Example: Spherical and cartesian grids

Published 2009-05-01 | Author: Marco Miani

Representation of spherical (red) and cartesian (black) computational grids used by SWAN. Latter gives an example of unstructured grids. Conversion from former to latter involves a deformation factor which is acceptable within a given spatial limit.

The drawing is based on Tomas M. Trzeciak’s Stereographic and cylindrical map projections example.

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Spherical and cartesian grids

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% Author: Marco Miani

\documentclass[12pt]{article}
\usepackage{tikz}
\usetikzlibrary{positioning}
%% helper macros

% The 3D code is based on The drawing is based on Tomas M. Trzeciak's 
% `Stereographic and cylindrical map projections example`: 
% http://www.texample.net/tikz/examples/map-projections/
\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,thin,black] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,thin,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}%this is fake: for drawing the grid
\newcommand\DrawLongitudeCirclered[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,red,thick] (150:1) arc (150:180:1);
  %\draw[current plane,dashed] (-50:1) arc (-50:-35:1);
}%for drawing the grid
\newcommand\DLongredd[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,black,dashed, ultra thick] (150:1) arc (150:180:1);
}
\newcommand\DLatred[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane,dashed,black,ultra thick] (-50:1) arc (-50:-35:1);

}
\newcommand\fillred[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,red,thin] (\angVis:1) arc (\angVis:\angVis+180:1);

}

\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane,thin,black] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,thin,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}%Defining functions to draw limited latitude circles (for the red mesh)
\newcommand\DrawLatitudeCirclered[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  %\draw[current plane,red,thick] (-\angVis-50:1) arc (-\angVis-50:-\angVis-20:1);
\draw[current plane,red,thick] (-50:1) arc (-50:-35:1);

}

\tikzset{%
  >=latex,
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}
\usepackage{amsmath}
\usetikzlibrary{arrows}
\pagestyle{empty}
\usepackage{pgfplots}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}

\begin{document}
\begin{figure}[ht!]
	\begin{tikzpicture}[scale=1,every node/.style={minimum size=1cm}]
	%% some definitions
	
	\def\R{4} % sphere radius
	
	\def\angEl{25} % elevation angle
	\def\angAz{-100} % azimuth angle
	\def\angPhiOne{-50} % longitude of point P
	\def\angPhiTwo{-35} % longitude of point Q
	\def\angBeta{30} % latitude of point P and Q
	
	%% working planes
	
	\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
	\LongitudePlane[xzplane]{\angEl}{\angAz}
	\LongitudePlane[pzplane]{\angEl}{\angPhiOne}
	\LongitudePlane[qzplane]{\angEl}{\angPhiTwo}
	\LatitudePlane[equator]{\angEl}{0}
	\fill[ball color=white!10] (0,0) circle (\R); % 3D lighting effect
	\coordinate (O) at (0,0);
	\coordinate[mark coordinate] (N) at (0,\H);
	\coordinate[mark coordinate] (S) at (0,-\H);
	\path[xzplane] (\R,0) coordinate (XE);
	
    %defining points outsided the area bounded by the sphere
	\path[qzplane] (\angBeta:\R+5.2376) coordinate (XEd);
	\path[pzplane] (\angBeta:\R) coordinate (P);%fino alla sfera
	\path[pzplane] (\angBeta:\R+5.2376) coordinate (Pd);%sfora di una quantità pari a 10 dopo la sfera
	\path[pzplane] (\angBeta:\R+5.2376) coordinate (Td);%sfora di una quantità pari a 10 dopo la sfera
	\path[pzplane] (\R,0) coordinate (PE);
    \path[pzplane] (\R+4,0) coordinate (PEd);
	\path[qzplane] (\angBeta:\R) coordinate (Q);
	\path[qzplane] (\angBeta:\R) coordinate (Qd);%sfora di una quantità pari a 10 dopo la sfera
	
	\path[qzplane] (\R,0) coordinate (QE);
    \path[qzplane] (\R+4,0) coordinate (QEd);%sfora di una quantità 10 dalla sfera sul piano equat.


    \DrawLongitudeCircle[\R]{\angPhiOne} % pzplane
    \DrawLongitudeCircle[\R]{\angPhiTwo} % qzplane
    \DrawLatitudeCircle[\R]{\angBeta}
    \DrawLatitudeCircle[\R]{0} % equator
	%labelling north and south
	\node[above=8pt] at (N) {$\mathbf{N}$};
	\node[below=8pt] at (S) {$\mathbf{S}$};
	
	\draw[-,dashed, thick] (N) -- (S);
	\draw[->] (O) -- (P);
	\draw[dashed] (XE) -- (O) -- (PE);
	\draw[dashed] (O) -- (QE);
	%connecting Points outside the sphere
	\draw[-,dashed,black,very thick] (O) -- (Pd);
	\draw[-,dashed,black,very thick] (O) -- (PEd);
    \draw[-,dashed,black,very thick] (O) -- (QEd);
    \draw[-,dashed,black,very thick] (O) -- (XEd);
    \draw[dashed] (XE) -- (O) -- (PE);
    %draw black thick flat grid
    \draw[-,ultra thick,black] (Pd) -- (PEd) node[below, left] {$P_1$};%verticale sinistro
    \draw[-,ultra thick,black] (PEd) -- (QEd)node[below, right] {$P_3$};%orizzontale inferiore
    \draw[-,ultra thick,black] (Pd)-- (XEd)node[above, right] {$P_2$};%orizzontale superiore	
    \draw[-,ultra thick,black] (XEd) -- (QEd);	
    		
	\draw[pzplane,->,thin] (0:0.5*\R) to[bend right=15]
	    node[midway,right] {$\lambda$} (\angBeta:0.5*\R);
	\path[pzplane] (0.5*\angBeta:\R) node[right] {$$};
	\path[qzplane] (0.5*\angBeta:\R) node[right] {$$};
	\draw[equator,->,thin] (\angAz:0.5*\R) to[bend right=30]
	    node[pos=0.4,above] {$\phi_1$} (\angPhiOne:0.5*\R);
	\draw[equator,->,thin] (\angAz:0.6*\R) to[bend right=35]
	    node[midway,below] {$\phi_2$} (\angPhiTwo:0.6*\R);
			\path[xzplane] (0:\R) node[below] {$$};
	\path[xzplane] (\angBeta:\R) node[below left] {$$};
    \foreach \t in {0,2,...,30} { \DrawLatitudeCirclered[\R]{\t} }
	\foreach \t in {130,133,...,145} { \DrawLongitudeCirclered[\R]{\t} }
	
	%drawing grids on the spere invoking DLongredd and DrawLongitudeCirclered
	
	\foreach \t in {130,145,...,145} { \DLongredd[\R+3]{\t} }
	\foreach \t in {130,133,...,145} { \DrawLongitudeCirclered[\R+3]{\t} }

	\foreach \t in {0,30,...,30} { \DLatred[\R+3]{\t} }
    \foreach \t in {0,2,...,30} { \DrawLatitudeCirclered[\R+3]{\t} }
	
    %labelling
    \draw[-latex,thick](4,-5.5)node[left]{$\mathsf{Grid(s)\ in\ Fig. \ (\ref{fig:Grid})}$}
    	         to[out=0,in=270] (5.8,-2.3);
    \draw[thick](3.6,-6)node[left]{$[\mathsf{Rectilinear}]$};
    	
\end{tikzpicture}
	\caption[Representation of spherical and regular computational grids used by SWAN]
    {Representation of spherical (red) and cartesian (black) co-ordinate systems. Latter 
    gives an example of unstructured grids. Both unstructured. Conversion from former 
    to latter involves a deformation factor which is acceptable within a given spatial limit. 
    For my case, only unstructured flat meshes are employed (\textit{Lisboa} Geodetic 
    datum: black grid on the right). Confront above represented points ($P_1,P_2,P_3$) with 
    Fig.(\ref{fig:Grid}). \\Mathematically frames change accordingly: see Eq.(\ref{eq:actbal2sph}).}
	\label{fig:frames}
\end{figure}



\end{document} 

Comments

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