Geography Reference
In-Depth Information
2
R
. In contrast, Fig.
4.2
presents the
helicoid
and the
one-sheeted
hyperboloid of revolution
as alternative examples of a ruled surface.
the circular cone of the sphere
S
2
R
cos
Φ
0
2
Example 4.1 (Circular cone
C
of the sphere
S
R
).
2
Let us construct a circular cone of the sphere
R
as a ruled surface. First, we choose the parallel
circle, also called
small circle
, of the parameterized sphere as the reference curve or directrix.
Second, we attach locally to any point of the directrix a vector field which is generated by a
ruler moving along the reference curve. Consult Box
4.1
and Fig.
4.1
for a more detailed analysis.
In terms of spherical coordinates
S
, we parameterize the “position vector”
X
(
Λ, Φ, R
)
with respect to an orthonormal frame of reference
{
E
1
,
E
2
,
E
3
|O}
, spanning a three-dimensional
Euclidean space
E
{
Λ, Φ, R
}
2
R
of radius
R
.
{C
Λ
,C
Φ
|Λ, Φ}
is the local frame of reference i.e. Cartan's moving frame (“repere mobile”),
attached to a point
{Λ, Φ}∈
S
3
, attached to the origin
O
, which is the center of the sphere
S
2
R
. As the reference curve, we have chosen the parallel circle
Φ
0
=
constant, namely the directrix
x
(
U
), where
U
=
Λ
is the parameter of the reference curve. The
generator or the ruler of the surface is the vector field
Y
(
U
):=
C
Φ
(
U
), the unit vector which is
normal to
C
Λ
(
Λ, Φ
0
), directed towards North. The linear manifold
V
Y
(
U
), also called the
bundle
of straight lines
, is forming the circular cone
of radius
R
cos
Φ
0
as soon as the ruler moves
around the parallel circle. Finally, we have gained the parameterized ruled surface
X
(
U, V
). Its
typical matrix of the metric, G, has been computed.
C
2
R
cos
Φ
0
End of Example.
Additionally, let us more precisely define a ruled surface in Definition
4.1
, which follows after
Box
4.1
summarizing the vector definitions of Example
4.1
.
2
R
: ruled surface).
Spherical coordinates and Cartan's frame of reference:
2
R
cos
Φ
0
Box 4.1 (Circular cone
C
of the sphere
S
⎡
⎤
R
cos
Φ
cos
Λ
R
cos
Φ
sin
Λ
R
sin
Φ
⎣
⎦
,
X
(
Λ, Φ
)=[
E
1
,
E
2
,
E
3
]
(4.1)
⎡
⎤
−
sin
Λ
+cos
Λ
0
D
Λ
X
(
Λ, Φ
)
⎣
⎦
,
C
Λ
:=
=[
E
1
,
E
2
,
E
3
]
(4.2)
D
Λ
X
(
Λ, Φ
)
⎡
⎤
−
sin
Φ
cos
Λ
−
sin
Φ
sin
Λ
cos
Φ
D
Φ
X
(
Λ, Φ
)
⎣
⎦
.
C
Φ
:=
=[
E
1
,
E
2
,
E
3
]
D
Φ
X
(
Λ, Φ
)
Directrix
x
(
U
) (parallel circle, small circle, curve of constant latitude
Φ
0
,U
=
Λ
):
⎡
⎤
R
cos
Φ
0
cos
U
R
cos
Φ
0
sin
U
R
sin
Φ
0
⎣
⎦
.
x
(
U
):=[
E
1
,
E
2
,
E
3
]
(4.3)
Generator or ruler of the surface (
U
=
Λ
):
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