GeodesicLinesOnSphere::usage="Find the geodesic lines of a sphere, remembering that one of the two conditions must be met: a) The curvature of the line is equal to zero. b) The normal to the surface is the main normal of the line. References: Fedenko, A.S. Problemas de Geometría Diferencial. Moscú: Editorial Mir: p.110, 1981.; Personal.";
Development:
In[]:=
s[u_,v_]:={4Cos[v]Cos[u],4Cos[v]Sin[u],4Sin[v]};
The maximum circumferences in the sphere are those that have the same center of the sphere:
In[]:=
f[x_,y_]:=
16-
2
x
-
2
y
;​​g[x_,y_]:=-x-2y;​​f[rCos[t],rSin[t]]==g[rCos[t],rSin[t]]​​roft=r/.First@Solve[%,r]//Simplify​​z[t_]:=g[roftCos[t],roftSin[t]];​​γ[t_]:={roftCos[t],roftSin[t],z[t]};
Out[]=
16-
2
r
2
Cos[t]
-
2
r
2
Sin[t]
-rCos[t]-2rSin[t]
Out[]=
-
4
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
ζ[t_]:=-
4Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
,-
4Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
,
4Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
8Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
;​​
In[]:=
​​ManipulateClearAll;s[u_,v_]:={4Cos[v]Cos[u],4Cos[v]Sin[u],4Sin[v]};ζ[t_]:=-
4Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
,-
4Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
,
4Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
8Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
;​​ShowParametricPlot3D[s[u,v],{u,0,t1/2},{v,0,t1},PlotStyle->Opacity[0.2],ColorFunction"BlueGreenYellow",MeshFalse,PerformanceGoal"Quality"],​​Graphics3DBlack,Ball0,-
4
5
,
8
5
,0.15,ParametricPlot3D[ζ[t],{t,0,t1},PlotStyleBlack],Graphics3DBlue,Point[{-1,0,3π}],Arrowheads[.05],ArrowTube0,-
4
5
,
8
5
,0,-
4
5
,
8
5
+t10,
1
5
,-
2
5
,​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,-3,3},ImageSize500,Style["Geodesic line:",Bold,Large],{{t1,1,"Value (t1)"},0.01,2π,0.01},ControlPlacementTop
Out[]=
​
Geodesic line:
Value (t1)
A point about the maximum circumference is:
In[]:=
Printζ
π
2

0,-
4
5
,
8
5

0,-
4
5
,
8
5

Verification that is a geodesic line:
In[]:=
ResourceFunction["Curvature"][ζ[t],t]
Out[]=
2
2
-
64
6
Cos[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
192
5
Cos[t]
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
80
4
Cos[t]
2
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
80
2
Cos[t]
4
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
192Cos[t]
5
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
64
6
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
48
4
Cos[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
128
3
Cos[t]
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
128Cos[t]
3
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
48
4
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
16
2
Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
16
2
Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
2
-
128
6
Cos[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
384
5
Cos[t]
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
160
4
Cos[t]
2
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
160
2
Cos[t]
4
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
384Cos[t]
5
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
128
6
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
96
4
Cos[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
256
3
Cos[t]
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
256Cos[t]
3
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
96
4
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
32
2
Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
32
2
Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]

3/2
2
2Sin[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
4Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
2
-
2Cos[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
4Sin[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
8Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
-
4Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
2
2Cos[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
4Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
In[]:=
2
2
-
64
6
Cos[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
192
5
Cos[t]
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
80
4
Cos[t]
2
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
80
2
Cos[t]
4
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
192Cos[t]
5
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
64
6
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
48
4
Cos[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
128
3
Cos[t]
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
128Cos[t]
3
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
48
4
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
16
2
Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
16
2
Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
2
-
128
6
Cos[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
384
5
Cos[t]
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
160
4
Cos[t]
2
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
160
2
Cos[t]
4
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
384Cos[t]
5
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
128
6
Sin[t]
3
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
96
4
Cos[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
256
3
Cos[t]
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
256Cos[t]
3
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
96
4
Sin[t]
2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
32
2
Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
32
2
Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]

3/2
2
2Sin[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
4Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
2
-
2Cos[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
4Sin[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
8Cos[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
-
4Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
+
2
2Cos[t](4
2
Cos[t]
+6Cos[t]Sin[t]-4
2
Sin[t]
)
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
+
4Sin[t]
2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
/.t
π
2
Out[]=
1
4
It is not fulfilled (a), it must be fulfilled (b):
In[]:=
ResourceFunction["NormalVector"][ζ[t],t]
Out[]=

5764801
9
Cos[t]
1
6
(7-3Cos[2t]+4Sin[2t])
1
2
(7-3Cos[2t]+4Sin[2t])
36
9/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
-
1647086
9
Cos[t]
Cos[2t]
1
6

⋯1⋯

1
2

⋯1⋯

3
9/2
2
2
Cos
⋯1⋯
⋯1⋯
]
+
⋯1⋯
+5
⋯1⋯

+
⋯1884⋯
+
5120Cos[t]
2
⋯1⋯
8
⋯1⋯
1
6
⋯1⋯
1
2
⋯1⋯
9
3/2
(
⋯1⋯
)
-
2048
3
Sin[t]
8
Sin[2t]
1
6
7-3
⋯1⋯
+4
⋯1⋯

1
2

⋯1⋯

9
3/2
(2
2
Cos[t]
+4Cos[t]Sin[t]+5
2
Sin[t]
)
,-
⋯1⋯
36
⋯1⋯
+
⋯1892⋯
+
⋯1⋯
9
⋯1⋯
,
⋯1⋯

Full expression not available
(
original memory size:
10 MB)
All that remains is to verify that it is equal to the normal surface:
Normal to the surface:
The normal of the curve is identical to the normal of the surface therefore it is a geodesic line as expected, it was fulfilled (b).
Now an animation with the rest of the geodesics (based on ResourceFunction[“Geodesic”]):