ABSTRACT (original article): We show that in Euclidean 3-space any closed curve which lies outside the unit sphere and contains the sphere within its convex hull has length . Equality holds only when γ is composed of four semicircles of length π, arranged in the shape of a baseball seam, as conjectured by V.A. Zalgaller in 1996. CITATION (original article): Ghomi, M. & Wenk, J. (2021). Shortest closed curve to inspect a sphere. Journal für die reine und angewandte Mathematik (Crelles Journal), 2021(781), 57-84. https://doi.org/10.1515/crelle-2021-0049
γ
≥4π
Here we verify some of the longer symbolic computations in the above paper, and also include the codes for generating the plots. The enumeration of the sections below correspond to the sections in the above paper.
Section 3: The Integral Formula for Efficiency
Section 3: The Integral Formula for Efficiency
Here we check the computations in Proposition 3.1 for the Jacobian of the map
F[t_,θ_]=λ[t]e0[t]+ρ[t](Cos[θ]e1[t]+Sin[θ]e2[t]);
where we have set e0:=.
γ
3.1 The case where γ’(t_0)≠0.
3.1 The case where ’(t_0)≠0.
γ
In[]:=
D[F[t,θ],t]/.{e0'[t]ve1[t],e1'[t]-ve0[t],e2'[t]0}
Out[]=
ve1[t]λ[t]-vCos[θ]e0[t]ρ[t]+e0[t][t]+(Cos[θ]e1[t]+e2[t]Sin[θ])[t]
′
λ
′
ρ
So the coordinates of the first partial derivative of F with respect to the frame (e0, e1, e2) are
{[t]-vCos[θ]ρ[t],vλ[t]+[t]Cos[θ],[t]Sin[θ]}
′
λ
′
ρ
′
ρ
In[]:=
D[F[t,θ],θ]
Out[]=
(Cos[θ]e2[t]-e1[t]Sin[θ])ρ[t]
The coordinates of the second partial derivative of F with respect to (e0, e1, e2) are
ρ[t]{0,-Sin[θ],Cos[θ]}
Now we take the cross product of the partial derivatives:
In[]:=
ρ[t]{[t]-vCos[θ]ρ[t],vλ[t]+[t]Cos[θ],[t]Sin[θ]}{0,-Sin[θ],Cos[θ]}//FullSimplify
′
λ
′
ρ
′
ρ
Out[]=
{ρ[t](vCos[θ]λ[t]+[t]),Cos[θ]ρ[t](vCos[θ]ρ[t]-[t]),Sin[θ]ρ[t](vCos[θ]ρ[t]-[t])}
′
ρ
′
λ
′
λ
The square of the norm of the Jacobian of F is given by
In[]:=
Dot[%,%]//FullSimplify
Out[]=
2
ρ[t]
2
v
2
Cos[θ]
2
λ[t]
2
(-vCos[θ]ρ[t]+[t])
′
λ
′
ρ
2
′
ρ
Next we rewrite the Jacobian as follows
2
ρ[t]
2
v
2
Cos[θ]
2
λ[t]
2
ρ[t]
′
λ
′
ρ
2
′
ρ
2
′
λ
By definition we have
λ[t][t_]=
1-
;2
ρ[t]
So
2
λ[t]
2
ρ[t]
Furthermore, we have
ρ[t][t]-λ[t][t]//Simplify
′
λ
′
ρ
Out[]=
′
λ
1-
2
λ[t]
In[]:=
2
′
λ
2
′
ρ
Out[]=
2
′
λ
1-
2
λ[t]
So it follows that the the square of the Jacobian of F is given by
2
ρ[t]
2
v
2
Cos[θ]
′
λ
ρ[t]
2
′
λ
2
ρ[t]
2
ρ[t]
2
vCos[θ]-[t]
′
λ
ρ[t]
2
(ρ[t]vCos[θ]+[t])
′
λ
3.2 The case where γ’(t_0)=0
3.2 The case where ’(t_0)=0
γ
Section 6: Efficiency of Line Segments
Section 6: Efficiency of Line Segments
The formula for the intersection of two spherical caps with radii , , and distance d between the centers is as follows. Note that this formula is valid only if the boundaries of the caps intersect.
ρ
0
ρ
1
In[]:=
A[ρ0_,ρ1_,d_]=2ArcCos-ArcCosCos[ρ0]-ArcCosCos[ρ1];
-Cos[d]+Cos[ρ0]Cos[ρ1]
Sin[ρ0]Sin[ρ1]
Cos[ρ1]-Cos[d]Cos[ρ0]
Sin[d]Sin[ρ0]
Cos[ρ0]-Cos[d]Cos[ρ1]
Sin[d]Sin[ρ1]
Next we set
In[]:=
H1[ρ0_,ρ1_,d_]=4π+4π-2A[ρ0,ρ1,d];
2
Sin
ρ0
2
2
Sin
ρ1
2
So we add the areas of two caps, and subtract twice the area on the intersection.
The horizon of a line segment with h0, h1 being the heights of its vertices and l the length of the segment is given by
In[]:=
FullSimplifyH1ArcCos,ArcCos,ArcCos,{h1>h0≥1,l>0,0<α<π}
1
h0
1
h1
l^2-h0^2-h1^2
-2h0h1
Out[]=
2π+4ArcSin--
2--+
2
h0
2
h1
2
l
2
(-1+)(-1+)
2
h0
2
h1
4ArcSin-+
2
h0
2
h1
2
l
-(-1+)(h0-h1-l)(h0+h1-l)(h0-h1+l)(h0+h1+l)
2
h0
h0
4ArcSin
-++
2
h0
2
h1
2
l
-(-1+)(-h0+h1-l)(h0+h1-l)(-h0+h1+l)(h0+h1+l)
2
h1
h1
So after some simplifications we set
In[]:=
H2[h0_,h1_,l_]=4ArcCos+--2+ArcSin--+ArcSin--;
2
h0
2
h1
2
l
2
(-1)(-1)
2
h0
2
h1
1
h0
2
h1
2
h0
2
l
(-1)(-)(-)
2
h0
2
l
2
(h1-h0)
2
(h0+h1)
2
l
1
h1
2
h0
2
h1
2
l
(-1)(-)(-)
2
h1
2
l
2
(h1-h0)
2
(h0+h1)
2
l
Next note that
In[]:=
FullSimplifyH2h0,++2h0lCos[α],l,{h0≥1,l>0,0<α<π}
2
h0
2
l
Out[]=
4ArcCos+-++2h0lCos[α]
-1++h0lCos[α]
2
h0
(-1+)(-1+++2h0lCos[α])
2
h0
2
h0
2
l
ArcSin
Cot[α]
-1+
2
h0
h0
ArcSin
(l+h0Cos[α])Csc[α]
h0
-1+++2h0lCos[α]
2
h0
2
l
2
h0
2
l
So letting ,
In[]:=
h1[l_,α_]=++2h0lCos[α];
2
h0
2
l
we have
In[]:=
H3[h0_,l_,α_]=4ArcCos+ArcSin+ArcSin;
-1++h0lCos[α]
2
h0
(-1+)(-1+)
2
h0
2
h1[l,α]
-1
h1[l,α]
(l+h0Cos[α])
h0Sin[α]
-1+
2
h1[l,α]
1
h0
Cot[α]
-1+
2
h0
In particular, note that
In[]:=
FullSimplifyH3h0,l,,{h0≥1,l>0}
π
2
Out[]=
4ArcCos-+
-1+
2
h0
-1++
2
h0
2
l
ArcSin
l
h0
-1++
2
h0
2
l
2
h0
2
l
So we obtain the following formula for one-edge spirals
In[]:=
H4[h0_,l_]=4ArcCos-+ArcSin;
-1+
2
h0
-1++
2
h0
2
l
1
2
h0
2
l
l
h0
-1++
2
h0
2
l
We may also rewrite the above formula in terms of h0 and h1:
In[]:=
FullSimplifyH4h0,-,{h1≥h0≥1}
2
h1
2
h0
Out[]=
4ArcCos-
-1+
2
h0
-1+
2
h1
4ArcCsch0
-1+
2
h1
-+
2
h0
2
h1
h1
So we set
In[]:=
H5[h0_,h1_]=4ArcCos-1-1-ArcSin-;
2
h0
2
h1
1
h1
2
h1
2
h0
h0-1
2
h1
Finally we define
In[]:=
Eff[h0_,h1_]=-;
H5[h0,h1]
2
h1
2
h0
In[]:=
Plot3D[Eff[h0,h1],{h0,1,3},{h1,h0,3}]
Out[]=
Section 7: Efficiency of Spirals
Section 7: Efficiency of Spirals
Section 8: Instantaneous Efficiency
Section 8: Instantaneous Efficiency
If we set
then the instant efficiency is given by
Set
then
which simplifies to
So we set
Next we generate some plots:
In Lemma 8.2 we compute
Section 9: Lifting and Splitting of Spirals
Section 9: Lifting and Splitting of Spirals
which is equivalent to:
Finally we have:
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Shortest closed curve to inspect a sphere
by Mohammad Ghomi and James Wenk
Wolfram Community, STAFF PICKS, March 13, 2025
https://community.wolfram.com/groups/-/m/t/3416183
by Mohammad Ghomi and James Wenk
Wolfram Community, STAFF PICKS, March 13, 2025
https://community.wolfram.com/groups/-/m/t/3416183