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Proposition 2.
An
x
y
stadium of flat paper mapped onto an
R
-sphere may
occupy no more surface area than
A
(
x, y
)=2
R
ˀR
×
.
y
2
R
+
x
sin
y
2
R
−
ˀR
cos
Proof.
To bound
A
(
x, y
) we will first establish
A
(0
,y
) and then bound the deriv-
ative
dA/dx
. This will allow us to bound the areas of all mapped stadiums.
First, consider an 0
y
stadium: a radius-
y/
2 disk. By definition, the disk
must fall within
y/
2 of its center on the sphere. A radius-
y/
2 spherical cap has
area 2
ˀR
2
(1
×
cos
2
R
), proving the claim for
A
(0
,y
).
Now consider an
x
−
y
stadium
S
with major path
P
.Let
f
be a contractive
map to the sphere and
A
be the area of points within distance
y/
2of
f
(
P
)on
the sphere. From Proposition
1
, it suces to bound
A
. Extend
S
by some length
dx
(see Fig.
3
). For suciently small
dx
, the extension of
f
(
P
) runs along a
geodesic. Call the added area
dA
. In Fig.
3
, this is the dotted region.
Now let
ʸ
=
dx/R
. This is the central angle corresponding to a geodesic of
length
dx
on the sphere. Extending
P
by
dx
will affect the latitudes within
y/
2
of our geodesic. Each latitude can be extended by at most an angle of
ʸ
.Let
r
a
be the radius of a circle of latitude at a spherical distance
a
from the equator.
It is well-known that
r
a
=
R
cos
R
. This yields:
×
ʸr
a
da
=
y/
2
y/
2
dx
R
R
cos
a
R
da
=2
R
sin
y
2
R
dx
dA
≤
−y/
2
−y/
2
so
dA/dx
2
R
sin
y/
(2
R
). For a stadium of length
x
, the area on the sphere is
bounded by
A
(
x, y
) as desired.
≤
A(1
/x
−
x
)
×
x
stadium can be inscribed within any
x
×
1
/x
paper rectangle.
By Proposition
2
, this stadium only occupies
A
(1
/x
−
x, x
) area on the sphere.
The remaining paper only has an area of
x
2
−
ˀx
2
/
4.
Upper Bound 3.
x
×
1
/x
paper can wrap an
R
-sphere only if
4
ˀR
2
x
2
ˀx
2
/
4+
A
(1
/x
≤
−
−
x, x
)
.
3.2
n
Circumscribed Circles on Spheres
Cutting the paper or adding more material can only increase the ability of the
paper to wrap a sphere. With this as inspiration, we transform the paper into
n
congruent disks, and then relate upper bounds on spherical cap coverings back
to rectangular sphere wrappings.
Proposition 3.
If an
R
-sphere can be wrapped by an
x
×
1
/x
paper, then it can
also be wrapped by
n
congruent disks of diameter
x
2
+(
nx
)
−
2
.
Proof.
Consider a arbitrary wrapping from an
x
×
1
/x
paper to an
R
-sphere.
Partition the flat paper into
n
small
x
×
1
/
(
nx
) r
ectangles. C
ircumscribe each
1
/
(
nx
) rectangle to get
n
disks of diameter
x
2
+(
nx
)
−
2
. These disks can
contractively map into the original paper.
x
×
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