[Tex/LaTex] Draw tangent vectors of a path on a sphere

asymptotetikz-pgf

I am trying to draw tangent and normal vectors to describe parallel transport on a sphere.
I have longitude and latitude circles, but I'm stuck with the vectors.
They have to adapt to the angles of the circles making the path, too…

I cheated and used arcs with a bigger radius to "simulate" tangent vectors, but it only works on the one circle.
Can you help me complete the drawing with the normal vectors ?

Thank you !

Here's a picture and a MWE :

\documentclass[12pt]{article}
\usepackage{tikz}

\usepackage{verbatim}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{5pt}%

\usetikzlibrary{positioning}


\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/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}

%Defining function to draw complete latitude circles 
\newcommand\DrawLongitudeCircle[2][1]{
    \LongitudePlane{\angEl}{#2}
    \tikzset{current plane/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)},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);
}

%Defining function to draw limited longitude circles 
\newcommand\DrawLongitudeCirclered[2][1]{
    \LongitudePlane{\angEl}{#2}
    \tikzset{current plane/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)},scale=#1}}
    % angle of "visibility"
    \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
    \draw[current plane,red,thick] (90:1) arc (90:180:1);   
}


%Defining function to draw complete latitude circles 
\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 function to draw limited latitude circles 
\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] (\angPhiTwo:1) node[below right] {$$} arc (\angPhiTwo:\angPhiOne:1) node[below left] {$$}; %Point Q suivi du point P

    \foreach \r in {-130,-110,...,-50}{
    \draw[current plane,blue,ultra thick,->] (\r:1) arc (\r:\r+2:20);
    }
}

\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{-130} % longitude of point P
\def\angPhiTwo{-50} % longitude of point Q
\def\angBeta{30} % latitude of point P and Q

%Sphere
\fill[ball color=white!10] (0,0) circle (\R); % 3D lighting effect

%Meridiens et équateur
\DrawLongitudeCircle[\R]{\angPhiOne} % pzplane
\DrawLongitudeCircle[\R]{\angPhiTwo} % qzplane
\DrawLatitudeCircle[\R]{0} % equator

%Poles nord et sud
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\coordinate[mark coordinate] (S) at (0,-\H);
\node[above=8pt] at (N) {$\mathbf{N}$};
\node[below=8pt] at (S) {$\mathbf{S}$};


%Trajectoires
\DrawLongitudeCirclered[\R]{180+\angPhiOne}
\DrawLongitudeCirclered[\R]{180+\angPhiTwo}
\DrawLatitudeCirclered[\R]{0}

\end{tikzpicture}
\end{figure}
\end{document}

Best Answer

You can use the turn option of the coordinate which locally shifts the coordinate system so that the last point reached is at the origin and the coordinate system is also "turned" so that the x-axis points in the direction of a tangent entering the last point.

You can replace your hacky \draw[current plane,blue,ultra thick,->] (\r:1) arc (\r:\r+2:20); by

\draw[current plane,blue,ultra thick,->] (\r:1) -- ([turn]90:.5); % tangent vectors
\draw[current plane,red,->] (\r:1) -- ([turn]0:.5); % normal vectors

Having:

The complete MWE:

\documentclass[12pt]{article}
\usepackage{tikz}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{5pt}%

\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/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}

%Defining function to draw complete latitude circles 
\newcommand\DrawLongitudeCircle[2][1]{
    \LongitudePlane{\angEl}{#2}
    \tikzset{current plane/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)},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);
}

%Defining function to draw limited longitude circles 
\newcommand\DrawLongitudeCirclered[2][1]{
    \LongitudePlane{\angEl}{#2}
    \tikzset{current plane/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)},scale=#1}}
    % angle of "visibility"
    \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
    \draw[current plane,red,thick] (90:1) arc (90:180:1);   
}


%Defining function to draw complete latitude circles 
\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 function to draw limited latitude circles 
\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] (\angPhiTwo:1) node[below right] {$$} arc (\angPhiTwo:\angPhiOne:1) node[below left] {$$}; %Point Q suivi du point P

    \foreach \r in {-130,-110,...,-50}{
    \draw[current plane,blue,ultra thick,->] (\r:1) -- ([turn]90:.5);
    \draw[current plane,red,->] (\r:1) -- ([turn]0:.5);
    }
}

\tikzset{%
>=latex,
inner sep=0pt,%
outer sep=2pt,%
mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
fill=black,circle}%
}

\usetikzlibrary{arrows}
\pagestyle{empty}
\usepackage{pgfplots}
\usetikzlibrary{calc,fadings,decorations.pathreplacing,positioning}
\pgfplotsset{compat=1.14}

\begin{document}
\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{-130} % longitude of point P
\def\angPhiTwo{-50} % longitude of point Q
\def\angBeta{30} % latitude of point P and Q

%Sphere
\fill[ball color=white!10] (0,0) circle (\R); % 3D lighting effect

%Meridiens et équateur
\DrawLongitudeCircle[\R]{\angPhiOne} % pzplane
\DrawLongitudeCircle[\R]{\angPhiTwo} % qzplane
\DrawLatitudeCircle[\R]{0} % equator

%Poles nord et sud
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\coordinate[mark coordinate] (S) at (0,-\H);
\node[above=8pt] at (N) {$\mathbf{N}$};
\node[below=8pt] at (S) {$\mathbf{S}$};


%Trajectoires
\DrawLongitudeCirclered[\R]{180+\angPhiOne}
\DrawLongitudeCirclered[\R]{180+\angPhiTwo}
\DrawLatitudeCirclered[\R]{0}

\end{tikzpicture}
\end{document}

Related Solutions

[Tex/LaTex] How to draw tangent line of an arbitrary point on a path in TikZ

Here's an approach that uses the decorations.markings library to place support coordinates at a specified distance along a path, which can be used to set a local coordinate system at a later time to draw tangents or orthogonal lines.

You specify a tangent point by using tangent=<pos> in your first path. In a later path, you can then set use tangent to set a local coordinate system for that path: (0,0) is the tangent point itself, (1,0) is 1 unit along the tangent line, (0,1) is one unit along the orthogonal line.

You can specify tangent=<pos> multiple times in your original path. You can then specify which tangent point to use for a new path by calling use tangent=<count>, where the tangent points are numbered in the order they were specified in the original path.

Using these styles, the following code

\draw [
    tangent=0.4,
    tangent=0.56
] (0,0)
    to [out=20,in=120] (5,2)
    to [out=-60, in=110] (9,1);
\draw [blue, thick, use tangent] (-3,0) -- (3,0);
\draw [orange, thick, use tangent=2] (-2,0) -- (2,0) (0,0) -- (0,1);

would yield this

Here's the complete code:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\begin{document}

\begin{tikzpicture}[
    tangent/.style={
        decoration={
            markings,% switch on markings
            mark=
                at position #1
                with
                {
                    \coordinate (tangent point-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,0pt);
                    \coordinate (tangent unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (1,0pt);
                    \coordinate (tangent orthogonal unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,1);
                }
        },
        postaction=decorate
    },
    use tangent/.style={
        shift=(tangent point-#1),
        x=(tangent unit vector-#1),
        y=(tangent orthogonal unit vector-#1)
    },
    use tangent/.default=1
]
\draw [
    tangent=0.4,
    tangent=0.56
] (0,0)
    to [out=20,in=120] (5,2)
    to [out=-60, in=110] (9,1);
\draw [blue, thick, use tangent] (-3,0) -- (3,0);
\draw [orange, thick, use tangent=2] (-2,0) -- (2,0) (0,0) -- (0,1);
\end{tikzpicture}
\end{document}

[Tex/LaTex] How to draw tangent vectors and component vectors on a curve

To give you a correctly answer, I need to add a code to put together all the next parts . The trick here: we place a node on the path with option sloped so we a get a tangent. To show this tangent line, we place the node with anchor=south west. The corner of the bow is now correctly placed,and finally we give correct size to the node.

All the parts together:

enter image description here

\documentclass[]{scrartcl}  
\usepackage{tikz}   
\usetikzlibrary{arrows}       
\begin{document}

 \begin{tikzpicture}[allow upside down]
\draw[red,line width=1pt] (0,0) .. controls ++(1,6) and ++(-1,2) .. (10,4)
      node[sloped,inner sep=0cm,above,pos=.5,
      anchor=south west,
      minimum height=(10.5)*0.3cm,minimum width=(10.5)*.3cm](N){};

\path (N.south west)%
           edge[-stealth',blue] node[left] {$\vec{ n}$} (N.north west)
           edge[-stealth',blue] node[above] {$\vec{ t}$} (N.south east);
\draw[-stealth',gray]  (N.south west)  --%
      node[below] {$\vec{t_a}$} (N.south west -| N.south east); 
\draw[-stealth',gray]  (N.south west)  --%
      node[right] {$\vec{t_o}$} (N.south west |- N.south east);

  \end{tikzpicture}    
 \end{document}

Complement about tangents and normals

\documentclass[11pt]{scrartcl}   
\usepackage{tikz}       
 \usetikzlibrary{arrows}      
\begin{document}

 \begin{tikzpicture}[allow upside down]
    \draw[red,line width=1pt] (0,0)
    .. controls +(right:6cm) and +(left:4cm) .. (2,6)
    \foreach \p in {0,5,...,100} {
      node[sloped,inner sep=0cm,above,pos=\p*0.01,
      anchor=south west,
      minimum height=(10+\p)*0.03cm,minimum width=(10+\p)*0.03cm]
      (N \p){}
    }
    .. controls +(right:2cm) and +(right:3cm) .. (6,2)
    \foreach \p in {5,10,...,100} {
      node[sloped,inner sep=0cm,above,pos=\p*0.01,
      anchor=south west,
      minimum height=(110-\p)*0.03cm,minimum width=(110-\p)*0.03cm]
      (N2 \p){}
    }
    ;
    \foreach \p in {0,5,...,100} {
      \draw[-latex,blue] (N \p.south west) -- (N \p.north west);
      \draw[-latex,color=green!50!black] (N \p.south west) -- (N \p.south east);
    }
    \foreach \p in {5,10,...,100} {
      \draw[-latex,blue] (N2 \p.south west) -- (N2 \p.north west);
      \draw[-latex,color=green!50!black] (N2 \p.south west) -- (N2 \p.south east);
    }
  \end{tikzpicture}
\vspace{2cm}
\begin{tikzpicture}[>=latex']
\coordinate (A) at (1,1);
\coordinate  (B) at (3,4);
\draw
    [->]
    (A) --  node [above] {\(\vec v\)}(B)  
    ; 
    \draw[red,->] (A)-- node [above] {\(\vec a\)}(A|-B) ; 
    \draw[red,->] (A|-B)--node [left] {\(\vec b\)}(B) ;
\end{tikzpicture}

 \end{document}

enter image description here

Next a code to show how to draw component vector (it's possible to create a macro to do this. Instead of \draw [->](A) -- (B) node [above,midway] {\(\vec v\)};you can write \draw [->](A) -- node [above] {\(\vec v\)}(B);.

With pgf 2.1 CVS now it's possible to use a new option edge label or edge node to place directly the label (see the last picture)

\documentclass[11pt]{scrartcl}   
\usepackage{tikz}       
 \usetikzlibrary{arrows}      
\begin{document}

\begin{tikzpicture}[>=latex']
  \coordinate (A) at (1,1);
\coordinate  (B) at (3,4);     
    \draw [->](A) --  node [above] {\(\vec v\)}(B);
    \path (A)--(A|-B) coordinate (C) ;
    \draw[red,->] (A) -- node[left] {\(\vec a\)} (C) ; 
    \draw[red,->] (C)-- node[above] {\(\vec b\)}(B) ;  
\end{tikzpicture} 

 \end{document}

enter image description here

Only with PGF 2.10 CVS

enter image description here

Best Answer

Related Solutions

[Tex/LaTex] How to draw tangent line of an arbitrary point on a path in TikZ

[Tex/LaTex] How to draw tangent vectors and component vectors on a curve

Related Question