Civil Engineering Reference
In-Depth Information
and
be the two principal directions and κ
1
and κ
2
be the respec-
tive principal curvatures. If
q
is a point on the ellipse on the tangent plane T
p
with axes κ
1
and κ
2
, we have ‖
pq
‖ ≥ κ, as shown in Figure 4.9a.
Let
p
be a point on S,
ˆ
v
v
ˆ
1
2
pq
2
2
2
22
2
2
22
=
κ
cos
θ κ
+
sin
θ κκ θκ θ
and
=
( os
+
sin)
1
2
1
2
pq
2
2
2
2
2
22
2
2
2
−= −
κκ θ
cos( cos)
1
θ
+
κ
sin( sin)
θ θκκ
1
−
−
2
cos in
θθ
1
2
1
2
2
2
2
22
2
2
2
22
2
=
κ
cos in
θ θκ θθκκ
+
sin os
−
2
c
os
θθκκ θθ
sin(
=−
)sin
cos
≥
0
1
2
12
1
2
This result shows that the curvature at a point
p
in any direction is bounded by the ellipse
centred at
p
with κ
1
and κ
2
as axes. Similarly, as shown in Figure 4.9b, for ellipse drawn with
axes in terms of radius of curvature ρ = 1/κ, we have ‖
pq
‖ ≤ ρ.
As the locus of a metric is an ellipse, a metric M can be defined on the tangent plane T
p
at
p
such that the length in any direction from
p
will not exceed the radius of curvature in that
direction. Such a metric is given by
1
0
κ
2
0
v
v
ˆ
ˆ
ρ
2
v
v
ˆ
ˆ
1
1
1
1
M =
vv
ˆˆ
=
v
ˆˆ
v
12
1
2
2
1
0
κ
2
0
2
2
2
ρ
2
Let AB be a straight line segment on the tangent plane T
p
. The length of edge AB with respect
to M(
p
) is given by
vv
⋅
ˆ
2
κ
0
1
1
L
=
BMAB
T
=
AB
vv
T
M
=
B
vvvv
ˆ
⋅
ˆ
⋅
1
2
2
0
κ
vv
⋅
ˆ
2
2
cos
θ
2
AB
θθ
κ
0
1
2
2
2
2
=
=
AB
κ
cos
θ κ
+
sin
θ
≥
κ
AB
cos in
1
2
2
0
κ
sin
θ
2
BL
AB
AB
AB
L
≥
κ
≥
where
v
=
ρ
(a)
(b)
v
2
·
v
2
·
q
q
ρ
2
κ
2
ρ
κ
ρ
1
κ
1
θ
θ
v
1
·
v
1
·
p
p
Figure 4.9
Curvature at a point is bounded by an ellipse: (a) ellipse of surface curvature; (b) ellipse of radius
of curvature.
