The tikz-3dplot-circleofsphere Package: Drawing circles of a sphere with tikz-3dplot

Abstract

A circle of a sphere is a circle drawn on a spherical surface like, for instance, circles of latitude or longitude. Circles in arbitrary 3D positions can be drawn with TikZ very easily using a transformed coordinate system provided by the tikz-3dplot package (that is because TikZ can only draw circles on the xy-plane). However, automatically distinguishing the parts of the circle lying on the front and back sides of the sphere, e.g. by drawing a solid arc on the front side and a dashed one on the back side, is a somewhat tricky feat. The tikz-3dplot-circleofsphere package will perform that feat for you.

Full text

The tikz-3dplot-circleofsphere Package:
Drawing circles of a sphere with tikz-3dplot
Matthias Wolff [0000–0002–3895–7313]
BTU Cottbus-Senftenberg
August 7, 2018
Abstract
Acircle of a sphere is a circle drawn on a spherical surface like, for instance, circles of
latitude or longitude. Circles in arbitrary 3D positions can be drawn with TikZ [2] very
easily using a transformed coordinate system provided by the tikz-3dplot package [1]
(that is because TikZ can only draw circles on the xy-plane). However, automatically
distinguishing the parts of the circle lying on the front and back sides of the sphere, e.g.
by drawing a solid arc on the front side and a dashed one on the back side, is a somewhat
tricky feat. The tikz-3dplot-circleofsphere package will perform that feat for you.
1\documentclass{standalone}
2\usepackage{tikz-3dplot-circleofsphere}
3\begin{document}
4\centering
5\def\r{3}
6\tdplotsetmaincoords{60}{125}
7\begin{tikzpicture}[tdplot_main_coords]
8\draw[tdplot_screen_coords,thin,black!30] (0,0,0) circle (\r);
9\foreach \a in {-75,-60,...,75}
10 {\tdplotCsDrawLatCircle[thin,black!30]{\r}{\a}}
11 \foreach \a in {0,15,...,165}
12 {\tdplotCsDrawLonCircle[thin,black!30]{\r}{\a}}
13 \tdplotCsDrawLatCircle%
14 [thick,tdplotCsFill/.style={opacity=0.05}]{\r}{0}
15 \tdplotCsDrawGreatCircle%
16 [red,thick,tdplotCsFill/.style={opacity=0.1}]{\r}{105}{-23.5}
17 \end{tikzpicture}
18 \end{document}
1\documentclass{standalone}
2\usepackage{tikz-3dplot-circleofsphere}
3\begin{document}
4\centering
5\def\r{3}
6\tdplotsetmaincoords{60}{125}
7\begin{tikzpicture}[tdplot_main_coords]
8\def\e{80};
9\draw[tdplot_screen_coords,very thin] (0,0,0) circle (\r);
10 \foreach \a in {0,5,...,175} {
11 \tdplotCsDrawGreatCircle%
12 [thin, tdplotCsBack/.style={thin,blue!40}]%
13 {\r}{\a}{90*sin(\a)*sin(\e)}
14 }
15 \end{tikzpicture}
16 \end{document}
1

Contents
1 Just Looking for the Minimalist Code? 3
2 The tikz-3dplot-circleofsphere Package 4
2.1 Installation ........................................ 4
2.2 DrawingCommands ................................... 4
\tdplotCsDrawCircle[style]{r}{alpha}{beta}{epsilon}............ 4
\tdplotCsDrawGreatCircle[style]{r}{alpha}{beta}............... 5
\tdplotCsDrawLatCircle[style]{r}{epsilon}................... 5
\tdplotCsDrawLonCircle[style]{r}{alpha}.................... 6
\tdplotCsDrawPoint[style]{r}{alpha}{beta}................... 7
2.3 AuxiliaryCommands................................... 7
\tdplotCsFrontsidePoint ............................... 7
\tdplotCsBacksidePoint ................................ 7
\tdplotCsComputeTransformRotScreen ........................ 7
2.4 KnownIssues....................................... 8
3 Implementation Details 8
3.1 TheMaths ........................................ 8
CirclesonaSphere.................................... 8
Coordinate Transforms with tikz-3dplot ....................... 8
DrawingCirclesofaSphere............................... 11
3.2 ThePackageSourceCode ................................ 12
3.3 AnAuxiliaryMatlabScript ............................... 16
References 18
2

1 Just Looking for the Minimalist Code?
There you go!
1\documentclass{standalone}
2\usepackage{tikz,tikz-3dplot}
3%% >> MINIMALIST CIRCLE OF SHPERE DRAWING CODE _______________________________________________________________________
4\newcommand\scircle[4]{%
5\tdplotsetrotatedcoords{#2}{#3}{0} % Rotate coordinate system
6\let\a\tdplotalpha % alpha (rotated coord. system)
7\let\b\tdplotbeta % beta (rotated coord. system)
8\let\p\tdplotmainphi % phi (main coord. system)
9\let\t\tdplotmaintheta % theta (main coord. system)
10 \pgfmathsetmacro\azx{cos(\a)*cos(\b)*sin(\p)*sin(\t) - sin(\b)*cos(\t) - cos(\b)*cos(\p)*sin(\a)*sin(\t)}
11 \pgfmathsetmacro\azy{-cos(\a)*cos(\p)*sin(\t) - sin(\a)*sin(\p)*sin(\t)}
12 \pgfmathsetmacro\azz{cos(\b)*cos(\t) + cos(\a)*sin(\b)*sin(\p)*sin(\t) - cos(\p)*sin(\a)*sin(\b)*sin(\t)}
13 \pgfmathsetmacro\re {#1*cos(#4)} % Radius of circle
14 \pgfmathsetmacro\ze {#1*sin(#4)} % z-coordinate of drawing plane
15 \pgfmathsetmacro\coX{\ze*cos(#2)*sin(#3)} % x-coordinate offset for ze
16 \pgfmathsetmacro\coY{\ze*sin(#2)*sin(#3)} % y-coordinate offset for ze
17 \pgfmathsetmacro\coZ{\ze*cos(#3)} % z-coordinate offset for ze
18 \coordinate (coffs) at (\coX,\coY,\coZ); % Offset as coordinate value
19 \tdplotsetrotatedcoordsorigin{(coffs)} % Offset coordinate system
20 \begin{scope}[tdplot_rotated_coords] % Drawing scope >>
21 \pgfmathsetmacro\tanEps{tan(#4)} % Tangent of elevation angle
22 \pgfmathsetmacro\bOneside{((\tanEps)^2)>=(((\azx)^2+(\azy)^2)/(\azz)^2)} % Circle entirely on one side?
23 \ifthenelse{\bOneside=1}{% % Circle on one side of sphere >>
24 \pgfmathsetmacro\bFrontside{(\azx*\re+\azz*\ze)>=0} % Circle entirely on front side?
25 \ifthenelse{\bFrontside=1} % |
26 {\draw (0,0) circle (\re);} % Draw on front side
27 {\draw[dashed] (0,0) circle (\re);} % Draw on back side
28 }{% % << Circle on both sides >>
29 \pgfmathsetmacro\u{\azy} % Substitution u=...
30 \pgfmathsetmacro\v{sqrt( (\azx)^2 + (\azy)^2 - (\azz)^2*(\tanEps)^2 )} % Substitution v=...
31 \pgfmathsetmacro\w{\azx - \azz*\tanEps} % Substitution w=...
32 \pgfmathsetmacro\phiBf{2*atan2(\u-\v,\w)} % Back->front crossing angle
33 \pgfmathsetmacro\phiFb{2*atan2(\u+\v,\w)} % Front->back crossing angle
34 \pgfmathsetmacro\bUnwrapA{(\phiFb-\phiBf)>360} % Unwrap front->back angle #1?
35 \pgfmathsetmacro\bUnwrapB{\phiBf>\phiFb} % Unwrap front->back angle #2?
36 \ifthenelse{\bUnwrapA=1}{\pgfmathsetmacro\phiBf{\phiBf+360}}{} % Unwrap front->back angle #1
37 \ifthenelse{\bUnwrapB=1}{\pgfmathsetmacro\phiBf{\phiBf-360}}{} % Unwrap front->back angle #2
38 \draw[dashed] (\phiFb:\re) arc (\phiFb:{\phiBf+360}:\re); % Draw back side arc
39 \draw (\phiBf:\re) arc (\phiBf:\phiFb:\re); % Draw back side arc
40 }% <<
41 \end{scope} % << (Drawing scope)
42 }
43 %% << ________________________________________________________________________________________________________________
44 \begin{document}
45 \tdplotsetmaincoords{60}{125} % Set main coordintate system
46 \begin{tikzpicture}[thick,tdplot_main_coords] % TikZ picture >>
47 \begin{scope}[black!30] % Draw in gray >>
48 \draw[tdplot_screen_coords] (0,0,0) circle (2.5); % Sphere outline
49 \scircle{2.5}{0}{0}{0} % Equator
50 \end{scope} % <<
51 \scircle{2.5}{-40}{40}{30} % Draw another sphere circle
52 \end{tikzpicture} % <<
53 \end{document}
Want some more convenience or interested in what we did? Read on. ..
3

2 The tikz-3dplot-circleofsphere Package
2.1 Installation
Download tikz-3dplot-circleofsphere.sty from [3] file into your project folder and include
the package with \usepackage{tikz-3dplot-circleofsphere}.
2.2 Drawing Commands
\tdplotCsDrawCircle[style]{r}{alpha}{beta}{epsilon}
Draws a circle of a sphere.
Parameters
style TikZ style
•use tdplotCsFront/.style={...}to style the front side arc
•use tdplotCsBack/.style={...}to style the back side arc
•use tdplotCsFill/.style={...}to style the circle filling
•use tdplotCsDrawAux to draw some auxiliary information
rRadius of sphere
alpha Azimuthal angle of drawing plane.
Passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
beta Polar angle of drawing plane.
Passed as beta to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
epsilon Elevation angle of circle above the drawing plane. Permissible values are −90 <
epsilon <90. Use 0 for drawing a great circle.
Output
–none–
Example
xy
z1\def\r{1.5}
2\tdplotsetmaincoords{60}{125}
3\begin{tikzpicture}[tdplot_main_coords]
4\begin{scope}[thin,black!30]
5\draw[->] (-1.3*\r,0,0) -- (1.3*\r,0,0) node[anchor=north east] {$x$};
6\draw[->] (0,-1.3*\r,0) -- (0,1.3*\r,0) node[anchor=north] {$y$};
7\draw[->] (0,0,-1.3*\r) -- (0,0,1.3*\r) node[anchor=south east] {$z$};
8\draw[tdplot_screen_coords] (0,0,0) circle (\r);
9\tdplotCsDrawLatCircle{\r}{0}
10 \end{scope}
11 \tdplotCsDrawCircle{\r}{-40}{40}{30}
12 \end{tikzpicture}
4

\tdplotCsDrawGreatCircle[style]{r}{alpha}{beta}
Draws a great circle.
Equivalent to \tdplotCsDrawCircle[style]{r}{alpha}{beta}{0}.
Parameters
style TikZ style
•use tdplotCsFront/.style={...}to style the front side arc
•use tdplotCsBack/.style={...}to style the back side arc
•use tdplotCsFill/.style={...}to style the circle filling
•use tdplotCsDrawAux to draw some auxiliary information
rRadius of sphere
alpha Azimuthal angle of drawing plane.
Passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
beta Polar angle of drawing plane.
Passed as beta to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
Output
–none–
Example
xy
z1\def\r{1.5}
2\tdplotsetmaincoords{60}{125}
3\begin{tikzpicture}[tdplot_main_coords]
4\begin{scope}[thin,black!30]
5\draw[->] (-1.3*\r,0,0) -- (1.3*\r,0,0) node[anchor=north east] {$x$};
6\draw[->] (0,-1.3*\r,0) -- (0,1.3*\r,0) node[anchor=north] {$y$};
7\draw[->] (0,0,-1.3*\r) -- (0,0,1.3*\r) node[anchor=south east] {$z$};
8\draw[tdplot_screen_coords] (0,0,0) circle (\r);
9\tdplotCsDrawLatCircle{\r}{0}
10 \end{scope}
11 \tdplotCsDrawGreatCircle[tdplotCsFill/.style={green,opacity=0.2}]{\r}{-40}{40}
12 \end{tikzpicture}
\tdplotCsDrawLatCircle[style]{r}{epsilon}
Draws a circle of latitude.
Equivalent to \tdplotCsDrawCircle[style]{r}{0}{0}{epsilon}.
Parameters
style TikZ style
•use tdplotCsFront/.style={...}to style the front side arc
•use tdplotCsBack/.style={...}to style the back side arc
•use tdplotCsFill/.style={...}to style the circle filling
•use tdplotCsDrawAux to draw some auxiliary information
rRadius of sphere
5

epsilon Elevation angle of circle above the drawing plane. Permissible values are −90 <
epsilon <90. Use 0 for drawing the sphere equator.
Output
–none–
Example
xy
z1\def\r{1.5}
2\tdplotsetmaincoords{60}{125}
3\begin{tikzpicture}[tdplot_main_coords]
4\begin{scope}[thin,black!30]
5\draw[->] (-1.3*\r,0,0) -- (1.3*\r,0,0) node[anchor=north east] {$x$};
6\draw[->] (0,-1.3*\r,0) -- (0,1.3*\r,0) node[anchor=north] {$y$};
7\draw[->] (0,0,-1.3*\r) -- (0,0,1.3*\r) node[anchor=south east] {$z$};
8\draw[tdplot_screen_coords] (0,0,0) circle (\r);
9\tdplotCsDrawLatCircle{\r}{0}
10 \end{scope}
11 \tdplotCsDrawLatCircle[tdplotCsFront/.style={green}]{\r}{40}
12 \end{tikzpicture}
\tdplotCsDrawLonCircle[style]{r}{alpha}
Draws a circle of longitude.
Equivalent to \tdplotCsDrawCircle[style]{r}{alpha}{90}{0}.
Parameters
style TikZ style
•use tdplotCsFront/.style={...}to style the front side arc
•use tdplotCsBack/.style={...}to style the back side arc
•use tdplotCsFill/.style={...}to style the circle filling
•use tdplotCsDrawAux to draw some auxiliary information
rRadius of sphere
alpha Azimuthal angle of drawing plane.
Passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
Output
–none–
Example
xy
z1\def\r{1.5}
2\tdplotsetmaincoords{60}{125}
3\begin{tikzpicture}[tdplot_main_coords]
4\begin{scope}[thin,black!30]
5\draw[->] (-1.3*\r,0,0) -- (1.3*\r,0,0) node[anchor=north east] {$x$};
6\draw[->] (0,-1.3*\r,0) -- (0,1.3*\r,0) node[anchor=north] {$y$};
7\draw[->] (0,0,-1.3*\r) -- (0,0,1.3*\r) node[anchor=south east] {$z$};
8\draw[tdplot_screen_coords] (0,0,0) circle (\r);
9\tdplotCsDrawLatCircle{\r}{0}
10 \end{scope}
11 \tdplotCsDrawLonCircle[thick,tdplotCsBack/.style={thin,solid,green}]{\r}{0}
12 \end{tikzpicture}
6

\tdplotCsDrawPoint[style]{r}{alpha}{beta}
Draws a point on a sphere.
Parameters
style TikZ style
•use tdplotPtFront/.style={...}to style a front side point
•use tdplotPtBack/.style={...}to style a back side point
rRadius of sphere
alpha Azimuthal angle of drawing plane.
Passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
beta Polar angle of drawing plane.
Passed as beta to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
Output
–none–
Remarks
•Redefine \tdplotCsFrontsidePoint to customize drawing of a front side point.
•Redefine \tdplotCsBacksidePoint to customize drawing of a back side point.
Example
xy
z
•
1\def\r{1.5}
2\tdplotsetmaincoords{60}{125}
3\begin{tikzpicture}[tdplot_main_coords]
4\begin{scope}[thin,black!30]
5\draw[->] (-1.3*\r,0,0) -- (1.3*\r,0,0) node[anchor=north east] {$x$};
6\draw[->] (0,-1.3*\r,0) -- (0,1.3*\r,0) node[anchor=north] {$y$};
7\draw[->] (0,0,-1.3*\r) -- (0,0,1.3*\r) node[anchor=south east] {$z$};
8\draw[tdplot_screen_coords] (0,0,0) circle (\r);
9\tdplotCsDrawLatCircle{\r}{0}
10 \end{scope}
11 \tdplotCsDrawPoint{\r}{-40}{40}
12 \end{tikzpicture}
2.3 Auxiliary Commands
\tdplotCsFrontsidePoint
Invoked by \tdplotCsDrawPoint to draw a point on the front side of a sphere. Redefine to
customize.
\tdplotCsBacksidePoint
Invoked by \tdplotCsDrawPoint to draw a point on the back side of a sphere. Redefine to
customize.
\tdplotCsComputeTransformRotScreen
7

Computes the elements of the full rotation matrix
A=

axx axy axz
ayx ayy ayz
azx azy az z 
.
See Section 3.1 for details.
Parameters
none
Output
\axx Element axx of full rotation matrix
\axy Element axy of full rotation matrix
. . .
\azz Element azz of full rotation matrix
Remarks
The command uses some internal variables of tikz-3dplot, namely \tdplotalpha,\tdplotbeta,
\tdplotmainphi, and \tdplotmaintheta.
2.4 Known Issues
•The tdplotCsFill and tdplotCsDrawAux styles are only effective when specified directly
with the drawing command.
3 Implementation Details
3.1 The Maths
Circles on a Sphere
We consider circles on a sphere of radius ras illustrated in Fig. 1. For drawing a great circle, i.e. a
circle whose center coincides with the center of the sphere (blue in Fig. 1), we rotate the coordinate
system by two Euler angles, an azimuthal angle 0◦≤α < 360◦and a polar angle 0◦≤β < 360◦,
and draw on the new xryrplane. For drawing small circles (red in Fig. 1), we additionally elevate
the drawing plane by an angle −90◦<<90◦,6= 0, and draw on the rotated and elevated xroyro
plane. The tricky part of drawing circles of spheres is to determine which part is on the back side
of the sphere and to draw it a different style, e.g., dashed.
Coordinate Transforms with tikz-3dplot
We use the circle and arc path construction operations of TikZ for drawing. As TikZ will only
draw circles and arcs on the xy-plane, we need to rotate and possibly offset the coordinate system
as described above using the tikz-3dplot [1] package.
First, tikz-3dplot provides a main coordinate system which is basicly defining the view point
on a 3D coordinate system. Denote by P= (x y z)>a point in the 3D coordinate system.
tikz-3dplot transforms that point in to screen coordinates P0= (x0y0z0)>by


x0
y0
z0
=Rd(φ, θ)

x
y
z
(1)
8

x
y
z
~o
α
β
xr
yr
View angle of Fig. 2

xro
yro
Figure 1: A great circle (blue) and a small circle (red).
with the rotation matrix1
Rd(φ, θ) = Rz0(φ)Rx(θ)>(2)
=


cos φ−sin φ0
sin φcos φ0
0 0 1


1 0 0
0 cos θ−sin θ
0 sin θcos θ


>
=

cos φsin φ0
−cos θsin φcos θcos φ+sin θ
sin θsin φ−sin θcos φcos θ
.
We set the main coordinate system by \tdplotsetmaincoords{hφi}{hθi}.
Second, for drawing circles and arcs outside the xy-plane, we need to rotate the coordinate
system further. To this end, we use tikz-3dplot’s rotated coordinate system2


x0
y0
z0
=Rd(φ, θ)D(α, β, γ)

x
y
z
(3)
with the rotation matrix (cf. [1, p. 7])
D(α, β, 0) = Rz(α)Ry(β) (4)
=

cos α−sin α0
sin αcos α0
0 0 1


cos β0 sin β
0 1 0
−sin β0 cos β

=

cos αcos β−sin αcos αsin β
sin αcos βcos αsin αsin β
−sin β0 cos β

1Eqn. (2.1) in [1] seems to be incorrect. I used a version with changes marked in red: Since Rz
0(φ)Rx(θ)>=
Rx(θ)>Rz
0(φ)>, rotations are performed in reverse order and direction.
2Eqn. (2.4) in [1] seems to be incorrect. I used a version with changes marked in red: Rotations are performed
in reverse order.
9

where we deliberately omitted the last rotation Rz(γ) by choosing γ= 0. Thus, the full rotation
matrix for drawing a great circle is
A=

axx axy axz
ayx ayy ayz
azx azy az z 
=Rd(φ, θ)D(α, β, 0) (5)
=

cos φsin φ0
−cos θsin φcos θcos φsin θ
sin θsin φ−sin θcos φcos θ


cos αcos β−sin αcos αsin β
sin αcos βcos αsin αsin β
−sin β0 cos β

=

cos αcos βcos φ+ cos βsin αsin φ
cos βcos φsin αcos θ−cos αcos βcos θsin φ−sin βsin θ
cos αcos βsin φsin θ−sin βcos θ−cos βcos φsin αsin θ
cos αsin φ−cos φsin α
cos αcos φcos θ+ sin αcos θsin φ
−cos αcos φsin θ−sin αsin φsin θ
cos αcos φsin β+ sin αsin βsin φ
cos βsin θ−cos αsin βcos θsin φ+ cos φsin αsin βcos θ
cos βcos θ+ cos αsin βsin φsin θ−cos φsin αsin βsin θ

The rotated coordinate system is set by \tdplotsetrotatedcoords{hαi}{hβi}{0}.
With the coordinate transforms described so far, we can only draw great circles. For drawing
small circles, we additionally need to offset the origin of the rotated coordinate system. To this
end we define an elevation angle which defines the height
ze=rsin (6)
of the offset drawing plane over the rotated one as illustrated in Fig. 2 (ris still the radius of the
sphere). As the circle to be drawn must lie on the sphere, its radius
re=rcos (7)
decreases with increasing elevation.
xr
zr
yr

ze
re
Figure 2: Illustration of z-coordinate and radius of an elevated circle on a sphere
The offset vector (see Fig. 1) is given by
~o =D(α, β, 0) 

0
0
ze
=

zecos αsin β
zesin αsin β
zecos β
=

rsin cos αsin β
rsin sin αsin β
rsin cos β
(8)
and applied through \tdplotsetrotatedcoordsorigin{h~o i} command of tikz-3dplot.
10

Drawing Circles of a Sphere
The parametric representation of a circle of a sphere in the rotated and offset coordinate system is


x(ϕ)
y(ϕ)
z(ϕ)
=

recos ϕ
resin ϕ
0
.(9)
After applying the coordinate transforms described above, we could just draw this circle by
\draw (0,0) circle hrei;
However, as we want to visualize the parts of the circle lying on the front and back sides of the
sphere, we consider the parametric representation in the rotated but not not(!) offset coordinate
system


x(ϕ)
y(ϕ)
z(ϕ)
=

recos ϕ
resin ϕ
ze
.(10)
The respective screen coordinates are


x0(ϕ)
y0(ϕ)
z0(ϕ)
=A

x(ϕ)
y(ϕ)
z(ϕ)
=

axx axy axz
ayx ayy ayz
azx azy az z 


rcos cos ϕ
rcos sin ϕ
rsin 
(11)
=

axx ·rcos cos ϕ+axy ·rcos sin ϕ+axz ·rsin 
ayx ·rcos cos ϕ+ayy ·rcos sin ϕ+ayz ·rsin 
azx ·rcos cos ϕ+azy ·rcos sin ϕ+az z ·rsin 
.
By examining the z0(ϕ) coordinate we can determine which parts of the circle are
on the front side (or “above” the drawing paper) z0(ϕ)>0 and (12)
on the back side (or “below” the drawing paper) z0(ϕ)<0
of the sphere. We denote by ϕ0the crossing angles between the front and back sides. In order to
determine them we solve
0!
=z0(ϕ0) = azx ·rcos cos ϕ0+azy ·rcos sin ϕ0+az z ·rsin . (13)
I must admit that I was too lazy to puzzle this out myself.. . ;-) Matlab says:
tan ϕ0
2=
azy cos ±qa2
zx cos2+a2
zy cos2−a2
zz sin2
azx cos −azz sin (14)
=
azy ±qa2
zx +a2
zy −a2
zz tan2
azx −azz tan ,(15)
where
a2
zz sin2≥(a2
zx +a2
zy ) cos2 tan2≥a2
zx +a2
zy
a2
zz
(16)
must hold. With the substitutions
u=azy ,(17)
v=qa2
zx +a2
zy −a2
zz tan2and (18)
w=azx −azz tan (19)
we get
tan ϕ0
2=u±v
w ϕ0=(2 arctan2(u+v, w)
2 arctan2(u−v, w)(20)
11

Here we used the arctan2(x, y) function which is defined as
arctan2(x, y) =























arctan x
yy > 0
arctan x
y+π y < 0, x ≥0
arctan x
y−π y < 0, x < 0
π
2y= 0, x > 0
−π
2y= 0, x < 0
0y= 0, x = 0
(21)
Iff condition (16) holds, Eqn. (13) has exactly two solutions,3
ϕ0,bf : angle of back to front side crossing and
ϕ0,fb : angle of front to back side crossing,
As PGF’s atan2 function takes values in [−360◦,+360◦], we unwrap ϕ0,bf and ϕ0,fb as follows:
if ϕ0,fb −ϕ0,bf >360◦then ϕ0,bf ←ϕ0,bf + 360◦,(22)
if ϕ0,bf > ϕ0,fb then ϕ0,bf ←ϕ0,bf −360◦.(23)
Now we can draw the arc on the sphere’s back side in the rotated and offset coordinate system by
\draw (hϕ0,fbi:hrei) arc (hϕ0,fb i:hϕ0,bf + 360◦i:hrei);
and the arc on the sphere’s front side by
\draw (hϕ0,bfi:hrei) arc (hϕ0,bf i:hϕ0,fbi:hrei);
Otherwise, iff condition (16) does not hold, Eqn. (13) has no solutions, which means that the circle
lies entirely either on the front side or on the back side of the sphere. In order to determine which
is the case, we test if an arbitrary point on the circle lies above or below the drawing paper (cf.
Eq. (12)). We choose ϕ0= 0 and evaluate the right side of Eq. (13)
azx ·rcos +azz ·rsin (≥0 : front side circle,
<0 : back side circle. (24)
Depending on the result, we draw the circle
\draw (0,0) circle (hrei);
in the front side or back side style.
3.2 The Package Source Code
1%% == LaTeX PACKAGE tikz-3dplot-circleofsphere ================================
2%% Drawing circles of a sphere with tikz-3dplot
3%%
4%% Matthias Wolff, BTU Cottbus-Sentenberg
5%% July 27, 2018
6%%
7%% References:
8%% [1] J. Hein. The tikz-3dplot package. 2012. Online, retrieved July 20, 2018.
9%% http://mirror.ctan.org/graphics/pgf/contrib/tikz-3dplot/tikz-3dplot_documentation.pdf
10 %% [2] T. Tantau. TikZ & PGF - Manual for Version 3.0.1a. 2015. Online, retrieved July 22, 2018.
11 %% http://mirror.ctan.org/graphics/pgf/base/doc/pgfmanual.pdf
12 %% [3] Drawing Great Circles
13 %% https://tex.stackexchange.com/questions/168521/spherical-triangles-and-great-circles
14
15 %% == REQUIRED PACKAGES =======================================================
16
3which coincide iff the left and right sides of condition(16) are equal
12

17 \RequirePackage{xifthen}
18 \RequirePackage{tikz}
19 \RequirePackage{tikz-3dplot}
20
21 %% == TikZ STYLES =============================================================
22
23 \tikzset{
24 tdplotCsFront/.style={solid},
25 tdplotCsBack/.style={dashed},
26 tdplotCsFill/.style={opacity=0},
27 tdplotPtFront/.style={},
28 tdplotPtBack/.style={},
29 tdplotCsDrawAux/.style={}
30 }
31
32 %% == COMMANDS ================================================================
33
34 \newcommand{\tdplotCsComputeTransformRotScreen}{%
35 % Computes the elements of the full rotation matrix
36 %
37 % A = [\axx \axy \axz]
38 % [\ayx \ayy \ayz]
39 % [\azx \azy \azz].
40 %
41 % Ouput:
42 % \axx - Element A(1,1) of rotation matrix
43 % \axy - Element A(1,2) of rotation matrix
44 % ...
45 % \azz - Element A(3,3) of rotation matrix
46 %
47 \let\a\tdplotalpha
48 \let\b\tdplotbeta
49 \let\p\tdplotmainphi
50 \let\t\tdplotmaintheta
51 % Row 1: [\axx \axy \axz]
52 \pgfmathsetmacro\axx{cos(\a)*cos(\b)*cos(\p) + cos(\b)*sin(\a)*sin(\p)}
53 \pgfmathsetmacro\axy{cos(\a)*sin(\p) - cos(\p)*sin(\a)}
54 \pgfmathsetmacro\axz{cos(\a)*cos(\p)*sin(\b) + sin(\a)*sin(\b)*sin(\p)}
55 % Row 2: [\ayx \ayy \ayz]
56 \pgfmathsetmacro\ayx{cos(\b)*cos(\p)*sin(\a)*cos(\t) - cos(\a)*cos(\b)*cos(\t)*sin(\p) - sin(\b)*sin(\t)}
57 \pgfmathsetmacro\ayy{cos(\a)*cos(\p)*cos(\t) + sin(\a)*cos(\t)*sin(\p)}
58 \pgfmathsetmacro\ayz{cos(\b)*sin(\t) - cos(\a)*sin(\b)*cos(\t)*sin(\p) + cos(\p)*sin(\a)*sin(\b)*cos(\t)}
59 % Row 3: [\azx \azy \azz]
60 \pgfmathsetmacro\azx{cos(\a)*cos(\b)*sin(\p)*sin(\t) - sin(\b)*cos(\t) - cos(\b)*cos(\p)*sin(\a)*sin(\t)}
61 \pgfmathsetmacro\azy{-cos(\a)*cos(\p)*sin(\t) - sin(\a)*sin(\p)*sin(\t)}
62 \pgfmathsetmacro\azz{cos(\b)*cos(\t) + cos(\a)*sin(\b)*sin(\p)*sin(\t) - cos(\p)*sin(\a)*sin(\b)*sin(\t)}
63 }
64
65 % ------------------------------------------------------------------------------
66
67 \newcommand{\tdplotCsDrawCircle}[5][]{%
68 % Draws a circle of a sphere.
69 %
70 % Input:
71 % #1 - TikZ style
72 % - use tdplotCsFront/.style={...} to style the front side arc
73 % - use tdplotCsBack/.style={...} to style the back side arc
74 % - use tdplotCsFill/.style={...} to style the circle filling
75 % - use tdplotCsDrawAux to draw some auxiliary information
76 % #2 - Radius of sphere
77 % #3 - Azimuthal angle of drawing plane 1)
78 % #4 - Polar angle of drawing plane 2)
79 % #5 - Elevation angle of circle above the drawing plane. Permissible
80 % values are -90 < #5 < 90. Use 0 for drawing a great circle.
81 %
82 % Ouput:
83 % none
84 %
85 % Footnotes:
86 % 1) passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
87 % 2) passed as beta to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
88 \begin{scope}[#1] % Macro scope >>
13

89 % Do some computation # -----------------------------------
90 \pgfmathsetmacro\r {#2} % Parse radius
91 \pgfmathsetmacro\alp{#3} % Parse azimuthal angle (alpha)
92 \pgfmathsetmacro\bet{#4} % Parse polar angle (beta)
93 \pgfmathsetmacro\eps{#5} % Parse elevation angle (epsilon)
94 \pgfmathsetmacro\re {\r*cos(\eps)} % Radius of circle
95 \pgfmathsetmacro\ze {\r*sin(\eps)} % z-coordinate of drawing plane
96 \pgfmathsetmacro\coX{\ze*cos(\alp)*sin(\bet)} % x-coordinate offset for ze
97 \pgfmathsetmacro\coY{\ze*sin(\alp)*sin(\bet)} % y-coordinate offset for ze
98 \pgfmathsetmacro\coZ{\ze*cos(\bet)} % z-coordinate offset for ze
99 \coordinate (coffs) at (\coX,\coY,\coZ); % Offset as coordinate value
100 % Rotate and offset coordinate system % -----------------------------------
101 \tdplotsetrotatedcoords{\alp}{\bet}{0} % Rotate coordinate system
102 \tdplotsetrotatedcoordsorigin{(coffs)} % Offset coordinate system
103 % Draw % -----------------------------------
104 \begin{scope}[tdplot_rotated_coords] % Drawing scope >>
105 \tdplotCsComputeTransformRotScreen % Compute full rotation matrix
106 \pgfmathsetmacro\tanEps{tan(\eps)} % Tangent of elevation angle
107 \pgfmathsetmacro\bOneside{((\tanEps)^2)>=(((\azx)^2+(\azy)^2)/(\azz)^2)} % Circle entirely on one side?
108 \ifthenelse{\isin{tdplotCsFill}{#1}}{ % Fill style passed >>
109 \fill[tdplotCsFill] (0,0) circle (\re); % Draw filling of circle
110 }{} % <<
111 \ifthenelse{\bOneside=1}{ % Circle on one side of sphere >>
112 \pgfmathsetmacro\bFrontside{(\azx*\re+\azz*\ze)>=0} % Circle entirely on front side?
113 \ifthenelse{\bFrontside=1} % |
114 {\draw[tdplotCsFront] (0,0) circle (\re);} % Draw on front side
115 {\draw[tdplotCsBack] (0,0) circle (\re);} % Draw on back side
116 }{ % << Circle on both sides >>
117 \pgfmathsetmacro\u{\azy} % Substitution u=...
118 \pgfmathsetmacro\v{sqrt( (\azx)^2 + (\azy)^2 - (\azz)^2*(\tanEps)^2 )} % Substitution v=...
119 \pgfmathsetmacro\w{\azx - \azz*\tanEps} % Substitution w=...
120 \pgfmathsetmacro\phiBf{2*atan2(\u-\v,\w)} % Back->front crossing angle
121 \pgfmathsetmacro\phiFb{2*atan2(\u+\v,\w)} % Front->back crossing angle
122 \pgfmathsetmacro\bUnwrapA{(\phiFb-\phiBf)>360} % Unwrap front->back angle #1?
123 \pgfmathsetmacro\bUnwrapB{\phiBf>\phiFb} % Unwrap front->back angle #2?
124 \ifthenelse{\bUnwrapA=1}{\pgfmathsetmacro\phiBf{\phiBf+360}}{} % Unwrap front->back angle #1
125 \ifthenelse{\bUnwrapB=1}{\pgfmathsetmacro\phiBf{\phiBf-360}}{} % Unwrap front->back angle #2
126 \draw[tdplotCsBack] (\phiFb:\re) arc (\phiFb:{\phiBf+360}:\re); % Draw back side arc
127 \draw[tdplotCsFront] (\phiBf:\re) arc (\phiBf:\phiFb:\re); % Draw back side arc
128 }% <<
129 % Auxliliary drawing (for debugging and illustration) % - - - - - - - - - - - - - - - - -
130 \ifthenelse{\isin{tdplotCsDrawAux}{#1}}{ % Auxiliary drawing activated >>
131 \draw[red!40,->] (-\re,0,0) -- (\re,0,0) node[anchor=north] {$x_d$}; % x-axis of drawing corrd. system
132 \draw[red!40,->] (0,-\re,0) -- (0,\re,0) node[anchor=north] {$y_d$}; % y-axis of drawing corrd. system
133 \draw[red!40,->] (0,0,0) -- (0,0,\re) node[anchor=north] {$z_d$}; % z-axis of drawing corrd. system
134 \ifthenelse{\bOneside=0}{ % Circ.on both sides of sphere >>
135 \node[red] at (\phiBf:\re) {$\circ$}; % Indicate back-front crossing
136 \node[red] at (\phiFb:\re) {$\times$}; % Indicate front-back crossing
137 }{} % <<
138 \coordinate (coffs) at (-\coX,-\coY,-\coZ); % HACK: Forcibly reset ...
139 \tdplotsetrotatedcoordsorigin{(coffs)} % ... coordinate system
140 \begin{scope}[tdplot_rotated_coords] % Aux. display scope >>
141 \node[tdplot_screen_coords,red,anchor=north west] at (0.7*\r,-0.9*\r) % Make a litte display ...
142 {\parbox{200pt}{\footnotesize % ... >>
143 $\theta=\tdplotmaintheta^\circ,\phi=\tdplotmainphi^\circ$\\ % Main coord. sys. parameters
144 $\alpha=\alp^\circ,\beta=\bet^\circ,% Rot. coord. sys. parameters
145 \epsilon\!=\!\eps^\circ\!$\\ % Drawing plane elev. angle
146 $a_{zx}=\azx, a_{zy}=\azy, a_{zz}=\azz$\\ % Elems. of full rot. matrix
147 $r_e\!=\!\re, z_e\!=\!\ze$\\ % Radius and z-elevation
148 $\texttt{\textbackslash bOneside}\!=\!\bOneside$, % One-side circle flag
149 \ifthenelse{\bOneside=1}{ % One-side circle >>
150 $\texttt{\textbackslash bFrontside}\!=\!\bFrontside$\\ % Front-side flag
151 }{ % << Two-side circle >>
152 $\texttt{\textbackslash bUnwrapA}\!=\!\bUnwrapA$, % Angle unwrap flag #1
153 $\texttt{\textbackslash bUnwrapB}\!=\!\bUnwrapB$\\ % Angle unwrap flag #2
154 $\circ\!: \!\texttt{\textbackslash phiBf}\!=\!\phiBf^\circ\!, % Back-front crossing angle
155 \times\!:\!\texttt{\textbackslash phiFb}\!=\!\phiFb^\circ$\\ % Front-back crossing angle
156 }% <<
157 }}; % <<
158 \end{scope} % << (Aux. display scope)
159 }{} % << (Auxiliary drawing activated)
160 \end{scope} % << (Drawing scope)
14

161 \end{scope} % << (Macro scope)
162 }
163
164 % ------------------------------------------------------------------------------
165
166 \newcommand{\tdplotCsDrawGreatCircle}[4][]{%
167 % Draws a great circle.
168 %
169 % Input:
170 % #1 - TikZ style
171 % - use tdplotCsFront/.style={...} to style the front side arc
172 % - use tdplotCsBack/.style={...} to style the back side arc
173 % - use tdplotCsFill/.style={...} to style the circle filling
174 % - use tdplotCsDrawAux to draw some auxiliary information
175 % #2 - Radius of sphere
176 % #3 - Azimuthal angle of drawing plane 1)
177 % #4 - Polar angle of drawing plane 2)
178 %
179 % Ouput:
180 % none
181 %
182 % Footnotes:
183 % 1) passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
184 % 2) passed as beta to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
185 \tdplotCsDrawCircle[#1]{#2}{#3}{#4}{0}
186 }
187
188 % ------------------------------------------------------------------------------
189
190 \newcommand{\tdplotCsDrawLatCircle}[3][]{%
191 % Draws a circle of latitude.
192 %
193 % Input:
194 % #1 - TikZ style
195 % - use tdplotCsFront/.style={...} to style the front side arc
196 % - use tdplotCsBack/.style={...} to style the back side arc
197 % - use tdplotCsFill/.style={...} to style the circle filling
198 % - use tdplotCsDrawAux to draw some auxiliary information
199 % #2 - Radius of sphere
200 % #3 - Elevation angle of circle above the drawing plane. Permissible
201 % values are -90 < #5 < 90. Use 0 for drawing a great circle.
202 %
203 % Ouput:
204 % none
205 \tdplotCsDrawCircle[#1]{#2}{0}{0}{#3}
206 }
207
208 % ------------------------------------------------------------------------------
209
210 \newcommand{\tdplotCsDrawLonCircle}[3][]{%
211 % Draws a circle of longitude.
212 %
213 % Input:
214 % #1 - TikZ style
215 % - use tdplotCsFront/.style={...} to style the front side arc
216 % - use tdplotCsBack/.style={...} to style the back side arc
217 % - use tdplotCsFill/.style={...} to style the circle filling
218 % - use tdplotCsDrawAux to draw some auxiliary information
219 % #2 - Radius of sphere
220 % #3 - Azimuthal angle of drawing plane 1)
221 %
222 % Ouput:
223 % none
224 %
225 % Footnotes:
226 % 1) passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
227 \tdplotCsDrawCircle[#1]{#2}{#3}{90}{0}
228 }
229
230 % ------------------------------------------------------------------------------
231
232 \newcommand{\tdplotCsFrontsidePoint}{%
15

233 % Invoked by \tdplotCsDrawPoint to draw a point on the front side of a sphere.
234 % Redefine to customize.
235 \textbullet%
236 }
237
238 % ------------------------------------------------------------------------------
239
240 \newcommand{\tdtlotCsBacksidePoint}{%
241 % Invoked by \tdplotCsDrawPoint to draw a point on the back side of a sphere.
242 % Redefine to customize.
243 $\circ$%
244 }
245
246 % ------------------------------------------------------------------------------
247
248 \newcommand{\tdplotCsDrawPoint}[4][]{%
249 % Draws a point on a sphere.
250 %
251 % Input:
252 % #1 - TikZ style
253 % - use tdplotPtFront/.style={...} to style a front side point
254 % - use tdplotPtBack/.style={...} to style a back side point
255 % #2 - Radius of sphere
256 % #3 - Azimuthal angle of drawing plane 1)
257 % #4 - Polar angle of drawing plane 2)
258 %
259 % Ouput:
260 % none
261 %
262 % Remarks:
263 % - Redefine \tdplotCsFrontsidePoint to customize drawing of a front side
264 % point.
265 % - Redefine \tdplotCsBacksidePoint to customize drawing of a back side
266 % point.
267 %
268 % Footnotes:
269 % 1) passed as alpha to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
270 % 2) passed as beta to \tdplotsetrotatedcoords{alpha}{beta}{gamma}
271 \begin{scope}[#1] % Macro scope >>
272 \pgfmathsetmacro{\r}{#2} % Parse radius
273 \pgfmathsetmacro{\alp}{#3} % Parse alpha angle
274 \pgfmathsetmacro{\bet}{#4} % Parse beta angle
275 \tdplotsetrotatedcoords{\alp}{\bet}{0} % Set rotated coord. system
276 \begin{scope}[tdplot_rotated_coords] % Draw in rotated coord. system >>
277 \tdplotCsComputeTransformRotScreen % Get \azz
278 \pgfmathsetmacro{\bVisible}{\azz>0} % Test if point is on visible side
279 \ifthenelse{\bVisible=1}{% % Point on front side >>
280 \node[tdplotPtFront] at (0,0,\r) {\tdplotCsFrontsidePoint}; % Draw it
281 }{% % << Point on back side >>
282 \node[tdplotPtBack] at (0,0,\r) {\tdtlotCsBacksidePoint}; % Draw it
283 }% <<
284 \end{scope} % <<
285 \end{scope} % <<
286 }
287
288 %% == EOF =====================================================================
3.3 An Auxiliary Matlab Script
1%% == LaTeX PACKAGE tikz-3dplot-circleofsphere ================================
2% Drawing circles of a sphere with tikz-3dplot
3%
4% Matthias Wolff, BTU Cottbus-Sentenberg
5% July 26, 2018
6%
7% References:
8% [1] J. Hein. The tikz-3dplot package. 2012. Online, retrieved July 20, 2018.
9% https://mirror.hmc.edu/ctan/graphics/pgf/contrib/tikz-3dplot/tikz-3dplot_documentation.pdf
10 %
11
12 %% Rotation matrices ==========================================================
13 syms a b p t
16

14
15 % R rotation matrix -----------------------------------------------------------
16 Rz = [ cos(p) -sin(p) 0
17 sin(p) cos(p) 0
18 0 0 1 ];
19
20 Rx = [ 1 0 0
21 0 cos(t) -sin(t)
22 0 sin(t) cos(t) ];
23
24 % - [1] eq. (2.1) line 2
25 % R = Rz*Rx; disp(R);
26
27 % - [1] eq. (2.1) line 3
28 % R = [ cos(p) sin(p) 0
29 % -cos(t)*sin(p) cos(t)*cos(p) -sin(t)
30 % sin(t)*sin(p) -sin(t)*cos(p) cos(t) ];
31
32 % - [1] eq. (2.1) line 3, corrected
33 R = (Rz*Rx).’;
34
35 % -- D rotation matrix --------------------------------------------------------
36 Dz = [ cos(a) -sin(a) 0
37 sin(a) cos(a) 0
38 0 0 1 ];
39
40 Dy = [ cos(b) 0 sin(b)
41 010
42 -sin(b) 0 cos(b) ];
43
44 Dx = [ 1 0 0
45 0 cos(b) -sin(b)
46 0 sin(b) cos(b) ];
47
48 D = Dz*Dy; disp(D);
49
50 % -- Full rotation matrix -----------------------------------------------------
51 A = R*D; disp(A);
52 axx = A(1,1); axy = A(1,2); axz = A(1,3);
53 ayx = A(2,1); ayy = A(2,2); ayz = A(2,3);
54 azx = A(3,1); azy = A(3,2); azz = A(3,3);
55
56 %% == Transform a vector (world -> screen) ====================================
57 syms x y z
58 p = [ x
59 y
60 z ];
61 q=A*p;
62 disp(q);
63
64 %% == View angle ==============================================================
65 syms p0 r eps azx azy azz
66 assume(p0,’real’);
67 assume(r,’real’);
68 assume(eps,’real’);
69 assume(azx,’real’);
70 assume(azy,’real’);
71 assume(azz,’real’);
72 eqn = azx*r*cos(eps)*cos(p0) + azy*r*cos(eps)*sin(p0) + azz*r*sin(eps) == 0
73 solve(eqn,p0,’Real’,true)
74
75 %% == EOF =====================================================================
17

References
[1] Jeff Hein. The tikz-3dplot package. http://mirror.ctan.org/graphics/pgf/contrib/
tikz-3dplot/tikz-3dplot_documentation.pdf, 2012. Retrieved: July 27, 2018.
[2] Till Tantau. Tikz & pgf - manual for version 3.0.1a. http://mirror.ctan.org/graphics/
pgf/base/doc/pgfmanual.pdf, 2015. Retrieved: July 27, 2018.
[3] Matthias Wolff. The tikz-3dplot-circleofsphere package: Drawing circles of a sphere with
tikz-3dplot.https://github.com/matthias-wolff/tikz-3dplot-circleofsphere, 2018.
Retrieved: July 27, 2018.
18

x
y
z
xdyd
zd
◦
×
θ= 60.0◦, φ = 125.0◦
α=−40.0◦, β = 30◦,  = 30◦
azx =−0.05588, azy = 0.8365, azz = 0.54507
re=2.59808, ze= 1.5
\bOneside =0, \bUnwrapA = 0, \bUnwrapB = 1
◦:\phiBf =−18.22858◦
,×:\phiFb =205.86197◦
1\documentclass{standalone}
2\usepackage[dvipsnames]{xcolor}
3\usepackage{tikz-3dplot-circleofsphere}
4
5\begin{document}
6
7\def\elev{ 30} \pgfmathsetmacro{\tdpTheta}{90-\elev}
8\def\azim{ 35} \pgfmathsetmacro{\tdpPhi}{90+\azim}
9\def\R{3}
10 \tdplotsetmaincoords{\tdpTheta}{\tdpPhi}
11 \begin{tikzpicture}[scale=1,tdplot_main_coords]
12 \begin{scope}[black!30,name=auxiliary]
13 \draw[tdplot_screen_coords] (0,0,0) circle (\R);
14 \draw[->] (-1.3*\R,0,0) -- (1.3*\R,0,0) node[anchor=north east]{$x$};
15 \draw[->] (0,-1.3*\R,0) -- (0,1.3*\R,0) node[anchor=north west]{$y$};
16 \draw[->] (0,0,-1.3*\R) -- (0,0,1.3*\R) node[anchor=south]{$z$};
17 \tdplotCsDrawCircle{\R}{0}{0}{0};
18 \end{scope}
19 \begin{scope}
20 % \tdplotCsDrawLatCircle[tdplotCsDrawAux]{\R}{-30}
21 % --
22 \tdplotCsDrawCircle[tdplotCsDrawAux]{\R}{-40}{30}{30}
23 % --
24 % \foreach \a in {0,15,...,345}
25 % { \tdplotCsDrawCircle[very thin,gray]{\R}{\a}{90}{0} }
26 % \foreach \a in {-75,-60,...,75}
27 % { \tdplotCsDrawCircle[very thin,gray]{\R}{0}{0}{\a} }
28 % -- Pathologic cases -->
29 % \tdplotCsDrawCircle{\R}{35}{60}{0}
30 % <--
31 \end{scope}
32 \end{tikzpicture}
33
34 \end{document}
19