pst-coiltikz-pgf
I need help with drawing a 2D square lattice made of springs and balls (see the picture) on a convex surface, such as spherical one. Could you please let me know is there a possibility of making this in PSTricks or TikZ? Thanks a lot.
I have linked [1] a related example in PDF with a grid on a spherical surface.
[1] http://www.texample.net/media/tikz/examples/PDF/spherical-and-cartesian-grids.pdf
To give a correct answer, we need to define cross product and vector product (this work is done with metapost in cahier gutemberg 48 but it's in french)
I don't have enough time to define all these macros but it's possible to find a way to draw the arc. First we know that the arc PQ (blue) is in the plane OPQ and is a part of a circle of center O and radius OP. So I search a Coordinate system xyz with x=OP and y=OA'. A is a point of the equator of longitude = -20. Why ? because I want OP and OA radius of the equator and OP perpendicular at OA. Then I need to find A' of longitude-20 and latitude >30 but I need to calculate the value.
Update How to determine the latitude of A' ?
In the next pictures, H is the projection of Q on the plane (OPA). It's possible to calculate PH with two sides (OP=1 and OH=0.866) I find 1.001. Then the lines PH and OA have an intersection at the point I. Now i calculate OI=1.238 and PI=1.591. J is a point of the line OA' and I is the projection of J on the plane (OPA). P, Q, J are aligned and IJ= 0.795. IJ/OI=0.641=tan(32.7). The latitude of A' is 32.7. Now I can draw the circle of radius 1 that passes through P and A' with center O.
Now I need to draw the circle of center O and radius 1. The circle passes through P and A' but also by Q. I draw the diameter POP' and QOQ'.
Todo : Calculus to determine correctly the latitude of A', cross product to determine N'. A macro to place a point with longitude and latitude.
In my code, I redefined personal macro with names that I understand correctly.
\documentclass[11pt]{scrartcl}
\usepackage{tikz}
\usetikzlibrary{calc}
\tikzset{%
add/.style args={#1 and #2}{
to path={%
($(\tikztostart)!-#1!(\tikztotarget)$)--($(\tikztotarget)!-#2!(\tikztostart)$)%
\tikztonodes},add/.default={.2 and .2}}
}
\tikzset{%
mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=2pt,
fill=black,circle}%
}
\newcommand\pgfmathsinandcos[3]{%
\pgfmathsetmacro#1{sin(#3)}%
\pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[2][current plane]{%
\pgfmathsinandcos\sinEl\cosEl{\Elevation} % elevation
\pgfmathsinandcos\sint\cost{#2} % azimuth
\tikzset{#1/.estyle={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[2][current plane]{%
\pgfmathsinandcos\sinEl\cosEl{\Elevation} % elevation
\pgfmathsinandcos\sint\cost{#2} % latitude
\pgfmathsetmacro\ydelta{\cosEl*\sint}
\tikzset{#1/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\ydelta)}}} %
}
\newcommand\DrawLongitudeCircle[1]{
\LongitudePlane{#1}
\tikzset{current plane/.prefix style={scale=\R}}
\pgfmathsetmacro\angVis{atan(sin(#1)*cos(\Elevation)/sin(\Elevation))} %
\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);
}%
\newcommand\DrawLatitudeCircle[1]{
\LatitudePlane{#1}
\tikzset{current plane/.prefix style={scale=\R}}
\pgfmathsetmacro\sinVis{sin(#1)/cos(#1)*sin(\Elevation)/cos(\Elevation)}
\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);
}%
\newcommand\DrawPointOnSphere[3]{%
\pgfmathsinandcos\sinLoM\cosLoM{#1}
\pgfmathsinandcos\sinLaM\cosLaM{#2}
}
\begin{document}
\null\vfill
\begin{center}
\begin{tikzpicture}
\def\R{4} % sphere radius
\def\Elevation{25} % elevation angle
\def\angleLongitudeP{-110} % longitude of point P
\def\angleLongitudeQ{-45} % longitude of point Q
\def\angleLatitudeQ{30} % latitude Q ; 0 latitude of P
\def\angleLongitudeA{-20} % longitude of point A
\pgfmathsetmacro\H{\R*cos(\Elevation)} % distance to north pole
\LongitudePlane[PLongitudePlane]{\angleLongitudeP}
\LongitudePlane[QLongitudePlane]{\angleLongitudeQ}
\LongitudePlane[ALongitudePlane]{\angleLongitudeA}
\fill[ball color=white!10] (0,0) circle (\R); % 3D lighting effect
\coordinate (O) at (0,0);
\coordinate[] (N) at (0,\H);
\coordinate[] (S) at (0,-\H);
\DrawLongitudeCircle{\angleLongitudeP} % PLongitudePlane
\DrawLongitudeCircle{\angleLongitudeQ} % QLongitudePlane
\DrawLongitudeCircle{\angleLongitudeA}
\DrawLatitudeCircle{\angleLatitudeQ}
\DrawLatitudeCircle{0} % equator
\DrawLongitudeCircle{0}
%setup coordinates P and Q
\path[ALongitudePlane] (0:\R) coordinate (A);
\path[ALongitudePlane] (32.5:\R) coordinate (A');
\path[ALongitudePlane] (122.5:\R) coordinate (N');
\path[PLongitudePlane] (0:\R) coordinate (P);
\draw[dashed,add= 1 and 0] (O) to (P);
\path[QLongitudePlane] (\angleLatitudeQ:\R) coordinate (Q);
\draw[dashed,add= 1 and 0] (O) to (Q) ;
\path[QLongitudePlane] (0:\R) coordinate (B);
\draw [dashed] (O) -- (B) ;
\draw [dashed] (O) -- (N) ;
\foreach \v in {A,O,N,S,P,Q,A',B,N'} {\coordinate[mark coordinate] (v) at (\v);
\node [above] at (\v) {\v};}
\begin{scope}[ x={(P)}, y={(A')}, z={(N')}]
\draw[dashed,fill opacity=.3] circle (1);
\draw[blue] ( 0:1) arc (0:68:1) ;
\draw[] ( 68:1) arc (68:115:1) ;
\draw[] (-55:1) arc (-55:0:1);
\draw[red,->](0,0,0)--(0,0,1);
\draw[red,->](0,0,0)--(0,1,0);
\draw[red,->](0,0,0)--(1,0,0);
\end{scope}
\end{tikzpicture}
\end{center}
\vfill
\end{document}
A possible solution:
\documentclass[png,border=10pt,tikz]{standalone}
\usepackage{tikz,tkz-euclide,pdftexcmds}
\usetikzlibrary{backgrounds}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\usetkzobj{all}
\makeatletter
\pgfdeclareradialshading[tikz@ball]{my ball}{\pgfqpoint{5bp}{10bp}}{%
color(0bp)=(tikz@ball!10!white);
color(8bp)=(tikz@ball!65!white);
color(12bp)=(tikz@ball);
color(25bp)=(tikz@ball!70!black);
color(50bp)=(black)}
\pgfdeclareradialshading[tikz@ball]{new ball}{\pgfqpoint{12bp}{12bp}}{%
color(0cm)=(tikz@ball!10!white);
color(0.5cm)=(tikz@ball!50!white);
color(0.75cm)=(tikz@ball);
color(0.9cm)=(tikz@ball!80!black);
color(1cm)=(tikz@ball!60!black)}
% code by percusse:
% https://tex.stackexchange.com/a/74237/13304
\pgfdeclareradialshading{ball shadow}{\pgfpointorigin}%
{color(0cm)=(black);
color(2mm)=(gray!70);
color(3mm)=(gray!30);
color(7mm)=(white)
}
% to make possible use "myball color=..."
\tikzoption{my ball color}{\pgfutil@colorlet{tikz@ball}{#1}\def\tikz@shading{my ball}\tikz@addmode{\tikz@mode@shadetrue}}
\tikzoption{new ball color}{\pgfutil@colorlet{tikz@ball}{#1}\def\tikz@shading{new ball}\tikz@addmode{\tikz@mode@shadetrue}}
\tikzoption{ball shadow color}{\pgfutil@colorlet{tikz@ball}{#1}\def\tikz@shading{ball shadow}\tikz@addmode{\tikz@mode@shadetrue}}
\newcommand{\ball}[3][white]{%
\begin{scope}[shift = {(#2,#3)}]
\node[shading=ball shadow,xscale=2,yscale=0.3,circle,minimum size=7mm]
at (0.15,-0.225){};
% test to decide which shading use: special one for white balls
% not 100% efficient: maybe it's better to use directly a custom
% white ball shading, but this way one could always use
% "my ball color=green" say
\ifnum\pdf@strcmp{#1}{white}=\z@%
\shade[my ball color=#1] (0.15,0.25) circle (0.5);
\else
\shade[new ball color=#1] (0.15,0.25) circle (0.5);
\fi
\end{scope}
}
\makeatother
\begin{document}%
\begin{tikzpicture}
\def\height{0.75}
\def\length{3}
\tkzDefPoint(0,0){A} \tkzDefPoint(0.9*\length,-1.1\height){B}
\tkzDefPoint(1.9*\length,0){C} \tkzDefPoint(1.1*\length, \height){D}
%\tkzDrawPolygon(A,B,C,D) % Beware to put the right segments in foreground
% now this can not be a polygon: need to be split
% Same would apply to the shifted one if the balls touch the upper layer
\tkzDrawSegments(C,D D,A);
\begin{scope}[shift={(0,2)}]
\tkzDefPoint(0,0){A1}\tkzDefPoint(0.9*\length,-\height){B1}
\tkzDefPoint(1.9*\length,0){C1} \tkzDefPoint(1.1*\length, \height){D1}
\tkzDrawPolygon(A1,B1,C1,D1)
\end{scope}
\tkzDrawSegments(A,A1 C1,C D1,D);
\begin{pgfonlayer}{foreground}
\tkzDrawSegments(B,B1);
% other segments in foreground
\tkzDrawSegments(A,B B,C);
\end{pgfonlayer}
% balls: beware the order matters
\ball{1}{0.15}
\ball[orange]{2.6}{0.35}
\ball{1.85}{-0.2}
\ball[orange]{4.3}{0.25}
\ball[orange]{3.425}{-0.1}
\end{tikzpicture}
\end{document}
The result:
The tricks are:
\ball command, besides the usual capability of being shifted, can also set the color;B,B1 in the foreground layer: in such a way balls seem to be in background;pgfonlayer enviroment, the simplest thing to do is to first draw the balls behind, proceeding then with the other ones.
Best Answer
Welcome to TeX.SE! Here is a proposal.
It is straightforward to embed this into a spherical coordinate system. (Big thanks to @caverac for pointing out what the question really is.)
As for your additional request: yes, it is possible. However, things depend on what you really want to achieve in the end. Here I present some example that heavily relies on Fritz's stellar answer. However, when trying to polish it by drawing things on different layers, I encountered unexpected problems. In any case, I have no idea what your real aim is. (See e.g. this great post for additional possibilities.) I would kindly like to ask you to make additional requests in form of a new question, which list all the requirements. Asking questions is free, after all. I leave you with