How to draw Spherical Cone? - TeX

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How to draw Spherical Cone?

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Anonymous 1123

I am trying to draw a Spherical Cone [like this](https://mathworld.wolfram.com/SphericalCone.html) 
![ScreenHunter 50.png](/image?hash=53952a647168d62f18ba543dda8e7a9ca9992bb208c19276f3b148254509ddb6)

I only draw 

```
\documentclass[border=2mm,tikz]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc,backgrounds}
\usepackage{pgfplots}
\begin{document}
	%polar coordinates of visibility
	\pgfmathsetmacro\th{65}
	\pgfmathsetmacro\az{110}
	\tdplotsetmaincoords{\th}{\az}
	%parameters of the cone
	\pgfmathsetmacro\r{3} %radius of sphere
	\pgfmathsetmacro\R{(2 * sqrt(2) * \r) / 3} %radius of base
	\pgfmathsetmacro\v{(4 * \r) / 3} %hight of cone
	\begin{tikzpicture} [scale=1, tdplot_main_coords, axis/.style={blue,thick}]
	\path
	coordinate (O) at (0,0,0)
	coordinate (I) at (0,0,{sqrt(\r*\r - \R*\R)})
	coordinate (B) at (0,\R,0)
	coordinate (A) at (60:\R)
	coordinate (S) at (0,0,\v)
	coordinate (C) at  ($(O)-(A)$)
	coordinate (M) at ($(A)!.5!(B)$)
	coordinate (H) at ($(M)!1/2!(S)$)
	;
	
	\foreach \v/\position in { O/left,A/below,S/right,I/left} {\draw[draw =black, fill=black] (\v) circle (1pt) node [\position=0.2mm] {$\v$};
	}
	
		\draw[dashed] (S)--(O)  --(A)-- cycle (I) -- (A);
	\pgfmathsetmacro\cott{{cot(\th)}}
	\pgfmathsetmacro\fraction{\R*\cott/\v}
	
	\pgfmathsetmacro\fraction{\fraction<1 ? \fraction : 1}
	
	\pgfmathsetmacro\angle{{acos(\fraction)}}
	
	% % angles for transformed lines
	\pgfmathsetmacro\PhiOne{180+(\az-90)+\angle}
	\pgfmathsetmacro\PhiTwo{180+(\az-90)-\angle}
	
	% % coordinates for transformed surface lines
	\pgfmathsetmacro\sinPhiOne{{sin(\PhiOne)}}
	\pgfmathsetmacro\cosPhiOne{{cos(\PhiOne)}}
	\pgfmathsetmacro\sinPhiTwo{{sin(\PhiTwo)}}
	\pgfmathsetmacro\cosPhiTwo{{cos(\PhiTwo)}}
	
	% % angles for original surface lines
	\pgfmathsetmacro\sinazp{{sin(\az-90)}}
	\pgfmathsetmacro\cosazp{{cos(\az-90)}}
	\pgfmathsetmacro\sinazm{{sin(90-\az)}}
	\pgfmathsetmacro\cosazm{{cos(90-\az)}}
	
	\draw[dashed] (0,0,\v) -- (\R*\cosPhiOne,\R*\sinPhiOne,0);
	\draw[dashed] (0,0,\v) -- (\R*\cosPhiTwo,\R*\sinPhiTwo,0);
	
\begin{scope}[canvas is xy plane at z=0]
\draw[dashed] (\tdplotmainphi:\R) arc(\tdplotmainphi:\tdplotmainphi+180:\R);

\draw[thick] (\tdplotmainphi:\R) coordinate(BR) arc(\tdplotmainphi:\tdplotmainphi-180:\R) coordinate(BL);
\end{scope}
	
	
\begin{scope}[tdplot_screen_coords, on background layer] 
\draw[thick] (I) circle (\r); 
%\fill[ball color=orange,opacity=1] (T) circle (\myr); 
\end{scope}
\end{tikzpicture}
\end{document}
```
How to get the correct result?

Top Answer

user 3.14159

This is a code that works for many angles, but not for all. The reason that it does not work for all is that Ti*k*Z has conventions of returning angles that regularly cause headache for me. In principle it should be easy to fix this, but at this point I do not have time for that. The view angle chosen in your screen shot works, of course.

On the bright side I added some functional shading that allows one to shade cones. The shading is not really realistic, but perhaps better than nothing. I benefitted a lot from [this post](https://tex.stackexchange.com/a/333409) when creating the shading. You will need the latest version of `3dtools`.

```
\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{calc,decorations.pathreplacing,3dtools}%https://github.com/marmotghost/tikz-3dtools
\begin{document}
\pgfdeclarelayer{background} 
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\foreach \Angle in {5,15,...,235} 
{\begin{tikzpicture}[declare function={R=2;},line cap=round,line join=round,
	sphere/.style={ball color=blue,3d/screen coords,fill opacity=0.8,
		insert path={(0,0) circle[radius=R]}}]
 \begin{scope}[3d/install view={phi=110,psi=0,theta=\Angle}]
  \path  (0,0,0) coordinate (O) (0,0,1) coordinate (P);
  \draw[stored path/reset coordinate index,
  	3d/hidden/.style={store path=bg},
  	3d/visible/.append style={on layer=foreground}]   	
	pic{3d/circle on sphere={R=2,P={(P)}}};	
  \pgfmathtruncatemacro{\Nhid}{tikztdindexoflastcoordinate}
  \ifnum\Nhid>0
   \path[stored path/first coordinate of=bg] coordinate (bg0)
	 [stored path/last coordinate of=bg] coordinate (bg1);
  \fi	
  \draw[stored path/reset coordinate index,
  	3d/hidden/.append style={on layer=background},
  	3d/visible/.style={store path=fg}]		
	pic{3d/circle on sphere={R=2,P={(P)}}};	
  \pgfmathtruncatemacro{\Nvis}{tikztdindexoflastcoordinate}	
  \ifnum\Nvis>0
   \path[stored path/first coordinate of=fg] coordinate (fg0)
	 [stored path/last coordinate of=fg] coordinate (fg1);
  \fi	
  \pgfmathtruncatemacro{\itest}{(\Nhid>0?1:0)+(\Nvis>0?2:0)}
  \begin{scope}[3d/screen coords]
   \ifcase\itest
	% no path
   \or
	% only hidden
	\begin{scope}[on layer=background]
	 \clip[stored path/restore path=bg];
	 \path[sphere,ball color=gray];
	\end{scope}
   \or
	% only visible
	\begin{scope}[on layer=foreground]
	 \clip[stored path/restore path=fg];
	 \path[sphere];
	\end{scope} 
   \or
    \begin{scope}[on layer=background]
	 \clip let \p1=(bg0),\p2=(bg1),\n1={Mod(atan2(\y1,\x1),360)},
	 	\n2={Mod(atan2(\y2,\x2),360)}
	  in (bg0) [stored path/append path=bg] -- (bg1)
	  arc[start angle=\n2,end angle=\n1,radius=R] -- cycle;
	 \path[sphere,ball color=gray]; 
	\end{scope}
    \begin{scope}[on layer=foreground]
	 \clip let \p1=(fg0),\p2=(fg1),\n1={Mod(atan2(\y1,\x1),360)},
	 	\n2={Mod(atan2(\y2,\x2),360)}
	  in (fg0) [stored path/append path=fg] -- (fg1)
	  arc[start angle=\n2,end angle=\n1,radius=R] -- cycle;
	 \path[sphere]; 
	\end{scope}
   \fi
  \end{scope}
  \pgfmathsetmacro{\myh}{tddistance("(O)","(P)")}
  \pgfmathsetmacro{\myr}{sqrt(R*R-\myh*\myh)}
  \path (P) pic{3d/shaded cone={r=\myr,h=-\myh}};
  \draw[3d/visible,3d/screen coords] circle[radius=R];
 \end{scope} 
\end{tikzpicture}}
\end{document}
```
![ani.gif](/image?hash=55f00d5868b1392d193e32b8933cf2c29af6c63b295665b56221b6c6781b812a)

From this you can perhaps judge the complexity. This function draws great circle arcs. Perhaps somewhat reliably. These are not the most general arcs. The answer you link to draws longitude and latitude arcs. Even if you combine them one does not cover all cases. And, as I said, if one implements this, it is almost unavoidable to make errors, not just in corner cases. One complication is that, if you specify the normal of the plane of the plane the arc should be in, there are two possible arcs. So one also has to define an orientation. Possible, but more work, and even more room for errors.

This is a function of limited scope: ``` \documentclass[tikz,border=3mm]{standalone} \usetikzlibrary{3dtools}% https://github.com/marmotghost/tikz-3dtools \makeatletter \tikzset{3d/point on sphere/.code={\tikzset{3d/.cd,#1, /tikz/insert path={ ({\pgfkeysvalueof{/tikz/3d/R}*cos(\pgfkeysvalueof{/tikz/3d/longitude})*cos(\pgfkeysvalueof{/tikz/3d/latitude})}, {\pgfkeysvalueof{/tikz/3d/R}*sin(\pgfkeysvalueof{/tikz/3d/longitude})*cos(\pgfkeysvalueof{/tikz/3d/latitude})}, {\pgfkeysvalueof{/tikz/3d/R}*sin(\pgfkeysvalueof{/tikz/3d/latitude})})}}}, 3d/latitude/.initial=0,3d/longitude/.initial=0, 3d/arcs/O/.initial={(0,0,0)}, 3d/arcs/n/.initial={(0,0,1)}, 3d/arcs/A/.initial={(1,0,0)}, 3d/arcs/B/.initial={(0,1,0)}, pics/3d/great circle arc/.style={code={% \tikzset{3d/arcs/.cd,#1}% \def\pv##1{\pgfkeysvalueof{/tikz/3d/arcs/##1}}% \pgfmathsetmacro{\pgfutil@tmpa}{tddistance("\pgfkeysvalueof{/tikz/3d/arcs/A}","\pgfkeysvalueof{/tikz/3d/arcs/O}")}% \pgfmathsetmacro{\pgfutil@tmpb}{tddistance("\pgfkeysvalueof{/tikz/3d/arcs/B}","\pgfkeysvalueof{/tikz/3d/arcs/O}")}% \pgfmathtruncatemacro{\pgfutil@tmpi}{abs(\pgfutil@tmpa-\pgfutil@tmpb)<0.01?1:0}% \ifnum\pgfutil@tmpi=0\relax \PackageWarning{3dtools}{The points \pv{A} and \pv{B} do not sit on the same sphere around \pv{O}.}% \else \pgfmathsetmacro{\pgfutil@tmpA}{TD("\pgfkeysvalueof{/tikz/3d/arcs/A}")}% \pgfmathsetmacro{\pgfutil@tmpB}{TD("\pgfkeysvalueof{/tikz/3d/arcs/B}")}% \pgfmathsetmacro{\pgfutil@tmpO}{TD("\pgfkeysvalueof{/tikz/3d/arcs/O}")}% \pgfmathtruncatemacro{\pgfutil@tmpV}{sign(screendepth(\pgfutil@tmpA)-screendepth(\pgfutil@tmpO))}% \pgfmathtruncatemacro{\pgfutil@tmpW}{sign(screendepth(\pgfutil@tmpB)-screendepth(\pgfutil@tmpO))}% \pgfmathsetmacro{\pgfutil@tmpr}{sqrt(\pgfutil@tmpa*\pgfutil@tmpb)}% \pgfmathsetmacro{\pgfutil@tmpc}{Mod(360+acos(TD("\pv{A}-\pv{O}o\pv{B}-\pv{O}")/\pgfutil@tmpa/\pgfutil@tmpb),360)}% \tikzset{3d/define orthonormal dreibein={A={\pv{O}},B={\pv{A}},C={\pv{B}}}} \begin{scope}[x/.expanded={\pgfkeysvalueof{/tikz/3d/aux keys/ex}}, y/.expanded={\pgfkeysvalueof{/tikz/3d/aux keys/ey}}, z/.expanded={\pgfkeysvalueof{/tikz/3d/aux keys/ez}}] \pgfmathsetmacro{\pgfutil@tmpt}{Mod(360+atan2(-1*nscreenx,nscreeny),360)}% \pgfmathtruncatemacro{\pgfutil@tmpi}{(\pgfutil@tmpt>0&&\pgfutil@tmpt<\pgfutil@tmpc)+(\pgfutil@tmpt+180>0&&\pgfutil@tmpt+180<\pgfutil@tmpc)} \typeout{angle=\pgfutil@tmpc,radius=\pgfutil@tmpr,O=\pv{O}, tcrit=\pgfutil@tmpt,V=\pgfutil@tmpV,W=\pgfutil@tmpW,i=\pgfutil@tmpi} \ifcase\pgfutil@tmpi \ifnum\pgfutil@tmpV<0 \path[3d/hidden] (0:\pgfutil@tmpa) arc[start angle=0,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpa]; \else \path[3d/visible] (0:\pgfutil@tmpa) arc[start angle=0,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpa]; \fi \or \ifnum\pgfutil@tmpV<0 \path[3d/hidden] (0:\pgfutil@tmpa) arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpa]; \path[3d/visible] (\pgfutil@tmpt:\pgfutil@tmpa) arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpa]; \else \path[3d/visible] (0:\pgfutil@tmpa) arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpa]; \path[3d/hidden] (\pgfutil@tmpt:\pgfutil@tmpa) arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpa]; \fi \or \ifnum\pgfutil@tmpV<0 \path[3d/hidden] (0:\pgfutil@tmpa) arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpa]; \path[3d/visible] (\pgfutil@tmpt:\pgfutil@tmpa) arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpt+180,radius=\pgfutil@tmpa]; \path[3d/hidden] (\pgfutil@tmpt+180:\pgfutil@tmpa) arc[start angle=\pgfutil@tmpt+180,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpa]; \else \path[3d/visible] (0:\pgfutil@tmpa) arc[start angle=0,end angle=\pgfutil@tmpt,radius=\pgfutil@tmpa]; \path[3d/hidden] (\pgfutil@tmpt:\pgfutil@tmpa) arc[start angle=\pgfutil@tmpt,end angle=\pgfutil@tmpt+180,radius=\pgfutil@tmpa]; \path[3d/visible] (\pgfutil@tmpt+180:\pgfutil@tmpa) arc[start angle=\pgfutil@tmpt+180,end angle=\pgfutil@tmpc,radius=\pgfutil@tmpa]; \fi \fi \end{scope} \fi }}} \makeatother \begin{document} \begin{tikzpicture}[3d/install view={phi=0,theta=70},line join = round, line cap = round, declare function={R=3;}] \path (0,0,0) coordinate (O) [3d/point on sphere={R=R,longitude=30,latitude=20}] coordinate (A) [3d/point on sphere={R=R,longitude=-120,latitude=50}] coordinate (B); \path pic{3d/great circle arc={A={(A)},B={(B)},O={(O)}}}; \draw[3d/visible,3d/screen coords] (O) circle[radius=R]; \end{tikzpicture} \end{document} ```

I did not implement it. In principle, this problem is solved with the `circle on sphere` pic, which computes the critical angle. However, I never got comfortable with how Ti*k*Z deals with angle ranges, so I am sorry to tell you that there is no `3dtools` realization, and likely there will never be because I am not enjoying to distinguish many cases, just to miss some possibility.

@marmot, re: [your answer](#a1934), This is an old question [here](https://tex.stackexchange.com/questions/46850/how-can-i-draw-an-arc-from-point-a-b-on-a-3d-sphere-in-tikz). How can I use `3dtools` to make it?

1 Answer