How to draw a square from a list of squares which vertices lie on a sphere? - TeX

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How to draw a square from a list of squares which vertices lie on a sphere?

tikz add tag

Anonymous 1123

I have 40 squares which vertices (have integer coordinates) lie on the sphere, 
```
\documentclass[12pt,a4paper]{article}
\usepackage{enumitem}

\begin{document}

	\[ (x-2)^2 + (y-4)^2 + (z-6)^2 = 225? \]
	\begin{enumerate}[label=\arabic*)]
		\item $ (-12, -1, 4), \quad (-9, 14, 8), \quad (0, 9, 20), \quad (-3, -6, 16)$. 
		\item  $ (-12, -1, 8), \quad (-9, 14, 4), \quad (0, 9, -8), \quad (-3, -6, -4)$. 
		\item $ (-12, -1, 8), \quad (-8, 14, 11), \quad (4, 9, 20), \quad (0, -6, 17)$.
		\item $ (-12, 2, 1), \quad (-9, 6, 16), \quad (0, 18, 11), \quad (-3, 14, -4)$. 
		\item $ (-12, 2, 1), \quad (-8, -1, 16), \quad (4, -10, 11), \quad (0, -7, -4)$. 
		\item $ (-12, 2, 11), \quad (-9, 6, -4), \quad (0, 18, 1), \quad (-3, 14, 16)$.
		\item $ (-12, 2, 11), \quad (-8, -1, -4), \quad (4, -10, 1), \quad (0, -7, 16)$, 
		\item $ (-12, 6, 1), \quad (-9, 2, 16), \quad (0, -10, 11), \quad (-3, -6, -4)$, 
		\item $ (-12, 6, 1), \quad (-8, 9, 16), \quad (4, 18, 11), \quad (0, 15, -4)$, 
		\item $ (-12, 6, 11), \quad (-9, 2, -4), \quad (0, -10, 1), \quad (-3, -6, 16)$. 
		\item $ (-12, 6, 11), \quad (-8, 9, -4), \quad (4, 18, 1), \quad (0, 15, 16)$.
		\item $ (-12, 9, 4), \quad (-9, -6, 8), \quad (0, -1, 20), \quad (-3, 14, 16)$. 
		\item $ (-12, 9, 8), \quad (-9, -6, 4), \quad (0, -1, -8), \quad (-3, 14, -4)$. 
		\item $ (-12, 9, 8), \quad (-8, -6, 11), \quad (4, -1, 20), \quad (0, 14, 17)$.
		\item $ (-9, 2, -4), \quad (0, -10, 11), \quad (13, 6, 16), \quad (4, 18, 1)$. 
		\item $ (-9, 2, 16), \quad (0, -10, 1), \quad (13, 6, -4), \quad (4, 18, 11)$. 
		\item $ (-9, 6, -4), \quad (0, 18, 11), \quad (13, 2, 16), \quad (4, -10, 1)$. 
		\item $ (-9, 6, 16), \quad (0, 18, 1), \quad (13, 2, -4), \quad (4, -10, 11)$. 
		\item $ (-8, -7, 8), \quad (-3, 2, 20), \quad (12, -1, 16), \quad (7, -10, 4)$. 
		\item $ (-8, -6, 11), \quad (-3, 6, 20), \quad (12, 2, 17), \quad (7, -10, 8)$. 
		\item $ (-8, -1, -4), \quad (-3, 18, 4), \quad (12, 9, 16), \quad (7, -10, 8)$. 
		\item $ (-8, -1, -4), \quad (0, 18, 1), \quad (12, 9, 16), \quad (4, -10, 11)$. 
		\item $ (-8, -1, 16), \quad (-3, -10, 4), \quad (12, -7, 8), \quad (7, 2, 20)$. 
		\item $ (-8, -1, 16), \quad (-3, 18, 8), \quad (12, 9, -4), \quad (7, -10, 4)$. 
		\item $ (-8, -1, 16), \quad (0, 18, 11), \quad (12, 9, -4), \quad (4, -10, 1)$. 
		\item $ (-8, 2, -5), \quad (-3, 18, 8), \quad (12, 6, 17), \quad (7, -10, 4)$.
		\item $ (-8, 2, 17), \quad (-3, -10, 8), \quad (12, -6, 11), \quad (7, 6, 20)$. 
		\item $ (-8, 2, 17), \quad (-3, 18, 4), \quad (12, 6, -5), \quad (7, -10, 8)$. 
		\item $ (-8, 6, -5), \quad (-3, -10, 8), \quad (12, 2, 17), \quad (7, 18, 4)$. 
		\item $ (-8, 6, 17), \quad (-3, -10, 4), \quad (12, 2, -5), \quad (7, 18, 8)$. 
		\item $ (-8, 6, 17), \quad (-3, 18, 8), \quad (12, 14, 11), \quad (7, 2, 20)$. 
		\item $ (-8, 9, -4), \quad (-3, -10, 4), \quad (12, -1, 16), \quad (7, 18, 8)$.
		\item $ (-8, 9, -4), \quad (0, -10, 1), \quad (12, -1, 16), \quad (4, 18, 11)$. 
		\item $ (-8, 9, 16), \quad (-3, -10, 8), \quad (12, -1, -4), \quad (7, 18, 4)$. 
		\item $ (-8, 9, 16), \quad (-3, 18, 4), \quad (12, 15, 8), \quad (7, 6, 20)$.  
		\item $ (-8, 9, 16), \quad (0, -10, 11), \quad (12, -1, -4), \quad (4, 18, 1)$.  
		\item $ (-8, 14, 11), \quad (-3, 2, 20), \quad (12, 6, 17), \quad (7, 18, 8)$.  
		\item $ (-8, 15, 8), \quad (-3, 6, 20), \quad (12, 9, 16), \quad (7, 18, 4)$.  
		\item $ (0, -1, 20), \quad (4, 14, 17), \quad (16, 9, 8), \quad (12, -6, 11)$.  
		\item $ (0, 9, 20), \quad (4, -6, 17), \quad (16, -1, 8), \quad (12, 14, 11)$.
	\end{enumerate}
\end{document}
```
How can I draw one or some that squares?

Top Answer

user 3.14159

This may be a starting point. The challenge seems to be to draw great circle arcs on a sphere in such a way that visible and hidden stretches get distinguished. Some time ago I posted a code that does that on TeX.SE. This code got resurrected here, and can definitely get more streamlined. But it seems to work at least.
```
\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3dtools}
\makeatletter
\pgfmathdeclarefunction{isfore}{3}{%
\begingroup%
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed}%
\pgfmathparse{%
sign(((\the\pgf@yx)*(\the\pgf@zy)-(\the\pgf@yy)*(\the\pgf@zx))*(#1)+
((\the\pgf@zx)*(\the\pgf@xy)-(\the\pgf@xx)*(\the\pgf@zy))*(#2)+
((\the\pgf@xx)*(\the\pgf@yy)-(\the\pgf@yx)*(\the\pgf@xy))*(#3))}%
\pgfmathsmuggle\pgfmathresult\endgroup%
}%
\tikzset{3d/great circle arc/.cd,
	theta1/.initial=0,phi1/.initial=0,theta2/.initial=0,phi2/.initial=30,
	r/.initial=R,fore/.style={draw=white,semithick},back/.style={draw=gray,very thin}}
	
\tikzset{pics/3d/great circle arc/.style={code={%
\tikzset{3d/great circle arc/.cd,#1}%
\def\pv##1{\pgfkeysvalueof{/tikz/3d/great circle arc/##1}}%
 % Cartesian coordinates of the first point (A) 
\pgfmathsetmacro\pgfutil@tempAx{\pv{r}*cos(\pv{theta1})*cos(\pv{phi1})}%
\pgfmathsetmacro\pgfutil@tempAy{\pv{r}*cos(\pv{theta1})*sin(\pv{phi1})}%
\pgfmathsetmacro\pgfutil@tempAz{\pv{r}*sin(\pv{theta1})}%
 % Cartesian coordinates of the second point (B) 
\pgfmathsetmacro\pgfutil@tempBx{\pv{r}*cos(\pv{theta2})*cos(\pv{phi2})}%
\pgfmathsetmacro\pgfutil@tempBy{\pv{r}*cos(\pv{theta2})*sin(\pv{phi2})}%
\pgfmathsetmacro\pgfutil@tempBz{\pv{r}*sin(\pv{theta2})}%
 % cross product C=AxB 
\pgfmathsetmacro@td@FPU\pgfutil@tempCx{(\pgfutil@tempAy)*(\pgfutil@tempBz)-(\pgfutil@tempBy)*(\pgfutil@tempAz)}%
\pgfmathsetmacro@td@FPU\pgfutil@tempCy{(\pgfutil@tempAz)*(\pgfutil@tempBx)-(\pgfutil@tempBz)*(\pgfutil@tempAx)}%
\pgfmathsetmacro@td@FPU\pgfutil@tempCz{(\pgfutil@tempAx)*(\pgfutil@tempBy)-(\pgfutil@tempBx)*(\pgfutil@tempAy)}%
 % normalize C to have length r
\pgfmathsetmacro@td@FPU\pgfutil@tempa{sqrt((\pgfutil@tempCx)*(\pgfutil@tempCx)+(\pgfutil@tempCy)*(\pgfutil@tempCy)+(\pgfutil@tempCz)*(\pgfutil@tempCz))/\pv{r}}%
\pgfmathsetmacro@td@FPU\pgfutil@tempCx{\pgfutil@tempCx/\pgfutil@tempa}%
\pgfmathsetmacro@td@FPU\pgfutil@tempCy{\pgfutil@tempCy/\pgfutil@tempa}%
\pgfmathsetmacro@td@FPU\pgfutil@tempCz{\pgfutil@tempCz/\pgfutil@tempa}%
 % angle between A and B
\pgfmathsetmacro@td@FPU\pgfutil@tempAdotB{((\pgfutil@tempAx)*(\pgfutil@tempBx)+
	(\pgfutil@tempAy)*(\pgfutil@tempBy)+(\pgfutil@tempAz)*(\pgfutil@tempBz))/(\pv{r}*\pv{r})}% 
\pgfmathsetmacro@td@FPU\pgfutil@tempangle{acos(\pgfutil@tempAdotB)}%	
 % cross product D=AxC
\pgfmathsetmacro@td@FPU\pgfutil@tempDx{(\pgfutil@tempAy)*(\pgfutil@tempCz)-(\pgfutil@tempCy)*(\pgfutil@tempAz)}%
\pgfmathsetmacro@td@FPU\pgfutil@tempDy{(\pgfutil@tempAz)*(\pgfutil@tempCx)-(\pgfutil@tempCz)*(\pgfutil@tempAx)}%
\pgfmathsetmacro@td@FPU\pgfutil@tempDz{(\pgfutil@tempAx)*(\pgfutil@tempCy)-(\pgfutil@tempCx)*(\pgfutil@tempAy)}%
\pgfmathsetmacro@td@FPU\pgfutil@tempa{sqrt((\pgfutil@tempDx)*(\pgfutil@tempDx)+(\pgfutil@tempDy)*(\pgfutil@tempDy)+(\pgfutil@tempDz)*(\pgfutil@tempDz))/\pv{r}}%
\pgfmathsetmacro@td@FPU\pgfutil@tempDx{\pgfutil@tempDx/\pgfutil@tempa}%
\pgfmathsetmacro@td@FPU\pgfutil@tempDy{\pgfutil@tempDy/\pgfutil@tempa}%
\pgfmathsetmacro@td@FPU\pgfutil@tempDz{\pgfutil@tempDz/\pgfutil@tempa}%
 %\typeout{A=(\pgfutil@tempAx,\pgfutil@tempAy,\pgfutil@tempAz),B=(\pgfutil@tempBx,\pgfutil@tempBy,\pgfutil@tempBz),C=(\pgfutil@tempCx,\pgfutil@tempCy,\pgfutil@tempCz)}
 %\typeout{\pgfutil@tempAdotB,\pgfutil@tempangle}
\edef\pgfutil@tempa{0}%
\pgfmathtruncatemacro{\pgfutil@tempd}{isfore(\pgfutil@tempAx,\pgfutil@tempAy,\pgfutil@tempAz)}%
\ifnum\pgfutil@tempd=-1\relax
\edef\pgfutil@templsthidcoords{(\pgfutil@tempAx,\pgfutil@tempAy,\pgfutil@tempAz)}%
\edef\pgfutil@templstviscoords{}%
\else
\edef\pgfutil@templsthidcoords{}%
\edef\pgfutil@templstviscoords{(\pgfutil@tempAx,\pgfutil@tempAy,\pgfutil@tempAz)}%
\fi
\pgfmathtruncatemacro\pgfutil@tempb{acos(\pgfutil@tempAdotB)}%
\pgfmathtruncatemacro\pgfutil@tempc{sign(\pgfutil@tempb)}%
\loop
\pgfmathsetmacro{\tmpx}{cos(\pgfutil@tempa)*\pgfutil@tempAx-\pgfutil@tempc*sin(\pgfutil@tempa)*\pgfutil@tempDx}%
\pgfmathsetmacro{\tmpy}{cos(\pgfutil@tempa)*\pgfutil@tempAy-\pgfutil@tempc*sin(\pgfutil@tempa)*\pgfutil@tempDy}%
\pgfmathsetmacro{\tmpz}{cos(\pgfutil@tempa)*\pgfutil@tempAz-\pgfutil@tempc*sin(\pgfutil@tempa)*\pgfutil@tempDz}%
\pgfmathtruncatemacro{\pgfutil@tempd}{isfore(\tmpx,\tmpy,\tmpz)}%
\ifnum\pgfutil@tempd=-1\relax
\edef\pgfutil@templsthidcoords{\pgfutil@templsthidcoords\space(\tmpx,\tmpy,\tmpz)}%
\else
\edef\pgfutil@templstviscoords{\pgfutil@templstviscoords\space(\tmpx,\tmpy,\tmpz)}%
\fi
\edef\pgfutil@tempa{\the\numexpr\pgfutil@tempa+1}%
\ifnum\pgfutil@tempa<\the\numexpr\pgfutil@tempc*\pgfutil@tempb\relax
\repeat
\pgfmathtruncatemacro{\pgfutil@tempd}{isfore(\pgfutil@tempBx,\pgfutil@tempBy,\pgfutil@tempBz)}%
\ifnum\pgfutil@tempd=-1\relax
\edef\pgfutil@templsthidcoords{\pgfutil@templsthidcoords\space(\pgfutil@tempBx,\pgfutil@tempBy,\pgfutil@tempBz)}%
\else
\edef\pgfutil@templstviscoords{\pgfutil@templstviscoords\space(\pgfutil@tempBx,\pgfutil@tempBy,\pgfutil@tempBz)}%
\fi
\ifx\pgfutil@templsthidcoords\pgfutil@empty%
\else
\draw[3d/hidden] plot coordinates {\pgfutil@templsthidcoords};%
\fi
\ifx\pgfutil@templstviscoords\pgfutil@empty%
\else
\draw[3d/visible] plot coordinates {\pgfutil@templstviscoords};%
\fi
}}}
\makeatother
\begin{document}
\foreach \Angle in {5,15,...,355}
{\begin{tikzpicture}[declare function={R=15;},scale=0.3]
\begin{scope}[3d/install view={phi=\Angle,theta=70}]
 \path (2,4,6) coordinate (O')
 	(-12,-1,4) coordinate (A) (-9,14,8) coordinate (B) 
 	(0,9,20) coordinate (C) (-3,-6,16) coordinate (D);
 \path[3d/screen coords,shift={(O')},
 	use as bounding box] (-R-1,-R-1) rectangle (R+1,R+1);	
 \shade[3d/screen coords,ball color = black!80,
 transform canvas={rotate around={-35:(O')}}]  
 (O') circle[radius=R];
 \path foreach \X in {A,B,C,D} 
 {[3d coordinate={(\X')=(\X)-(O')}]%(\X) node[circle,draw,blue]{}
 };
 \tikzset{get spherical angles/.code={%
 \expandafter\pgfmathsetmacro\csname lat##1\endcsname{asin(TD("(##1')o(0,0,1)")/R)}
 \expandafter\pgfmathsetmacro\csname az##1\endcsname{atan2(TD("(##1')o(0,1,0)"),TD("(##1')o(1,0,0)"))}}}
 \tikzset{get spherical angles/.list={A,B,C,D}}
 \path[shift={(O')},transform shape,
 	3d/visible/.append style={yellow},
	3d/hidden/.append style={yellow}] 
  pic{3d/great circle arc={theta1={\latA},phi1={\azA},theta2={\latB},phi2={\azB}}}
  pic{3d/great circle arc={theta1={\latB},phi1={\azB},theta2={\latC},phi2={\azC}}}
  pic{3d/great circle arc={theta1={\latC},phi1={\azC},theta2={\latD},phi2={\azD}}}
  pic{3d/great circle arc={theta1={\latD},phi1={\azD},theta2={\latA},phi2={\azA}}};
 %
\end{scope}
\end{tikzpicture}}
\end{document}
```
![ani.gif](/image?hash=a59e12c1aa1dcef0cd10a9d12aa6563219ac05712ff30f13e2682b4b2d0dcd41)

1 Answer