spheres. The tangent space of $M$ at
a point $p=X(u,v)$ is spanned by
\begin{eqnarray}
X_{u} &=&\left( 1-\kappa (u)r\cos v\right) T,  \label{e3} \\
X_{v} &=&-r\sin vN_{1}+r\cos vN_{2}.  \notag
\end{eqnarray
Then coefficients of the first fundamental form are
\begin{equation}
E=\left( 1-\kappa (u)r\cos v\right) ^{2},\text{ \ \ }F=0,\text{ \ \ }G=r^{2},
\label{e10}
\end{equation
where $W^{2}=EG-F^{2}=\left( 1-\kappa (u)r\cos v\right) ^{2}r^{2}$ \cit
{OBBA}.

\begin{proposition}
	\cite{OBBA} $X(u,v)$ is a regular tubular surface patch if and only if \
	1-\kappa (u)r\cos v\neq 0$.
\end{proposition}

The unit normal vector field and the second partial derivatives of $M$ are
obtained a
\begin{equation*}
N=-\cos vN_{1}-\sin vN_{2}
\end{equation*
an
\begin{eqnarray}
X_{uu} &=&-\kappa ^{\prime }(u)r\cos vT+\kappa (u)\left( 1-\kappa (u)r\cos
v\right) N_{1},  \notag \\
X_{uv} &=&\kappa (u)r\sin vT,  \label{e8} \\
X_{vv} &=&-r\cos vN_{1}-r\sin vN_{2},  \notag
\end{eqnarray
respectively.

Then the coefficients of the second fundamental form becom
\begin{equation}
l=-\kappa (u)\left( 1-\kappa (u)r\cos v\right) \cos v,\text{ \ \ }m=0,\text{ \
	\ }n=r.  \label{e9}
\end{equation
Thus, from the equations (\ref{e10}) and (\ref{e9}),\ the Gaussian and mean
curvature functions of $M$ are calculated a
\begin{equation*}
K=\frac{-\kappa (u)\cos v}{r\left( 1-\kappa (u)r\cos v\right) },
\end{equation*
an
\begin{equation*}
H=\frac{1-2\kappa (u)r\cos v}{2r\left( 1-\kappa (u)r\cos v\right) },
\end{equation*
respectively.

The shape operator matrix of the surface $M$ is as follows
\begin{equation}
A_{N}=\left[
\begin{array}{cc}
\frac{-\kappa (u)\cos v}{1-\kappa (u)r\cos v} & 0 \\
0 & \frac{1}{r
\end{array
\right]  \label{z4}
\end{equation
\cite{OBBA}.

From now on, we can give the parametrization of the focal surface $M^{\ast }$
of $M$ by obtaining the principal curvature functions of $M$.

Using (\ref{z4}), we get the principal curvature functions a
\begin{equation}
\kappa _{1}=\frac{1}{r},\ \kappa _{2}=\frac{-\kappa (u)\cos v}{1-\kappa
	(u)r\cos v}.  \label{z5}
\end{equation
From the definition of the focal surface of a given surface and using the
equation (\ref{z5}), we obtain the focal surface $M^{\ast }$ of $M$ a
\begin{equation}
X^{\ast }\left( u,v\right) =\gamma \left( u\right) +\frac{1}{\kappa (u)\cos
}\left( \cos vN_{1}\left( u\right) +\sin vN_{2}\right) .  \label{z3}
\end{equation
The tangent space of the focal surface $M^{\ast }$ is spanned by
\begin{equation}
\left( X^{\ast }\right) _{u}=\frac{-\kappa ^{\prime }(u)}{\kappa ^{2}(u)
\left( N_{1}+\tan vN_{2}\right) ,  \label{e14}
\end{equation
an
\begin{equation*}
\left( X^{\ast }\right) _{v}=\frac{1}{\kappa (u)\cos ^{2}v}N_{2}.
\end{equation*
Thus from (\ref{e14}), the coefficients of the first fundamental form are
obtained as
\begin{equation}
E^{\ast }=\frac{\left( \kappa ^{\prime }(u)\right) ^{2}}{\kappa ^{4}(u)\cos
	^{2}v},\text{ \ \ }F^{\ast }=\frac{-\kappa ^{\prime }(u)\sin v}{\kappa
	^{3}(u)\cos ^{3}v},\text{ \ \ }G^{\ast }=\frac{1}{\kappa ^{2}(u)\cos ^{4}v},
\label{16}
\end{equation
where $\left( W^{\ast }\right) ^{2}=\frac{\left( \kappa ^{\prime }(u)\right)
	^{2}}{\kappa ^{6}(u)\cos ^{4}v}\neq 0$. Further, from (\ref{A2}) we find the
normal vector of $M^{\ast }$
\begin{equation}
N^{\ast }=-T.  \label{e17}
\end{equation}
The second partial derivatives of $X^{\ast }\left( u,v\right) $ are as in
the following
\begin{eqnarray}
\left( X^{\ast }\right) _{uu} &=&\frac{\kappa ^{\prime }(u)}{\kappa (u)}T
\frac{-\kappa ^{\prime \prime }(u)\kappa (u)+2\left( \kappa ^{\prime
	}(u)\right) ^{2}}{\kappa ^{3}(u)}N_{1}  \notag \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ }+\frac{-\kappa ^{\prime \prime }(u)\kappa
	(u)+2\left( \kappa ^{\prime }(u)\right) ^{2}}{\kappa ^{3}(u)}\tan vN_{2}
\notag \\
\left( X^{\ast }\right) _{uv} &=&-\frac{\kappa ^{\prime }(u)}{\kappa ^{2}(u)
\left( 1+\tan ^{2}v\right) N_{2},  \label{e18} \\
\left( X^{\ast }\right) _{vv} &=&\frac{2\sin v}{\kappa (u)\cos ^{3}v}N_{2}.
\notag
\end{eqnarray
Hence, from (\ref{e17}) and (\ref{e18}), we get the coefficients of the
second fundamental form of the focal surface $M^{\ast }$ a
\begin{equation}
l^{\ast }=-\frac{\kappa ^{\prime }(u)}{\kappa (u)},\text{ \ \ }m^{\ast }=0
\text{ \ \ }n^{\ast }=0.  \label{e19}
\end{equation}

Thus, we can give the following results:

\begin{theorem}
	Let $M$ be a tubular surface with Frenet frame given with the
	parametrization (\ref{e1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{z3}) in $\mathbb{E}^{3}$. Then the Gaussian
	curvature of $M^{\ast }$ vanishes, so the focal surface is flat.
\end{theorem}

\begin{proof}
	Let $M^{\ast }$ be the focal surface of $M$ with the parametrization (\re
	{z3}) in $\mathbb{E}^{3}$. Using the equations (\ref{ilim18}), (\ref{16})
	and (\ref{e19}), we get $K^{\ast }=0$, which completes the proof.
\end{proof}

\begin{theorem}
	Let $M$ be a tubular surface with Frenet frame given with the
	parametrization (\ref{e1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{z3}) in $\mathbb{E}^{3}$. Then the mean curvature
	of $M^{\ast }$ is
	\begin{equation}
	H^{\ast }=-\frac{\kappa ^{3}(u)}{2\kappa ^{\prime }(u)}.  \label{e20}
	\end{equation}
\end{theorem}

\begin{proof}
	Let $M^{\ast }$ be the focal surface of $M$ with the parametrization (\re
	{z3}) in $\mathbb{E}^{3}$. Using the equations (\ref{ilim19}), (\ref{16})
	and (\ref{e19}), we get the result.
\end{proof}

\begin{theorem}
	There is no minimal focal surface $M^{\ast }$ of the tubular surface$\ M$.
\end{theorem}

\begin{proof}
	Assume that the focal surface $M^{\ast }$ of the surface $M$ is minimal.
	From the equation (\ref{e20}), the curvature function $\kappa $ according to
	Frenet frame vanishes identically, this is a contradiction. Thus, there is
	no minimal focal surface of the tubular surface $M$.
\end{proof}

\begin{theorem}
	Let $M$ be a tubular surface with Frenet frame given with the
	parametrization (\ref{e1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{z3}) in $\mathbb{E}^{3}$. Then the focal surface
	M^{\ast }$ has constant mean curvature if and only if the curvature function
	$\kappa (u)$ satisfies
	\begin{equation}
	\kappa (u)=\pm \frac{\sqrt{\left( u+c_{1}c\right) c}}{u+c_{1}c},  \label{e21}
	\end{equation}
	where $c,c_{1}$ are real constants.
\end{theorem}

\begin{proof}
	Let $M^{\ast }$ be the focal surface of $M$ with the parametrization (\re
	{z3}) in $\mathbb{E}^{3}$. From the equation (\ref{e20}), we get the
	differential equatio
	\begin{equation*}
	\kappa ^{3}(u)+2\kappa ^{\prime }(u)c=0,
	\end{equation*}
	which has a non-trivial solution (\ref{e21}).
\end{proof}

\begin{example}
	Let us consider the unit speed planar curve with the parameterization
	\begin{equation*}
	\gamma (u)=\left( \left( \frac{u}{\sqrt{2}}+1\right) \cos \left( \ln \left(
	\frac{u}{\sqrt{2}}+1\right) \right) ,\left( \frac{u}{\sqrt{2}}+1\right) \sin
	\left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right) ,0\right) .
	\end{equation*
	The Frenet apparatus of this curve are determined by
	\begin{equation*}
	\begin{array}{c}
	T(u)=\gamma ^{\prime }(u)=\frac{1}{\sqrt{2}}\left( \cos \left( \ln \left(
	\frac{u}{\sqrt{2}}+1\right) \right) -\sin \left( \ln \left( \frac{u}{\sqrt{2
	}+1\right) \right) ,\right. \\
	\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \text{\ \ }\left.
	\sin \left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right) +\cos \left( \ln
	\left( \frac{u}{\sqrt{2}}+1\right) \right) ,0\right)
	\end{array
	\end{equation*
	\begin{equation*}
	\begin{array}{c}
	N_{1}(u)=\frac{1}{\sqrt{2}}\left( \sin \left( \ln \left( \frac{u}{\sqrt{2}
	+1\right) \right) -\cos \left( \ln \left( \frac{u}{\sqrt{2}}+1\right)
	\right) ,\right. \\
	\text{ \ \ \ }\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\ \text{\ }\left. \cos
	\left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right) -\sin \left( \ln
	\left( \frac{u}{\sqrt{2}}+1\right) \right) ,0\right)
	\end{array
	\end{equation*
	\begin{equation*}
	N_{2}(u)=\left( 0,0,1\right) ,
	\end{equation*
	\begin{equation*}
	\kappa (u)=\frac{1}{u+\sqrt{2}},\text{ \ \ }\tau (u)=0.
	\end{equation*
Hence, the parameterization of tubular surface around the curve $\gamma (u)$ can be written with the Frenet frame as
{\scriptsize \begin{equation*}
		\begin{array}{c}
		X\left( u,v\right) =\left( \left( \frac{u}{\sqrt{2}}+1\right) \cos \left(
		\ln \left( \frac{u}{\sqrt{2}}+1\right) \right) -\frac{r}{\sqrt{2}}\cos
		v\left( \sin \left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right) +\cos
		\left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right) \right) ,\right. \\
		\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left( \frac{u}{\sqrt{2}}+1\right)
		\sin \left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right) +\frac{r}{\sqrt{2
		}\cos v\left( \cos \left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right)
		-\sin \left( \ln \left( \frac{u}{\sqrt{2}}+1\right) \right) \right) , \\
		\text{ \ \ \ \ \ \ \ }\left. r\sin v\right)
		\end{array
		\end{equation*}}
Also, the parameterization of the focal surface of this surface is obtained as
\begin{equation*}
\begin{array}{c}
	X^{\ast }(u,v)=\left( \left( \frac{u^{2}+2\sqrt{2}u+1}{\sqrt{2}\left( u
		\sqrt{2}\right) }\right) \cos \left( \ln \left( \frac{u}{\sqrt{2}}+1\right)
	\right) -\frac{1}{\sqrt{2}\left( u+\sqrt{2}\right) }\sin \left( \ln \left(
	\frac{u}{\sqrt{2}}+1\right) \right) ,\right. \\
	\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left( \frac{u^{2}+2\sqrt{2}u+1}{\sqrt
			2}\left( u+\sqrt{2}\right) }\right) \sin \left( \ln \left( \frac{u}{\sqrt{2}
	+1\right) \right) +\frac{1}{\sqrt{2}\left( u+\sqrt{2}\right) }\cos \left(
	\ln \left( \frac{u}{\sqrt{2}}+1\right) \right) , \\
	\left. \frac{1}{u+\sqrt{2}}\tan v\right
\end{array}.
\end{equation*
	Further, by taking radius $r=\sqrt{2},$ we plot the tubular surface and its
	focal surface in $\mathbb{E}^{3}$:
	\begin{figure}[tbph]
		\centering
		\includegraphics[width =10cm, angle=0]{a1}
		\caption{Tubular surface $M$ and the focal surface $M^{\ast }$}
	\end{figure}
\end{example}

Now, we obtain the following results for parameter curves on the
focal surface $M^{\ast }$.

\begin{theorem}
	Let $M$ be a tubular surface with Frenet frame given with the
	parametrization (\ref{e1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{z3}) in $\mathbb{E}^{3}$. Then,

	i) $u$-parameter curves of the focal surface $M^{\ast }$ cannot be
	asymptotic curves.

	ii) $v$-parameter curves of the focal surface $M^{\ast }$ are asymptotic
	curves.
\end{theorem}

\begin{proof}
	Let $M^{\ast }$ be the focal surface of $M$ with the parametrization (\re
	{z3}) in $\mathbb{E}^{3}$.

	i) From the definition of an asymptotic curve, using (\ref{e17}) and (\re
	{e18}), we get $\left\langle \left( X^{\ast }\right) _{uu},N^{\ast
	}\right\rangle =\frac{\kappa ^{\prime }(u)}{\kappa (u)}=0$ if and only if
	\kappa $ is constant which contradicts the regularity of $M^{\ast }$. Thus,
	u$-parameter curves of the focal surface $M^{\ast }$ cannot be asymptotic
	curves.

	ii) Again using the same equations, we obtain $\left\langle \left( X^{\ast
	}\right) _{vv},N^{\ast }\right\rangle =0$ which means $v$-parameter curves
	of the focal surface $M^{\ast }$ are asymptotic curves.
\end{proof}

\begin{theorem}
	Let $M$ be a tubular surface with Frenet frame given with the
	parametrization (\ref{e1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{z3}) in $\mathbb{E}^{3}$. Then,

	i) $u$-parameter curves of the focal surface $M^{\ast }$ are geodesic curves
	if and only if the equatio
	\begin{equation}
	\kappa (u)=-\frac{1}{c_{1}u+c_{2}},  \label{ozgu1}
	\end{equation
	held for the curvature of $\gamma $, where $c_{1},c_{2}$ are real constants.

	ii) $v$-parameter curves of the focal surface $M^{\ast }$ are geodesic
	curves if and only if \ $v=t\pi ,$ $t\in \mathbb{Z}$.
\end{theorem}

\begin{proof}
	i) From (\ref{e17}) and (\ref{e18}), we get $\left( X^{\ast }\right)
	_{uu}\wedge N^{\ast }=0$ if and only if $-\kappa ^{\prime \prime }\kappa
	+2(\kappa ^{\prime })^{2}=0$, which has a solution (\ref{ozgu1}) for real
	constants $c_{1}$ and $c_{2}$.

	ii) Again using the same equations, we obtain $\left( X^{\ast }\right)
	_{vv}\wedge N^{\ast }=0$ if and only if $\sin v=0$ which completes the proof.
\end{proof}

\section{\protect\smallskip \textbf{FOCAL SURFACE OF A TUBULAR SURFACE WITH DARBOUX FRAME}}

Tubular surface with the Darboux frame was studied by Do\u{g}an and Yayl\i\
in \cite{DY2}. In this part, we handle the tubular surface according to
the Darboux frame and give the focal surface of this surface in $\mathbb{E
^{3}$.

Let $\gamma \left( u\right) =\left( \gamma _{1}\left( u\right) ,\gamma
_{2}\left( u\right) ,\gamma _{3}\left( u\right) \right) $ be a unit speed
curve on a surface $S.$ The tubular surface with respect to Darboux frame has
the following parametrization
\begin{equation}
M:X\left( u,v\right) =\gamma \left( u\right) +r\left( \cos vY\left( u\right)
+\sin vU\left( u\right) \right) ,  \label{d1}
\end{equation
where $r=const.$ is the radius of the spheres and $U$ is the unit normal of
the surface $S$ along the curve $\gamma $. Then the tangent space of $M$ is spanned by $X_{u}$ and $X_{v}$ at a point $p=X(u,v)$
\begin{eqnarray}
X_{u} &=&\left( 1-br\right) T-r\tau _{g}\sin vY+r\tau _{g}\cos vU,
\label{d2} \\
X_{v} &=&-r\sin vY+r\cos vU,  \notag
\end{eqnarray
wher
\begin{equation}
b\left( u,v\right) =k_{g}\left( u\right) \cos v+k_{n}(u)\sin v.  \label{d2a}
\end{equation
Then coefficients of the first fundamental form becom
\begin{equation}
E=\left( 1-br\right) ^{2}+r^{2}\tau _{g}^{2},\ \ F=r^{2}\tau _{g},\ \
G=r^{2},  \label{d3}
\end{equation
where $W^{2}=EG-F^{2}=\left( 1-br\right) ^{2}r^{2}$ \cite{DY2}.

\begin{proposition}
	\cite{DY2} $X(u,v)$ is a regular tubular surface patch if and only if \
	b\neq \frac{1}{r}$.
\end{proposition}

The unit normal vector field and the second partial derivatives of $M$ are
obtained as
\begin{equation}
N=-\cos vY-\sin vU.  \label{d4}
\end{equation
an
\begin{eqnarray}
X_{uu} &=&\left( -b_{u}r-r\tau _{g}b_{v}\right) T+\left( k_{g}\left(
1-br\right) -r\tau _{g}^{\prime }\sin v-r\tau _{g}^{2}\cos v\right) Y  \notag
\\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }+\left(
k_{n}\left( 1-br\right) +r\tau _{g}^{\prime }\cos v-r\tau _{g}^{2}\sin
v\right) U,  \notag \\
X_{uv} &=&-b_{v}rT-r\tau _{g}\cos vY-r\tau _{g}\sin vU,  \label{d5} \\
X_{vv} &=&-r\cos vY-r\sin vU,  \notag
\end{eqnarray
respectively.

Then the coefficients of the second fundamental form becom
\begin{equation}
l=-\left( 1-br\right) b+r\tau _{g}^{2},\text{ \ \ }m=r\tau _{g},\text{ \ \
n=r.  \label{d6}
\end{equation}

\bigskip

Thus, from the equations (\ref{d3}) and (\ref{d6}),\ the Gaussian and mean
curvature functions of $M$ are calculated a
\begin{equation}
K=\frac{b}{r\left( br-1\right) },  \label{d7}
\end{equation
an
\begin{equation}
H=\frac{1-2br}{2\left( 1-br\right) r}  \label{d8}
\end{equation
respectively \cite{DY2}.

Now, we focus on the parametrization of the focal surface $M^{\ast }$ of $M$
by obtaining the principal curvature functions of $M$.

Using (\ref{d7}), (\ref{d8}), and the equation $\kappa _{i}=H\pm \sqrt
	H^{2}-K}$, $(i=1,2)$, we get the principal curvature functions as
\begin{equation}
\kappa _{1}=\frac{1}{r},\text{ \ }\kappa _{2}=\frac{b}{br-1}.  \label{d9}
\end{equation
From the definition of the focal surface of a given surface and using the
equation (\ref{d9}), we obtain the focal surface $M^{\ast }$ of $M$ as
\begin{equation}
X^{\ast }\left( u,v\right) =\gamma \left( u\right) +\frac{1}{b(u,v)}\left(
\cos vY(u)+\sin vU(u)\right) ,  \label{d10}
\end{equation
where $b\left( u,v\right) =k_{g}\left( u\right) \cos v+k_{n}(u)\sin v.$

The tangent space of the focal surface $M^{\ast }$ is spanned by the vectors
\begin{equation}
\left( X^{\ast }\right) _{u}=\left( -\frac{b_{u}}{b^{2}}\cos v-\frac{1}{b
\tau _{g}\sin v\right) Y+\left( -\frac{b_{u}}{b^{2}}\sin v+\frac{1}{b}\tau
_{g}\cos v\right) U,  \label{d11}
\end{equation
an
\begin{equation}
\left( X^{\ast }\right) _{v}=\left( -\frac{b_{v}}{b^{2}}\cos v-\frac{1}{b
\sin v\right) Y+\left( -\frac{b_{v}}{b^{2}}\sin v+\frac{1}{b}\cos v\right) U.
\label{d12}
\end{equation
Thus from (\ref{d11}) and (\ref{d12}), the coefficients of the first
fundamental form are obtained as follows
\begin{equation}
E^{\ast }=\frac{b_{u}^{2}+b^{2}\tau _{g}^{2}}{b^{4}},\text{ \ \ }F^{\ast }
\frac{b_{u}b_{v}+b^{2}\tau _{g}}{b^{4}},\text{ \ \ }G^{\ast }=\frac
	b_{v}^{2}+b^{2}}{b^{4}},  \label{d13}
\end{equation
where $\left( W^{\ast }\right) ^{2}=\frac{1}{b^{6}}\left( b_{u}-b_{v}\tau
_{g}\right) ^{2}$.

\begin{proposition}
	$X^{\ast }(u,v)$ is a regular tubular surface patch if and only if \ $\left(
	k_{g}^{\prime }-k_{n}\tau _{g}\right) \cos v+\left( k_{n}^{\prime
	}+k_{g}\tau _{g}\right) \sin v\neq 0$.
\end{proposition}

\begin{proof}
	Substituting the partial derivatives of the equation (\ref{d2a}) in $W^{\ast
	}$, we get the result.
\end{proof}

One can find the normal vector of the focal surface $M^{\ast }$ as
\begin{equation}
N^{\ast }=T.  \label{d14}
\end{equation
The second partial derivatives of $X^{\ast }\left( u,v\right) $ ar
{\footnotesize \begin{eqnarray}
	\left( X^{\ast }\right) _{uu} &=&\left( \frac{b_{u}-b_{v}\tau _{g}}{b
	\right) T  \notag \\
	&&+\left( \frac{-b_{uu}b^{2}\cos v+2bb_{u}^{2}\cos v+2b_{u}b^{2}\tau
		_{g}\sin v-b^{3}\tau _{g}^{2}\cos v-b^{3}\tau _{g}^{\prime }\sin v}{b^{4}
	\right) Y \notag \\
	&&+\left( \frac{-b_{uu}b^{2}\sin v+2bb_{u}^{2}\sin v-2b_{u}b^{2}\tau
		_{g}\cos v-b^{3}\tau _{g}^{2}\sin v+b^{3}\tau _{g}^{\prime }\cos v}{b^{4}
	\right) U,  \label{m1} \\
	\left( X^{\ast }\right) _{uv} &=&\left( \frac{-b_{uv}b^{2}\cos
		v+2bb_{u}b_{v}\cos v+b_{u}b^{2}\sin v+b_{v}b^{2}\tau _{g}\sin v-b^{3}\tau
		_{g}\cos v}{b^{4}}\right) Y  \notag \\
	&&+\left( \frac{-b_{uv}b^{2}\sin v+2bb_{u}b_{v}\sin v-b_{u}b^{2}\cos
		v-b_{v}b^{2}\tau _{g}\cos v-b^{3}\tau _{g}\sin v}{b^{4}}\right) U,  \notag
	\\
	\left( X^{\ast }\right) _{vv} &=&\left( \frac{-b_{vv}b^{2}\cos
		v+2bb_{v}^{2}\cos v+2b_{v}b^{2}\sin v-b^{3}\cos v}{b^{4}}\right) Y  \notag \\
	&&+\left( \frac{-b_{vv}b^{2}\sin v+2bb_{v}^{2}\sin v-2b_{v}b^{2}\cos
		v-b^{3}\sin v}{b^{4}}\right) U.  \notag
	\end{eqnarray}}

Hence, from (\ref{d13}) and (\ref{m1}), we get the coefficients of the
second fundamental form of the focal surface $M^{\ast }$ as
\begin{equation}
l^{\ast }=\frac{b_{u}-b_{v}\tau _{g}}{b},\text{ \ \ }m^{\ast }=0,\text{ \ \
n^{\ast }=0.  \label{d15}
\end{equation
Thus, we can give the following results:

\begin{theorem}
	Let $M$ be a tubular surface according to Darboux frame given with the
	parametrization (\ref{d1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{d10}) in $\mathbb{E}^{3}$. Then the Gaussian
	curvature of $M^{\ast }$ vanishes, so the focal surface is flat.
\end{theorem}

\begin{proof}
	Let $M^{\ast }$ be the focal surface of $M$ with the parametrization (\re
	{d10}) in $\mathbb{E}^{3}$. Using the equations (\ref{d13}), (\ref{d15}) and
	(\ref{ilim18}), we get $K^{\ast }=0$, which completes the proof.
\end{proof}

\begin{theorem}
	Let $M$ be a tubular surface according to Darboux frame given with the
	parametrization (\ref{d1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{d10}) in $\mathbb{E}^{3}$. Then the mean curvature
	of $M^{\ast }$ is
	\begin{equation}
	H^{\ast }=\frac{\left( b_{v}^{2}+b^{2}\right) b}{2\left( b_{u}-b_{v}\tau
		_{g}\right) }.  \label{d16}
	\end{equation}
\end{theorem}

\begin{proof}
	Let $M^{\ast }$ be the focal surface of $M$ with the parametrization (\re
	{d10}) in $\mathbb{E}^{3}$. Using the equations (\ref{ilim19}), (\ref{d13})
	and (\ref{d15}), we get the result.
\end{proof}

\begin{theorem}
	There is no minimal focal surface of the tubular surface$\ M$.
\end{theorem}

\begin{proof}
	Assume that the focal surface $M^{\ast }$ of the surface $M$ is minimal.
	From the equation (\ref{d16}), the curvature functions $H^{\ast
	}=0\Leftrightarrow b=0$, a contradiction. Thus, there is no minimal focal
	surface of the tubular surface $M$.
\end{proof}

\begin{example}
	Let us consider the unit speed curve with the parameterization
	\begin{equation*}
	\gamma (u)=\left( \cos \frac{u}{\sqrt{2}},\sin \frac{u}{\sqrt{2}},\frac{u}
		\sqrt{2}}\right) .
	\end{equation*
	The Darboux frame and curvatures $k_{g}$, $k_{n}$ of this curve are
	determined by
	\begin{equation*}
	\begin{array}{c}
	T(u)=\gamma ^{\prime }(u)=\frac{1}{\sqrt{2}}\left( -\sin \frac{u}{\sqrt{2}
	,\cos \frac{u}{\sqrt{2}},1\right),
	\end{array
	\end{equation*
	\begin{equation*}
	\begin{array}{c}
	Y(u)=\frac{1}{\sqrt{2}}\left( -\cos \frac{u}{\sqrt{2}}+\frac{1}{\sqrt{2}
	\sin \frac{u}{\sqrt{2}},-\sin \frac{u}{\sqrt{2}}-\frac{1}{\sqrt{2}}\cos
	\frac{u}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)
	\end{array
	\end{equation*
	\begin{equation*}
	U(u)=\frac{1}{\sqrt{2}}\left( \cos \frac{u}{\sqrt{2}}+\frac{1}{\sqrt{2}}\sin
	\frac{u}{\sqrt{2}},\sin \frac{u}{\sqrt{2}}-\frac{1}{\sqrt{2}}\cos \frac{u}
		\sqrt{2}},\frac{1}{\sqrt{2}}\right) ,
	\end{equation*
	\begin{equation*}
	k_{g}(u)=\frac{1}{2\sqrt{2}},\text{ \ \ }k_{n}(u)=\frac{1}{2\sqrt{2}}.
	\end{equation*
	Hence, the parameterization of tubular surface around the curve $\gamma (u)$ can be written according to Darboux frame as
	\begin{equation*}
	\begin{array}{c}
	X\left( u,v\right) =\left( \cos \frac{u}{\sqrt{2}}\left( 1-\cos v+\sin
	v\right) +\frac{1}{\sqrt{2}}\sin \frac{u}{\sqrt{2}}\left( \cos v+\sin
	v\right) ,\right. \\
	\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\sin \frac{u}{\sqrt{2}}\left(
	1-\cos v+\sin v\right) -\frac{1}{\sqrt{2}}\cos \frac{u}{\sqrt{2}}\left( \cos
	v+\sin v\right) , \\
	\text{ \ \ \ \ \ \ \ }\left. \frac{u+\cos v+\sin v}{\sqrt{2}}\right)
	\end{array
	\end{equation*
	Also, the parameterization of the focal surface of this surface is obtained
	as
	{\small \begin{equation*}
		\begin{array}{c}
		X^{\ast }(u,v)=\left( \cos \frac{u}{\sqrt{2}}\left( \frac{3\sin v-\cos v}{\cos v+\sin v}\right) +\sqrt{2}\sin \frac{u}{\sqrt{2}},\right.
		\sin \frac{u}{\sqrt{2}}\left( \frac
			3\sin v-\cos v}{\cos v+\sin v}\right) -\sqrt{2}\cos \frac{u}{\sqrt{2}},
		\left. \frac{u+2}{\sqrt{2}}\right
		\end{array}
		.
		\end{equation*}}
	Further, by taking radius $r=\sqrt{2},$ we plot the tubular surface and its
	focal surface in $\mathbb{E}^{3}$:
	\begin{figure}[tbph]
		\centering
		\includegraphics[width =10cm, angle=0]{c1}
		\caption{Tubular surface $M$ and the focal surface $M^{\ast }$}
	\end{figure}
\end{example}

\vskip 30mm

\begin{theorem}
	Let $M$ be a tubular surface with Darboux frame given with the
	parametrization (\ref{d1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{d10}) in $\mathbb{E}^{3}$. Then,

	i) $u$-parameter curves of the focal surface $M^{\ast }$ cannot be
	asymptotic curves.

	ii) $v$-parameter curves of the focal surface $M^{\ast }$ are asymptotic
	curves.
\end{theorem}

\begin{proof}
	Let $M^{\ast }$ be the focal surface of $M$ with the parametrization (\re
	{d10}) in $\mathbb{E}^{3}$.

	i) Using the equations (\ref{d14}), (\ref{m1}) and the definition of
	asymptotic curve, $\left\langle \left( X^{\ast }\right) _{uu},N^{\ast
	}\right\rangle =0$ if and only if $b_{u}-b_{v}\tau _{g}=0.$ But, this
	contradicts the regularity of $M^{\ast }$. Thus, $u$-parameter curves of the
	focal surface $M^{\ast }$ cannot be asymptotic curves.

	ii) Again using the same equations, we obtain $\left\langle \left( X^{\ast
	}\right) _{vv},N^{\ast }\right\rangle =0$ which means $v$-parameter curves
	of the focal surface $M^{\ast }$ are asymptotic curves.
\end{proof}

\begin{theorem}
	Let $M$ be a tubular surface with Darboux frame given with the
	parametrization (\ref{d1}) and $M^{\ast }$ be the focal surface of $M$ with
	the parametrization (\ref{d10}) in $\mathbb{E}^{3}$. Then,

	i) $u$-parameter curves of the focal surface $M^{\ast }$ are geodesic curves
	if and only if
	\begin{equation}
	-2\tau _{g}\left( b_{u}\tau _{g}^{\prime }+b\tau _{g}^{\prime \prime
	}\right) +4b\left( \tau _{g}^{\prime }\right) ^{2}-4b\tau _{g}^{4}=0
	\label{d17}
	\end{equation
	holds.

	ii) $v$-parameter curves of the focal surface $M^{\ast }$ cannot be geodesic
	curves.
\end{theorem}

\begin{proof}
	i) From (\ref{d14}) and (\ref{m1}), $\left( X^{\ast }\right) _{uu}\wedge
	N^{\ast }=0$ if and only i
	\begin{eqnarray}
	b\left( -b_{{u}u}b+2\left( b_{u}\right) ^{2}-b^{2}\left( \tau _{g}\right)
	^{2}\right) &=&0,  \label{d18} \\
	b^{2}\left( 2b_{u}\tau _{g}-b\tau _{g}^{\prime }\right) &=&0.  \label{d19}
	\end{eqnarray
	Therefore, by the use of these relations, we obtain the desired result.

	ii) Similarly, with the help of (\ref{d14}) and (\ref{m1}), $\left( X^{\ast
	}\right) _{vv}\wedge N^{\ast }=0$ if and only i
	\begin{eqnarray}
	b\left( b_{vv}b-2\left( b_{v}\right) ^{2}+b^{2}\right) &=&0,  \label{d20} \\
	b_{v} &=&0.  \label{d21}
	\end{eqnarray}

	Using these relations, we get $b=0$ which contradicts the representation of
	the focal surface. Hence, we get the result and this completes the proof.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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It's hard to beat a contender that's been good enough, long enough to thrive through at least five wars, 12 presidents, repeated national economic reversals and every imaginable skirt length for 55 years. In its entire history Inglis' ownership has changed just once; it's now in the hands of two, second-generation florists. They would be Olympic champions if there were an event for style-surfing. Ask any generation of Tucsonan.


text:
Il David è una scultura realizzata in marmo (altezza 520 cm incluso il basamento di 108 cm) da Michelangelo Buonarroti, databile tra il 1501 e l'inizio del 1504 e conservata nella Galleria dell'Accademia a Firenze.

Largamente considerato un capolavoro della scultura mondiale, è uno degli emblemi del Rinascimento nonché simbolo di Firenze e dell'Italia all'estero. L'opera, che ritrae l'eroe biblico nel momento in cui si appresta ad affrontare Golia, originariamente fu collocata in Piazza della Signoria, come simbolo della Repubblica fiorentina vigile e vittoriosa contro i nemici.

Da sempre considerato l'ideale di bellezza maschile nell'arte così come la Venere di Sandro Botticelli è considerata il canone di bellezza femminile, molti ritengono che il David sia l'oggetto artistico più bello mai creato dall'uomo.

Storia
Il 16 agosto del 1501 i consoli dell'Arte della Lana e l'Opera del Duomo di Firenze commissionarono a Michelangelo una statua di re David, da collocare in uno dei contrafforti esterni posti nella zona absidale della cattedrale di Santa Maria del Fiore. Si trattava di un'impresa che non aveva precedenti nell'arte rinascimentale e che era già stata tentata due volte. L'enorme blocco di marmo bianco destinato all'opera era infatti già stato abbozzato prima da Agostino di Duccio nel 1463-1464 e poi da Antonio Rossellino nel 1476, ma poi era stato abbandonato da entrambi per le caratteristiche non ottimali del pezzo, anche perché era stato sgrossato rozzamente e questo limitava le possibilità di intervento.

Il problema del marmo
Il problema principale dell'abbozzo era la fragilità del marmo, dovuta alla sua scarsa qualità, alla presenza di numerose fenditure e fori, detti taroli, e alla tendenza intrinseca, di quella tipologia di marmo, alla cottura, cioè alla perdita di coesione dei cristalli. Si riteneva che la forma del blocco fosse un altro ostacolo: troppo alta e stretta, insufficiente per un pieno sviluppo anatomico della figura. Il blocco era friabile specialmente nella zona sotto l'attuale braccio sinistro e si temeva che una volta scolpito non fosse in grado di reggere il peso della figura sulle sole gambe.

Nonostante le fonti tacciano al riguardo, è lecito pensare che il blocco dovesse presentare già alcune forme antropomorfe, per quanto parziali, tanto che i fiorentini erano soliti già chiamarlo "il Gigante".

La scolpitura

Nonostante le difficili premesse, Michelangelo, poco più che venticinquenne, non si scoraggiò e, conscio del prestigio che gli avrebbe garantito un successo, accettò la sfida, affrontando il blocco che era definito "male abbozatum et sculptum", all'interno dell'Opera (l'attuale cortile del Museo dell'Opera del Duomo).

L'inizio dei lavori risale al 9 settembre 1501, quando l'artista provò la durezza del blocco sbozzandolo con qualche colpo di scalpello. Si mise effettivamente all'opera il 13. Il 14 ottobre, probabilmente disturbato dagli occhi indiscreti di chi voleva vedere "il gigante" in lavorazione, fece costruire un recinto di tavole attorno al suo campo di lavoro.

Il soggetto sarebbe stato rappresentato nudo, come altre statue religiose dell'artista, e in un'iconografia innovativa, senza la testa di Golia ai piedi (presente come da tradizione nel David di Donatello e in quello di Verrocchio), quindi prima della micidiale sfida. Michelangelo provvide a stuccare e ricoprire le venature e i taroli della statua con malta di calce restituendo alla superficie la levigatezza tipica delle sue sculture giovanili.

L'esecuzione dovette essere circondata da un'aura di mistero e da una trepidante attesa nei fiorentini, consci dei successi romani dell'artefice e curiosi di sapere l'esito di una prova così difficoltosa. Lo stretto riserbo venne sciolto solo la vigilia della festa di San Giovanni, patrono cittadino, il 23 giugno 1503, quando venne aperto il recinto e invitata la popolazione ad ammirare il capolavoro ormai in via di completamento.

Il 26 gennaio 1504 la statua viene definita "quasi finita" e si procede a nominare una commissione per deciderne la collocazione.

A maggio del 1504 la statua fu trasportata nella sua sede definitiva, vi lavorò quindi tre anni, creando un'opera leggendaria che conteneva nella sua vicenda tutte le premesse per il mito: l'enorme difficoltà tecnica, l'innegabile bellezza del risultato e le numerose vicende che ne hanno segnato la storia.

La collocazione

Era chiaro che il risultato superava di gran lunga le aspettative e non era più adatto per i contrafforti del Duomo, ma esigeva una collocazione più ambiziosa, in piazza dei Priori, il cuore della vita politica cittadina: ciò venne proposto dal Gonfaloniere di Giustizia Pier Soderini, evidentemente rifacendosi a un proposito dello stesso Michelangelo, trasferendo il valore simbolico del David da un contesto religioso a uno civile. Evidentemente le autorità repubblicane avevano immediatamente colto la forte simbologia politica del David: egli incarnava il giusto che, armato solo di una fionda e della fede in Dio, riesce a prevalere sul forte ma iniquo, immagine facilmente accostabile a quella di un buon governo, garante delle libertà e del bene comune, protetto dal favore divino. Non si poteva chiedere una migliore "insegna" per la Repubblica appena restaurata e per i suoi valori, dopo un periodo di forti turbolenze.

Nella commissione che doveva scegliere il luogo di esposizione dell'opera figuravano, tra gli altri, gli artisti famosi attivi in città: Sandro Botticelli, Filippino Lippi, Leonardo da Vinci, Pietro Perugino, Lorenzo di Credi, Antonio e Giuliano da Sangallo, Simone del Pollaiolo, Andrea della Robbia, Cosimo Rosselli, Davide Ghirlandaio, Francesco Granacci, Piero di Cosimo, Andrea Sansovino. Le ipotesi plausibili erano diverse: Botticelli, isolatamente, preferiva una collocazione nei pressi del Duomo; l'araldo del Comune, sostenuto in primis da Filippino Lippi, prevedeva una collocazione a lato della porta principale di Palazzo Vecchio, affacciata sulla piazza; altri suggerirono anche al centro del suo cortile; un'altra strada indicava una collocazione sotto la Loggia della Signoria.

Le ragioni della seconda opzione, proposta da Giuliano da Sangallo, erano essenzialmente conservative, il marmo della statua era debole e "chotto", infatti come dal Sangallo ipotizzato, si sarebbe poi rovinato. A questa idea aderì Leonardo da Vinci, che prese parola per suggerire una collocazione dell'opera a ridosso della parete breve della loggia, incorniciata da una nicchia, "in modo che non guasti le cerimonie delli ufficiali". Si trattava di un'ipotesi che relegava la statua in posizione defilata, equivocandone l'essenza fisica e ribaltandone i valori formali. Vi si è voluto leggere uno spunto polemico tra i due geni, tra i quali dovevano correre pessimi rapporti. Leonardo infatti scrisse che non apprezzava gli "eccessi" anatomici che fanno parte dello stile michelangiolesco e dei suoi seguaci, pur senza mai citare direttamente il rivale: in uno schizzo che fece del David si vede chiaramente come l'enfasi muscolare, calata nel suo stile morbido e soffuso, appare quanto mai retorica e fuori luogo.

La posizione leonardesca rimase minoritaria e alla fine prevalse la posizione di Filippino Lippi, si optò per una collocazione di massimo risalto all'aperto, dominante e autorevole davanti a Palazzo Vecchio, al posto della Giuditta di Donatello, la statua doveva infatti simboleggiare la forza della nuova repubblica.

La sistemazione

L'enorme statua venne trasportata, in quattro giorni fino al 18 maggio 1504, con una partecipazione di più di quaranta uomini, incaricati di trainare e sorvegliare lo scorrimento all'interno di una gabbia lignea, che teneva il marmo prudentemente staccato dal fondo, scorrendo su travi unte di grasso di sego, per evitare vibrazioni che potessero danneggiarlo.

Durante il tragitto, in una pausa notturna un gruppo di giovani fedeli alla fazione filo-medicea, estromessa dal potere, aggredì la statua prendendola a sassate, in quanto simbolo riconosciuto del governo repubblicano: il valore simbolico dell'opera era già estremamente evidente.

Michelangelo rifinì la statua sul posto dipingendo in oro il tronco d'albero dietro la gamba destra e aggiungendo delle ghirlande di ottone con foglie in rame dorato che cingevano la testa e la cinghia della fionda.

La Giuditta venne spostata sotto la loggia l'8 giugno. Pochi giorni dopo, l'11, venne affidata l'esecuzione di una base adeguata a Simone del Pollaiolo e Antonio da Sangallo, così che l'8 settembre di quell'anno il David poteva essere collocato al suo posto, "fornito e scoperto di tutto", esposto tuttavia agli agenti atmosferici. Il David venne rivolto a sud-ovest, in segno di sfida alle popolazioni nemiche pronte ad attaccare Firenze. Accanto a lui doveva essere posta anche un'altra statua, raffigurante Ercole, a simboleggiare la forza sia fisica (Ercole) sia intellettuale (David) dei fiorentini e della Signoria, ma questa seconda statua non fu mai realizzata da Michelangelo e solo in seguito venne scolpita da Baccio Bandinelli.

Allo scultore vennero pagati 400 fiorini.

Il mito del David

Il successo del David di Michelangelo fu immediato. L'umanista Pomponio Gaurico nel suo dialogo De Sculptura del 1504 lo porta come esempio di arte eccelsa, lo stesso fece Benedetto Varchi anni dopo, mentre a testimonianza del mito che la statua incarnava nella cultura umanistica rinascimentale valgono le parole di Giorgio Vasari nelle Vite de' più eccellenti pittori, scultori e architettori nell'edizione del 1550 «[…] e veramente che questa opera ha tolto il grido a tutte le statue moderne et antiche, o greche o latine che elle si fossero […] perché in essa sono contorni di gambe bellissime et appiccature e sveltezza di fianchi divine; né mai più s'è veduto un posamento sì dolce né grazia che tal cosa pareggi, né piedi, né mani, né testa che a ogni suo membro di bontà d'artificio e di parità, né di disegno s'accordi tanto. E certo chi vede questa non dee curarsi di vedere altra opera di scultura fatta nei nostri tempi o ne gli altri da qualsivoglia artefice».

Sin dai tempi della sua prima apparizione la statua del David venne celebrata come l'opera capace di mutare il gusto estetico del suo tempo e di affermarsi quale espressione ideale del Rinascimento, grazie all'applicazione dello studio anatomico al fine di rendere con forme virili possenti e armoniche l'immagine del nudo eroico, la cui forma era la realizzazione fisica, di un complesso insieme di valori filosofici ed estetici. I Fiorentini si immedesimarono con l'aspetto atletico e fiero del giovane eroe, interpretandolo come espressione della forza e della potenza della città stessa nel momento del suo massimo splendore; per i sostenitori della Repubblica divenne il simbolo della vittoria della democrazia sulla tirannide esercitata in precedenza dalla famiglia Medici.

Danni e riparazioni
Nel 1512 una saetta colpì il basamento, che preoccupò per le "crettature", cioè i segni di cedimento, all'altezza delle caviglie, ma in definitiva non ci furono danni.

Il 26 aprile 1527, durante la terza cacciata dei Medici da Firenze, ci furono tumulti in città e un gruppo di repubblicani, asserragliati in Palazzo Vecchio, per difendersi dagli oppositori lanciarono dalle finestre pietre, tegole e mobili, che andarono a cadere anche sul David, causando gravi danni, quali la frantumazione del braccio sinistro in tre pezzi e la scheggiatura della fionda all'altezza della spalla. Giorgio Vasari e Francesco Salviati, devoti estimatori di Michelangelo, raccolsero i frammenti della statua e li nascosero in casa del padre del Salviati, nel chiasso dei Baroncelli. Con il ritorno del granduca Cosimo I si provvide al restauro. I segni dell'episodio sono ancora visibili.

Nel 1813 il dito medio della mano destra fu ricostruito in seguito a un danneggiamento. Nel 1843 lo scultore Lorenzo Bartolini, direttore delle "Regie Fabbriche", incaricò Aristodemo Costoli del restauro, che fu eseguito col metodo drastico che allora era comunemente adottato: pulitura a base di acido cloridrico e di ferri taglienti per togliere le croste superficiali. L'intervento nel corso degli anni si rivelò nefasto: si registrarono danni irreparabili alla superficie del marmo.

Il 29 agosto 1846 il fonditore Clemente Papi fece il calco in gesso che servì poi come base della futura gettatura in bronzo della copia, che attualmente si trova in piazzale Michelangelo, sulla terrazza che domina Firenze.

Il trasferimento

Nel 1872, viste le condizioni precarie di conservazione, fu deciso il ricovero della statua nella Galleria dell'Accademia a Firenze. Per accogliere la grande statua venne incaricato l'architetto Emilio De Fabris di costruire una nuova Tribuna scenograficamente posta al termine della Galleria dei Quadri antichi, con un'illuminazione propria, garantita in alto da un lucernario. Nell'agosto del 1873 la statua venne imbracata in un complesso carro ligneo, il cui modellino è visibile nel museo della Casa Buonarroti, e scorse su rotaie per le vie del centro fino all'Accademia, ancora una volta tra imponenti misure di sicurezza, accompagnata dal clamore popolare.

Nel museo restò però chiusa nella sua cassa per nove anni, in attesa del termine dei lavori alla Tribuna. Nel 1875, con le celebrazioni del IV centenario della nascita di Michelangelo si decise di creare una mostra con le riproduzioni in gesso dei suoi capolavori scultorei, e per l'occasione il David venne spacchettato provvisoriamente, entro la tribuna allestita con tendaggi che coprissero la zona al di sopra della trabeazione ancora in fase di edificazione.

Il 22 luglio 1882 il Museo michelangiolesco venne finalmente inaugurato e la statua rivelata alla fruizione del pubblico.

In piazza della Signoria, nel 1910, venne collocata una copia della statua, realizzata dallo scultore Luigi Arrighetti, vincitore di un apposito concorso indetto per la sua realizzazione.

Vicende recenti e il restauro
Nel 1991 un folle, Piero Cannata, danneggiò la statua con un martello. Un fatto simile era accaduto vent'anni prima alla Pietà vaticana, che fu pazientemente restaurata. In confronto a quell'atto, però, i danni al David furono assai più limitati: la scheggiatura dell'alluce e delle prime due dita del piede sinistro, che venne subito completato adoperando i frammenti originali e servendosi dei numerosi calchi esistenti per reintegrare la lacuna in maniera identica all'originale.

Una nuova copia della statua fu offerta da Firenze a Gerusalemme nel 2004 per celebrare il terzo millennio dalla conquista della città da parte di David. La proposta scatenò la protesta di alcuni religiosi ortodossi che consideravano il nudo michelangiolesco non degno di un eroe biblico e anzi troppo vicino a un ideale estetico classicista e dunque pagano; alla fine un compromesso fu raggiunto, optando per una riproduzione della versione di Verrocchio, che è vestita.

In occasione dell'Expo 2020, ritardato di un anno a causa della pandemia da COVID-19, è stata realizzata una fedele riproduzione della statua mediante la stampa in 3D, e la notizia di questo ambizioso progetto ha subito fatto il giro del mondo. Una delle difficoltà maggiori è stata dettata dalle dimensioni: le stampanti 3D infatti hanno la possibilità di stampare solo oggetti di dimensioni limitate. Essendo in scala 1:1 si è deciso quindi di stamparla in 14 parti diverse, che in totale hanno un peso di circa 400 kg. L'opera è stata realizzata utilizzando avanzate tecniche di scansione laser, che hanno permesso di riprodurre fedelmente anche i più piccoli dettagli. L'opera è stata esposta a Dubai, da ottobre 2021 a marzo 2022, nel padiglione italiano.

A partire dal 2003 il monumento è stato sottoposto a un accuratissimo lavaggio e restauro a cura del laboratorio di restauro dell'Opificio delle pietre dure di Firenze. Questo lungo lavoro è stato realizzato per celebrare il cinquecentenario della realizzazione dell'opera nel 2004. Il contestato metodo di restauro imposto dal comitato scientifico portò alle dimissioni della restauratrice incaricata, che rifiutò di eseguire restauri con liquidi sull'opera di Michelangelo. Critiche all'intervento di restauro, definito non necessario, sono arrivate anche dagli USA. Al termine dei lavori sono stati esposti accanto al David opere e installazioni di artisti contemporanei internazionali (fra i quali Jannis Kounellis), con un accostamento originale che ha suscitato clamore e interesse in tutto il mondo.

La disputa tra Comune e Stato
Nel 2009 il Comune di Firenze, guidato dall'allora sindaco Leonardo Domenici, propose un nuovo trasferimento del David per decongestionare il turismo nel centro cittadino, proponendo come sede alternativa la Stazione Leopolda. La proposta, presa più che altro come una provocazione, non venne ascoltata, ma fu all'origine di un contenzioso tuttora aperto tra Comune e Stato.

Con ricerche d'archivio fatte in quell'occasione, saltò infatti fuori che nell'atto di donazione di Palazzo Vecchio dallo stato al Comune (1871) era incluso l'Arengario (cioè la platea rialzata davanti al palazzo) e tutte le statue presenti, compreso quindi il David. Ciò ha indotto l'allora neoeletto sindaco Matteo Renzi a intraprendere una contesa legale con lo Stato rivendicando la proprietà della statua, che a oggi frutta circa 8 milioni d'euro all'anno in introiti legati agli ingressi alla Galleria dell'Accademia e al merchandising.

Dal canto suo lo Stato ha risposto, tramite i suoi legali, che la statua non era compresa nella donazione, non essendo elencata negli inventari, poiché era già sottinteso il suo trasferimento all'Accademia, al quale infatti il Comune allora non si oppose.

Descrizione e stile

Il soggetto del David, fortemente radicato nella tradizione figurativa fiorentina, venne rielaborato evitando gli schemi compositivi consolidati, scegliendo di rappresentare il momento di concentrazione prima della battaglia. I muscoli del corpo sono poderosi ma ancora a riposo, tuttavia capaci di trasmettere il senso di una straordinaria potenza fisica. L'espressione accigliata e lo sguardo penetrante rivelano la forte concentrazione mentale, manifestando quindi la potenza intellettuale che va a sommarsi a quella fisica.

L'eroe biblico è rappresentato nel momento in cui si appresta ad affrontare Golia, il gigante filisteo; nella mano destra, infatti, stringe il sasso con il quale sconfiggerà il nemico da lì a poco. Lo sguardo fiero e concentrato è rivolto al nemico, con le sopracciglia aggrottate, le narici dilatate e una leggera smorfia sulle labbra che forse tradisce un sentimento di disprezzo verso Golia.

Nella realizzazione degli occhi Michelangelo perfezionò la tecnica di perforare le pupille affinché potessero evitare la luce e creare un gioco di ombre che rende gli occhi molto più penetranti. Difficile è dire perché Michelangelo rompe con la tradizione che vuole Davide rappresentato con la testa di Golia tagliata oltre che non possente e muscoloso ma adolescelmente minuto (nelle opere di Ghiberti, Donatello e Verrocchio). Di certo ci sono che il masso era fragile e già sbozzato e quindi forse non permetteva la creazione agile della testa; che il David di Michelangelo doveva rappresentare la neonata Repubblica e si doveva distinguere dal David di Donatello che invece sanciva la consacrazione divina dei Medici tiranni (e non a caso con la nascita della repubblica era stato portato via da Palazzo Medici); che la religiosa conoscenza della sacre scritture in Michelangelo era approfondita e che la scelta di un eroe consapevolmente forte era ratificata dal passo della Bibbia in cui Davide dichiara di aver ucciso orsi e leoni con l'aiuto di Dio e di non aver timore di Golia (Samuele 17:34-37); la tesi è che la forza di David non proviene solo dalla fede religiosa in Dio, come altre versioni artistiche dell'eroe, gracile e quasi femmineo, sembrano avallare. In ogni caso Michelangelo non ha lasciato nessuno schizzo o documento né altre opere del periodo che testimonino le motivazioni delle scelte fatte, tra l'altro in autonomia, perché l'incarico non era per una statua di tale "grandezza" né per essere posizionata davanti a Palazzo Vecchio.

Per evitare di porre il peso della statua sulla parte sinistra del blocco, più debole, Michelangelo appoggia tutto il peso sulla gamba destra, rafforzata da un piccolo tronco che ha una funzione essenzialmente statica, come nella statuaria antica. La posa è quella tipica del contrapposto, che, tramandata anche nel medioevo, derivava dal canone di Policleto. Infatti il braccio destro e la gamba sinistra sono rilassate, al contrario delle altre due estremità.

Il corpo atletico, al culmine della forza giovanile, si manifesta tramite un accuratissimo studio dei particolari anatomici, dalla torsione del collo attraversato da una vena, alla struttura dei tendini, dalle venature su mani e piedi, alla tensione muscolare delle gambe, fino alla perfetta muscolatura del torso.

Per dare maggiore espressività e risalto Michelangelo ingrandisce leggermente la testa e le mani, nodi cruciali, perfezionati armonicamente con la veduta privilegiata dal basso. Questo effetto è adesso meno apprezzabile in seguito al suo trasferimento nel museo dove la statua è stata collocata su un piedistallo più basso di 63 centimetri. In queste variazioni di proporzione si possono leggere anche motivazioni di carattere filosofico: la testa rappresenta la ragione, quindi il mezzo che permette all'uomo di pensare e di distinguersi dalle bestie; le mani sono invece lo strumento di cui la ragione si serve per operare e creare.

Tra i riferimenti a statue precedenti gli storici colsero analogia coi Dioscuri di Montecavallo, con le rappresentazioni di Ercole su sarcofagi romani o sul pulpito del battistero di Pisa di Nicola Pisano; per quanto riguarda l'opera michelangiolesca, precedenti legati al trattamento della capigliatura e all'espressione concentrata e fiera sono ravvisabili, sebbene in piccolo, nel San Procolo dell'Arca di san Domenico a Bologna o nel San Paolo dell'Altare Piccolomini a Siena.

Il restauro moderno e lo stato di conservazione
L'ultimo intervento di restauro è del 2003 finanziato da Michael J. Collins Family, Kathe e John Dyson, The Sager Family Foundation, Glynn Cohen Villa Mangiacane Art Foundation e Denise O'Leary and Kent Thiry. Sono state utilizzate nella fase di diagnosi metodi moderni, come la scansione al laser dell'intera superficie, per simulare le zone di maggiore esposizione alla pioggia e agli altri agenti atmosferici e inquinanti. Il David è stato diviso in 68 zone fotografate digitalmente ognuna delle quali è stata analizzata secondo quattro tipi di problemi conservativi: difetti del marmo, depositi di materiali in superficie, rotture, residui di lavorazioni precedenti.

Per quanto riguarda la natura e la qualità del marmo, questo è attraversato da una grande quantità di venature e "taròli", inusuali nei marmi considerati buoni dagli artisti del XVI secolo; inoltre le zone maggiormente esposte alla pioggia (spalle, braccio destro, mano e piede sinistro) mostravano una erosione superficiale maggiore che nelle altre statue simili. Questa erosione è attribuibile anche in parte ai danni causati dall'incauto restauro del 1843 del Costoli, effettuato con metodi aggressivi, che arrivò a scavare la pelle del marmo in alcuni punti fino a 2 millimetri.

Vennero monitorati anche i "cretti", cioè le microscopiche crepe, concentrate all'altezza delle caviglie, apparse, secondo le fonti storiche, dopo il fulmine del 1512 e accentuate anche dal fatto che nella collocazione originaria la statua sporgeva di circa 28 centimetri rispetto al baricentro a causa del dissesto del terreno.

La statua inoltre mostrava ancora residui del calco in gesso, tracce di encausto steso in funzione protettiva nel 1813, macchie di cera depositata in occasione di cerimonie pubbliche durante i secoli di esposizione, e anche delle macchie brune di ossidi di ferro lasciati durante la costruzione di un gabbiotto protettivo nel 1872.

Alcuni particolari del David sono veri e propri rifacimenti di diverse epoche: parte della fionda e della mano destra sono successivi ai danneggiamenti del 1527, il dito medio della mano destra è del 1813, il mignolo del piede destro è del 1851.

Il lavoro di ripulitura del David è stato effettuato con tecniche molto semplici. I residui di gesso sono stati asportati con impacchi di acqua distillata su polpa di cellulosa e sepiolite con interposizione di carta giapponese. Le macchie di cera sono state asportate con un tampone imbevuto di un'essenza di petrolio, mentre l'encausto è stato eliminato meccanicamente con il bisturi e l'ausilio del microscopio. Con lo stesso sistema sono state tolte le stuccature della ricostruzione del braccio sinistro, sostituito da un impasto di calce e marmi polverizzati.

Oltre alla dovuta pulitura i restauratori hanno cercato di ottenere una maggiore uniformità cromatica e una migliore leggibilità al capolavoro michelangiolesco.

Note

Bibliografia

  Antonio Natali, Michelangelo. Agli Uffizi, dentro e fuori, Maschietto Editore, Firenze, 2014, ISBN 978-88-6394-084-8

Voci correlate
 Opere di Michelangelo
 Galleria dell'Accademia
 David bronzeo di Michelangelo

Altri progetti

Collegamenti esterni

 https://www.facebook.com/TheDavidByMichelangelo

Sculture di piazza della Signoria
Sculture di Michelangelo
Sculture marmoree
Sculture nella Galleria dell'Accademia
Michelangelo
Statue colossali
Nudi artistici scultorei

redpajama_set_name: RedPajamaWikipedia


text:
Commitments within the IUCN
The One Ocean Summit: three days to change the course of global marine conservation
This Wednesday, the One Ocean Summit will open its doors in Brest for three days that will prove pivotal for the future of our marine habitats. As the first international conference dedicated to the preservation of the oceans, the Summit will welcome representatives from across the world united by the urgent need to address human impacts on our seas.
Marine ecosystems provide precious contributions to human societies. Coastal ecosystems sequester 10 times more carbon per area than terrestrial ecosystems, making "blue carbon" an essential lever in the fight against catastrophic climate change. Over 200 million people worldwide depend on these marine ecosystems for their livelihoods, a figure that will only increase in the coming years as coastal urbanization intensifies.
These contributions are not mechanical processes but depend on the astonishing diversity of living organisms in dynamic interactions, from associations between the smallest microorganisms to currents traversing entire oceans.
Marine ecosystems, and all they bring us, are being irreversibly threatened by human activities. According to the IPBES report of 2018, over 250,000 square kilometers of coastal ecosystems have become "dead zones", suffocated by the leaching of fertilizers into our seas. Over 33% of coral reefs and 33% of marine mammals are threatened with extinction. Furthermore, recent research has reinforced the urgency to act on catastrophic climate change, showing that 57% of ocean surfaces recorded extreme heat in 2019, pushing the thermal tolerance of many ecosystems to their limits. All in all, 66% of marine environments are today severely altered by human activities, and 2% of "blue carbon" habitats are being lost every year.
The growing importance of oceans and coastal areas to international conservation efforts, as cemented in the IUCN World Congress in Marseille, is therefore most welcome. The One Ocean Summit, having been announced by President Macron during the Congress, is designed to build on the commitments made in Marseille. The IUCN French Committee therefore calls upon the Summit's participants at all levels to raise their ambitions for the protection of the oceans, in line with these recommendations.
The governance of the oceans will be a defining feature of the Summit. We call upon states to guarantee not only the long-term protection of marine ecosystems under their jurisdiction, but also to come together to define an ecologically and politically robust network of protected areas in the waters beyond their jurisdiction. Indeed, the IUCN goal for the protection of 30% of global land and seas will only come into being through concerted national and international governance efforts, notably by guaranteeing a high-level of protection, with strict standards established for protected areas.
The IUCN French Committee also calls upon the Congress to target specific threats to the health of marine ecosystems, which are often deeply rooted in land-based practices. This is certainly the case for the millions of tons of plastics that enter the oceans each year. The French government's "zero plastic waste in the sea" over the period 2020-2025 provides an instructive example of a systemic approach to this issue, implementing measures to reduce plastic pollution well-before it reaches the coast. We call for an approach that tackles each phase of the life-cycle of plastics, from reducing the demand and production of plastic-based products, to sorting and recycling plastic waste.
The devastating effects of overfishing must also be addressed, from depleted fish stocks to harmful methods of industrial fishing that threaten species such as sharks and marine mammals. The shocking images of over 100 000 dead fish shed by an industrial fishing trawler off the Atlantic coast of France last week are testament to the need for a drastic change to the status quo. We enlarge this focus on the resources of the sea to call for a moratorium on mineral exploitation of the seabed while such practices remain harmful to the marine environment. Finally, whilst recognizing the potential of off-shore wind farms to reduce global reliance on fossil fuels, we call for a better integration of biodiversity in the evaluation of such projects to ensure that the ecological transition is coherent with the protection of marine ecosystems.
More than any other habitat, the oceans remind us of our connectivity to other shores, and also with the natural processes that support and sustain us. It is time to recognize our duty towards the living beings that make up these processes and define and implement an international strategy to protect our oceans.
Previous post A tribute to three champions of nature
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Conference of the Parties to the UN CBD: Last call for action
Wetlands are our life insurance against the combined climate and biodiversity crises
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Adapting Paris (and the cities of the future) to global warming
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redpajama_set_name: RedPajamaCommonCrawl


Carclew continues to offer professional learning workshops for South Australian teachers. Over the past few years, workshops for teachers have had a visual arts focus, but if your school is interested in other art forms we would be happy to work with you, and a suitably experienced teaching artist to find a solution that meets your needs.
Carclew staff understand the growing need for teachers to build their skills and knowledge in the Arts as the Australian Curriculum: The Arts is implemented. Carclew works with many highly-skilled and experienced artists, across all artforms and would love to talk to schools about how we can support teachers in the classroom.
Power Drawing, is designed to bring the art of drawing into every classroom and to assist educators enhance their teaching in a very practical way.
Carclew's artists have worked with a number of staff groups to develop drawing skills and empower teachers with new and creative ways to engage students through drawing, at both primary and secondary level. The many drawing exercises the artists utilise, when working with staff groups are both relevant and enjoyable.
A two-hour workshop can take place after school during your staff meeting time at your school or at Carclew house. If the workshop is to take place at your school a suitable venue (artroom/wet area, activity space, library, hall etc) for the activity will be required. Dates and times are to be negotiated well in advance.
A three-hour workshop can take place on a student-free day at your school or at Carclew house. If the workshop is to take place at your school a suitable venue (artroom/wet area, activity space, library, hall etc) for the activity will be required. Dates and times are to be negotiated well in advance.
NB: The cost to the schools will depend on the artform and artist as well as the cost of materials. The artist/s can deliver the session at your school, or if you would prefer to come to Carclew's beautiful venue in North Adelaide there will be no additional cost. The estimated costs above are for visual arts workshops - performing arts workshops may be less, subject to materials requirements.
This has been so valuable for all of us. We are thinking of projects we can use the techniques for already. Thank you so much.
Thank you for the wonderful workshop...James and John were inspirational and we all had a really great time. They were so well prepared, dynamic and friendly to work with. I had people from a range of backgrounds and everyone was motivated and engaged. What an amazing duo.


The Alberni Project's HMCS Alberni Memorial will remain open to the public at the Comox Centre Mall through this Sunday.
The Alberni Project's HMCS Alberni Memorial will remain open to the public at the Comox Centre Mall through this Sunday.
Courtenay resident Lewis Bartholomew created the Alberni Project in 2000 while he was living in the USA. He became captivated by the story of the ship, which was torpedoed during the Second world War at the cost of 59 Canadian lives.
Bartholomew, who continues to develop a website, has taken a mobile exhibit to various schools, malls and public events around Vancouver Island.
This exhibit is the largest to date and longest it has been in one community at one time.
There will also be a raffle for a print of the painting of HMCS Alberni that inspired the creation of the project. A fundraising gift shop will offer T-shirts, pins and other memorabilia.
The exhibit is in Unit 200 near the mall office and drugstore during regular mall hours (Mondays to Saturdays from 10 a.m. to 4 p.m. and Sundays from 11 to 4).


Organizations, which implement and maintain the ISO 9001 Quality Management System, desire to implement the two other management systems, i.e. ISO 27001 Information Security and ISO 20000 IT Service Management System due to their altered perspective and new necessities.
Efficiency of the organization's existing ISO 9001 CGS will allow easy integration of the two other management systems into their organization and therefore, coordinated cooperation of relevant managers/owners so as to possess a more efficient Integrated Management System (IMS). Organizations having IMS will be able to save time and personnel costs, and achieve a unified perspective.
Participants will learn about the integration and management of all three management systems, and about the improvement of the Integrated Management System through practices and current examples.
Since this training targets participants who seek to integrate either one or both of ISO 27001 and ISO 20000 Management Systems into the ISO 9001 CGS in place at their organizations, participants should preferably have already taken ISO 9001:2000 CGS Awareness/Implementation Seminar.


text:
Klat_Davide_Monteleone_VII Photo_the Carmignac_Foundation_09
Republic of Chechnya, Russia, 03/2013. The special forces of the "Batallion Sever" (North Battalion), training for security operations. The "Kadyrovtsy", thousands of armed men devoted to the Kadyrov clan, including many former rebels from the First and Second Chechen Campaigns, were initially created as a personal security guard of Akhmad Kadyrov, despite at the time he had no formal legal status. After Akhmad Kadyrov was killed, the Security Service was formally liquidated and most of its units integrated into the system of the Ministry of the Interior of Chechnya. Two units were created: the "Akhmad Kadyrov" Second Road Patrol Regiment of the Police and the Oil Regiment, later renamed "Battalion North" and "Battalion South". As of 2008, Ramzan Kadyrov controls all of the Chechen forces, and has placed former commanders of his militia in the most important positions in the government. The two Security Units are involved in so-called "anti-terrorist operations", and according to several organizations, they have often violated human rights. Grozny.


Augmented reality (AR) is becoming more widespread and accepted by consumers as an immersive entertainment solution that is truly futuristic, so the Realfiction 'DeepFrame' has been unveiled at CES 2018 to identify what can be done to push things forward into new territory.
AR traditionally requires a headset or smartphone, but the 'DeepFrame' screen works by reflecting a display to bring digital elements into the real world without the need for projectors or goggles. This makes it applicable for educational and business purposes to enable visitors to enjoy the content without having to do anything but take a moment to enjoy what's being displayed.
The Realfiction 'DeepFrame' is actually tech-free and can be used with a wide array of displays including 4K and 8K options.


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That Guy by Kim Jones – Audiobook Review
That Guy by Kim Jones
Narrated by Samantha Cook and Aaron Shedlock
Standalone (for now) Romantic Comedy
I listened to That Guy by Kim Jones a few weeks ago and started the review but never finished it because I was waiting for Raegan to finish so we could both weigh in. But now, going back a few weeks later, what I mostly remember is laughing out loud at the silliness. I would say this falls near the top of the lists of RomComs where I literally laughed out loud through the entire book. In fact, it's near the top because it's admittedly over-the-top. If you know that going in, I think you will appreciate it so much more, because That Guy is a caricature of all the modern romance novels.
Penelope is a romance author struggling to pen her first book. She is also partially crazy (but in a funny way). Jake Swagger is "That Guy". You know, 'that guy' in every romance novel—rich, hot, bossy, aloof (at first), and of course, amazing in bed.
"In every romance novel, the hero, aka That Guy, always follows up a good backseat fingering with a move that sets the heroine's panties on fire. It ignites those feelings all over again, so that even as she comes down from her first orgasm, she's already anticipating the next one."
He's no longer fighting his smile. "So what did I do wrong?"
"You pulled your hand out of my pants and wiped your fingers on your pants like they were damp due to condensation from your water glass, rather than the sweet, sinful, innocently sexy kryptonite honey flowing from my vagina."
There is a very fine line between silly funny and over-the-top funny and That Guy straddles the line. Penelope is ridiculous, but in a funny way. I think that if you don't get that this is poking fun of the romance genre in a cute, inoffensive way, you will probably say it's over the top.
"Mother of fuck, woman. You're like a goddamn fungus!"
I smile. I can't help it. "Because I'm growing on you?"
Penelope was a bit exhausting, and I felt like she took a bit too much shit from Jake. I loved how she just wanted him to act like a book boyfriend but he never did what she expected. The sweet moments, however, were completely worth waiting for.
"I'm not easy to love, Jake."
His deep, rumble of laughter can be felt in my toes. Then, in a voice as equally sincere as his gaze, he tells me a truth that rocks my damn soul. "Loving you is the easiest thing I've ever done."
I laughed out loud from beginning to end.
It poked harmless fun at the romance genre.
Being in the head of a hysterically funny but totally crazy girl.
Secondary characters that need their own stories.
Banter! (My fave)
Love the cover.
HOT sex (though a bit too much at the end).
I don't remember the hero much even after just a few weeks.
She was hysterically funny at first but got a little annoying by the end.
A few things went over the top enough to be too OTT (with caps).
I think that Samantha Cook was really great for this. She made the book even better with her delivery, and she had great comedic timing. Though Aaron Shedlock is listed, he was more of a guest, only in the epilogue. I was a bit disappointed because I expected a dual narration book.
You have to realize going in that THAT GUY is satire. It makes fun of all the romance novel clichés and the romance genre as a whole but in a fun way. Penelope was over-the-top absurd, but she totally made me laugh, though I think it got to be a little much by the end. Jake Swagger (see? Satire.) was a bit too much of an asshole sometimes. Sometimes I just wanted him to be That Guy, you know? I also don't feel like, after 2 weeks, I remember Jake. I remember it being funny. I remember she was over the top, but I didn't like him enough to stick in my memory.
But all-in-all THAT GUY was a totally fun and outrageous romcom. There are two kinds of romantic comedies – the romance with some light funny moments, or the comedy with a little sweet romance. That Guy absolutely falls in the latter category. It seems like she may have been setting up another story. I will totally listen to that on release.
I hope Kim Jones keeps writing romantic comedies. I'd love to see her write one that isn't quite as ridiculous as this was, but even if they are all just like this, I'd listen to them all and enjoy the laughter, because That Guy was the rare rom-com that made me literally laugh out loud a bunch of times throughout the book.
Rating 4.25 Stars, 4.5 Heat, 4 Narration
Raegan's thoughts: 4 Stars
This was such a great book. Fun, witty, sarcastic and sexy. An excellent introduction to Kim Jones' writing. I enjoyed the POV from the aspiring author, but I'm not sure I will be running to read it again. If you're a part of the book community, whether you're a reader, blogger or author, this is a definite must read!
Purchase That Guy by Kim Jones
4.25 Stars, audiobook, Audiobook 4.25 Stars, Featured, Kim Jones, Reviews 0


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Q: How do I write a core location with MVVM architecture and perform unit testing? I want to develop an app to get device location and search near by areas. I have a basic understanding of MVVM architecture but so far I could not find a way to separate core location logic into view models . I also want to write sample unit test cases for the same.

A: You could create a LocationService protocol that defines all methods (for instance fetchCurrentPosition).
Then create a concrete class conforming to that protocol, that implements apple  CoreLocation services; this class will be used by the view model for example, and it will be injected in the initializer. Think something like:
class MyViewModel {

    let locationService: LocationService

    init(locationService: LocationService = LocationServiceImpl()) {
        self.locationService = locationService
    }

}

LocationServiceImpl is the class conforming to LocationService that actually uses apple CoreLocation; in this way, you can later mock every function of your location service by creating for example LocationServiceMock class that conforms to LocationService; in your unit tests, you'll create a view model with the mocked service.

redpajama_set_name: RedPajamaStackExchange


To complement our designs of standard and specialized Arc Welding or Resistance Welding Machinery, The Roueche Company designs, builds and integrates robust Special Tooling to suit each application.
Our engineering staff works closely with each customer to ensure the timely development of each system purchased and to further ensure that all customer expectations, requirements and tooling specifications are met.
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There are approximately 350,000 incidents of cardiac arrest that happen outside of hospitals in the US every year, 70% of them occur in a resident's home. The truth is, most cardiac arrests don't occur where immediate medical attention is available. With an overwhelming 70% of Americans who don't know the proper way to respond to cardiac emergencies, knowledge of .
Hiking can be quite the exciting adventure, whether its a short day hike or a month in the Appalachians. The thought of all that can go wrong isn't always the first thing that comes to mind when you're immersed in nature and making your way along the scenic route, but when a medical emergency happens, especially in the great .
What Should You Keep in your Emergency First Aid Kit?
Medical emergencies can happen anywhere at any time, and it's critical for the safety of you and those around you to be prepared. Whether you burn your hand on the stove at home or get a paper cut at work, we all have accidents that require first aid attention and if your home or business emergency first aid kit .
Accidents happen. Emergencies happen too. First aid training is not only essential to keep employees safe and healthy, it gives them peace of mind to handle medical emergencies calmly and effectively. Hopefully the worst-case scenario never happens, but it's better to be prepared and responsive than stunned and stupefied. First-aid training for employees prepares them for anything and everything .
Falling objects, heavy machinery, dangerous chemicals – depending on your occupation, workplace hazards are an everyday reality. And whether you've suffered a paper cut in the office, a broken bone in the factory, or a chemical burn in the lab, the next step should be the same for everyone – the first aid kit. But what happens if there .
Safety and wellness are part of an ongoing process that extends far beyond training and orientation. Organizations who embed safety rules into daily routines make it easy for workers to follow safe habits without having to think. Safety tips for the Workplace Safety is an integral part of any manufacturing operation. Plants should encourage an open-door policy where workers .


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Shanghai | Hidden Treasures of Shanghai
By Thilo_BottlesXO September 4, 2017 Blog, Shanghai No Comments
Shanghai is huge. Along with the hundreds of well-known tourist attractions and amazing restaurants, there are plenty of hidden treasures to discover. Here are a few of our favourites.
Our very own XO Bar is a cozy spot in the heart of the Former French Concession where you'll find our full selection of wines and craft beers to enjoy alongside a delicious selection of bites from our own kitchen. There's plenty of street-side patio space too, and we also use the space to host regular wine tastings, parties and more. Wines start at RMB 25 by the glass. Find it at 237 Hengshan Road, near Yongjia Road.
UnTour
UnTour gives simply the finest food tours in Shanghai. For any newbies to the city and it's unique, diverse and celebrated cuisine, this is the way to do it. And no, they won't make you wear a special cap or follow a flag around. The company, which also operates in Beijing, Chengdu and Hong Kong, offers tours of everything from late-night eats to breakfasts. Find out more at https://untourfoodtours.com/.
Shanghai Insiders
Shanghai is a city that it takes a while to get to know. There are loads of tour groups trying to be the ones to help you out on that, but one of the coolest is Shanghai Insiders. They offer motorbike sidecar tours of varying lengths around the city; it's a great way to not only get to know the city quickly, but also to feel like kind of a badass while you're doing it. Info can be found at www.shanghaiinsiders.com.
Shanghai Supper Club
A regular underground event for Shanghai's foodies, Shanghai Supper Club meets for covert reservation only dinners at a different location, with a different chef, every time. We'd say more, but we would want to spoil the allure. To get an invite you need to signup via email; you can do that by visiting www.shsupperclub.com. All reports say it's a great experience, if you can swing yourself an invite.
Architecture buffs will like this one; 1933 is a huge, awesome designed old abattoir, (aka slaughterhouse), designed by British architects way back when. Now it's been converted in to a space that houses restaurants, stores and offices and really rewards aimless wandering. Seriously, parts of it look like an Escher painting. It's at 10 Shajing Lu, up in Hongkou not far form Hailun Lu metro on Lines 4 and 10.
X Club Lazer Tag
A combination of drinking and lazer tag… what's not to love about that? X Club offers the full lazer tag experience, neon lit dystopian vibe and all – alongside a bar and club area. There are a few private functions rooms too, so it's basically an ideal spot for a nostalgic grown-up birthday party. Or just a Friday night, whatever. That's at 366 Zhaojiabang Lu.
The Pudong branch of Community Center Shanghai is a godsend to expats living on the east side, fresh-off-the-boat or no. They do tours, Chinese classes and general Shanghai orientation. Get down to one of their monthly Friday markets too, a chance to get hold of loads of international food and beverages. You'l find them on the third floor of Building A at 1146 Biyun Lu.
Binjiang Ice Bar
This one tickets two boxes; it's fun, and it's fancy. It's hidden away in a cellar of the Binjiang One complex, (the one that houses German fine dining restaurant Käfer, which you have to go through to get here), and it's so cold that they give you a big parka jacket at the door. Too cold for wine, obviously; the focus is on premium vodka and just enjoying the vibe of being in a makeshift urban igloo.
Himalayas Centre
OK, so this one isn't so "hidden". In fact, it's pretty hard to miss. But we don't think it gets as much love as it deserves. The massive hub contains an art museum, a hotel, a theatre and more. But what really sets it apart is its stunning, modern architecture; so striking and downright cool that it was one of the Shanghai locations used for Spike Jonze' visually gorgeous movie Her, (which used Shanghai as a stand in for a near-future L.A.). It's a little further from the riverside at 1188 Fangdian Lu, close to Huamu Lu station on Line 7.
La Parisienne is a French bakery and deli south of Lujiazui that's gotten a pretty healthy following thanks to its baguettes, sandwiches and expat fuel, aka imported cheese. Expats in Pudong will find there way here eventually, but that's no reason not to get a head start. Find the store at 88 ZhangyangLu, not far from Shangcheng Lu metro station.
Mu Lan Hua Ge Furniture Warehouse
This one's really out of the way, but also really worth the trek. This place, (the full name of which is Mu Lan Hua Ge Jia Ju Li Curio Company), is an incredible sea of antique furniture and weird curios. The dusty, dark warehouses, filled with one-of-a-kind bric-a-brac, feels like somewhere out of another world and gives you a little view in to an older, less hyper-modernized Shanghai. It's over at 1788 Jiyang Lu, not far from Lingzhao Xincun metro station on Line 8.
Daoshun Archery
This is an interesting one; late night Korean archery. Daoshun is a chain from South Korea that holds a lot of tournaments and has a nice, laidback vibe that makes it good if you feel like doing something vaguely sporty but also something new. You can find it in Building 6, 628 Zhangyang Lu, near Laoshan Lu, not far from Century Avenue metro station.
Gubei / Hongqiao
One of the best parts about heading out to Gubei and Hongqiao is getting to see a part of the city, however small, that is a genuine historical enclave for a foreign population. Gubei in particular is full of great Korean bars and restaurants. There's a small pedestrian street that runs between Jinhui Lu and Hongxin Lu that is packed with small, authentic spots. You seriously haven't lived until you've spent a night in the company of nothing but fried chicken, Korean BBQ and soju. The best way to get there is to take Line 10 to Longbai Xincun or Ziteng Lu; the area surrounding both is worth exploring if you have a spare evening or afternoon.
You can thank the area's East Asian expat population for this one too; that's a part of the world that has got the whole bathing thing down, so obviously this part of town is where some of Shanghai's best bathhouses are. 24-hour Korean space New Star at 258 Jinhui Nan Lu is a popular one. There's a large bounty of Japanese spots in the area too that aren't hard to find with a little research. This, though, is often considered the best of the bunch.
Hongqiao Flower & Bird Market
A big market with both indoor and outdoor areas, the Hongqiao Flower and Bird Market is a great slice of local life thats a lot of fun to wander around even if you aren't looking for any flora or new winged friends. This place is bright, colorful and literally full of life. It's at 718 Hongjing Lu, not far north of Longbai Xincun station.
New Hongqiao Center Garden
If you need a small oasis of green out in Shanghai's wild west, then the New Hongqiao Centre Garden is a blessing for those in the area looking for a bit of tranquility in which to take advantage of the sunlight. It's all based around a central lake and is great for that nature-in-the-heart-of-the-city vibe. You still can't escape the views of skyscrapers from inside, but they take on a different feel when viewed from a picnic blanket, wine in hand. You can find it right by Yili Lu Station on Line 10.
You can't have thought we'd end a list of the hottest things to do in Shanghai without including ourselves. Through the BottlesXO app, you can get chilled high quality wine and craft beer delivered instantly to wherever you are in Shanghai. In fact, our average delivery time is just 29 minutes. If you're new to town, get in the know – place your first order on the app and enter the coupon code BXONL2 to get RMB 50 OFF (that's valid on first-time orders only until ). If that's not a hidden treasure, we don't know what is. Download the app here.
Get settled into Shanghai with one of the hottest hidden treasures around. Download the BottlesXO wine and craft beer delivery app here.


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Pietro della Valle (* 2. April 1586 in Rom; † 20. oder 21. April 1652 ebenda) war ein italienischer Forschungsreisender, Reiseschriftsteller und Komponist.

Leben

Pietro della Valle beschäftigte sich schon früh mit den Wissenschaften und der Dichtkunst. 1611 nahm er mit einer spanischen Flotte an einem Zuge gegen die Barbareskenstaaten teil. Nachdem er in Neapel feierlich das Pilgerkleid genommen hatte, schiffte er sich 1614 in Venedig zu einer Wallfahrt in den Orient ein. Die Reise führte ihn in die Türkei, nach Ägypten und Arabien, von da nach Jerusalem, durch Syrien und Persien bis nach Indien.

Er blieb mehr als elf Jahre in diesen Gegenden und lernte deren Sprachen, Einwohner und geographische Verhältnisse genau kennen. Sein Hauptwerk ist jedoch nicht frei von Naivität und einer gewissen Legendenbildung.

Erst 1626 erreichte er mit einem stattlichen orientalischen Gefolge wieder Rom, wo ihn Papst Urban VIII. zum Ehrenkammerherrn ernannte. Valle widmete sich nun wieder den wissenschaftlichen Studien und der Ausarbeitung seiner Reisebeschreibung, die unter dem Titel: Viaggi descritti in lettere familiari ecc. (Rom 1650, 1653, 3 Bände) erschien und bald auch ins Französische (Paris 1661–63, 4 Bände, Rouen 1745, 8 Bände) und ins Deutsche (Genf 1674) übersetzt wurde. Das Werk besteht aus 54 Briefen an einen Freund und zeugt von des Verfassers vielseitiger Gelehrsamkeit und scharfer Beobachtungsgabe, obschon es auch von Leichtgläubigkeit und der Neigung, Wunderdinge zu erzählen, nicht frei ist. Er war auch ein gründlicher Kenner der Musik und wirkte im Musikleben Roms als Anreger und Ratgeber. Nachdem Valle wegen eines im Zorn begangenen Mordes eine Zeit als Flüchtling in Neapel lebte, kehrte er mit päpstlicher Bewilligung nach Rom zurück, wo er am 20. oder 21. April 1652 starb. sein Grab befindet sich in Santa Maria in Aracoeli. Auf der Reise nach Persepolis fesselten ihn vor allem Inschriften, die er kopierte. Er beschreibt sie in seinem fünfzehnten Brief und veröffentlicht die Zeichen. Es war die erste Keilschrift, die Europa zu sehen bekam.

Sein koptisches Wörterbuch wurde von Athanasius Kircher herausgegeben.

Werke
 Viaggi di Pietro della Valle il pellegrino con minuto ragguaglio di tutte le cose notabili osservate in essi : descritti da lui medesimo in 54 lettere familiari ... all'erudito ... Mario Schipano ; divisi in tre parti, cioe la Turchia, la Persia e l'India ... Vit. Mascardi, Roma 1650–1658
 Band 1, La Turchia, 1650 ()
Reiß-Beschreibung in unterschiedlichen Theile der Welt, nemlich in Türkey, Egypten, Palestina, Persien, Ost-Indien, und andere weit entlegene Landschafften: samt einer ausführlichen Erzehlung aller denck- und merckwürdigtster Sachen, so darinnen zu finden u. anzutreffen, erstl. V. d. Authore selbst […] in ital. Sprache beschrieben, u. in 54 Send-.Schreiben in 4 T. verf. Genff: Widerhold 1674.

Es können ihm nur wenige Kompositionen sicher zugeordnet werden, dies sind einige vokale und instrumentale Werke. Daneben trat er auch als Librettist in Erscheinung.

Siehe auch
 Works issued by the Hakluyt Society

Weblinks

Einzelnachweise

Autor
Literatur (17. Jahrhundert)
Literatur (Italienisch)
Reiseliteratur
Komponist (Barock)
Komponist (Italien)
Asienforscher
Musiker (Rom)
Mitglied der Accademia degli Umoristi
Historische Person (Italien)
Geboren 1586
Gestorben 1652
Mann


We are trying something new, a digital newsletter! We hope this new format will make it easier for our families to stay in touch with what's going on at Park Layne Elementary as we will be able provide links to videos, pictures and other digital content and resources directly to our families. Please be sure you've signed up for our remind text messages as this is the way the newsletter will initially be sent out each month. We would welcome any feedback on the newsletter as well.
We are so excited to see the number of volunteers at Park Layne increasing, especially in our Reading Buddy Program, which began last year under the guidance of our librarian, Mrs. Ray. Volunteers include central office personnel, parents, local church members, and other community members. Each volunteer commits to a minimum of 30 minutes per week. Volunteers read with students, review sight words with students and really help students gain confidence in their reading while providing a great support system. Our students love reading with our volunteers. If you or someone you know could commit to at least 30 minutes a week, please contact Mrs. Ray at gaybrielle.ray@tecumsehlocal.org. We would especially like to connect with our retired community members who may have more time and would benefit as much from helping and making connections with our great kids as the kids will benefit from the reading help and mentorship.
PTO is a great way to get involved and help out our schools. Our next PTO meeting will be Monday, December 4 at 7 pm in the Tecumseh Middle School library. Enter the front doors of the middle school and the library is about halfway back the main hallway on your right. If you would like to get involved with the planning of our PTO Spring Festival, join us at 6 pm on December 4 for a festival meeting prior to the regular PTO meeting. Did you miss out on our spirit wear sale of t-shirts, sweatshirts, etc. If so, stop by the school office to purchase extras that were ordered. You may want to consider these for Christmas gifts. We have a great variety of styles and sizes available in each elementary office. Thanks to everyone who sold cookie dough. Students who earned the bounce house reward or lunch with the principal will get these rewards in December. Sign up for Remind alerts for PTO by texting @TecPTO to 81010. Check out the PTO Facebook page for updates as well.
Following the program pictures of our veterans were taken with their student. All pictures can be found on the district website at this link.
Veterans introduced themselves and the students who invited them to the program.
All pictures can be found on the district website at this link.
Cadence Assembly - Check out the video from our awesome assembly.
Please be aware that House Bill 410 was passed in December of 2016 and as a result attendance is now measured in hours instead of days. Tecumseh elementary students are in attendance 6.5 hours per day.
The state of Ohio defines a student as being an habitual truant if they are absent 72 more more hours in one school year without legitimate excuse. By Tecumseh Local Schools Board Policy students may only be absent up to five days per semester with a parent note. All absences over five per semester require a doctor's note in order to be excused. If a doctor's note is not provided for absences over 5 per semester or if not absence note is provided for any absence, the absence is considered unexcused and counts towards the 72 hours described in HB410.
Please be aware of these requirements and make sure you provide proper documentation anytime your child is absent from school. If your child visits the doctor or dentist during the school day, please pick up an excuse and provide it to the school office upon your child's return to school.
Please avoid scheduling routine appointments and vacations while school is in session. If you already have a vacation planned that cannot be changed, please contact the school office and request a Student Leave Form as all vacation days are to be preapproved by the building principal. Days will only be approved if the absences will keep student under five days per semester.
Anytime a student is absent, important instructional time is lost that is difficult to make up. Please help support your child's education by ensuring he/she is at school on time each day unless they are too sick to attend.
The weather was too cold for an outside parade but our inside parade was just as fun.
Students were really able to see each other's costumes better inside. Follow the link to lots more pictures.
Our staff never miss an opportunity to have fun with our students as you can see from some of their costumes as well.