Civil Engineering Reference
In-Depth Information
4.2.3.1 Gaussian curvature and mean curvature
The Gaussian curvature K at a point on a surface is the product of the principal curva-
tures, and its value depends only on the geometrical shape of the surface but not on the
way it is embedded in space. On the other hand, the mean curvature H is the arithmetic
mean of the two principal curvatures. Like the principal curvatures, the Gaussian curva-
ture and the mean curvature are intrinsic properties of a curved surface at a point, and
the relationships between principal curvatures, Gaussian curvature and mean curvature
are given by
κκ
+
K
=
κκ
and
H
=
1
2
12
2
Based on these relationships, the principal curvatures can also be determined as follows:
2
LN
−
−
M
andH
GL
−
2
FM
+
EN
2
K
=
=
κκ
=± −
HHK
12
2
2
EG
F
2(
EG
−
F
)
Classification of surface by means of Gaussian curvature K (Smith and Farouki 2001):
K > 0: elliptic, K < 0: hyperbolic, K = 0: parabolic,
κ
1
= κ
2
: umbilic point
Most surfaces consist of regions of positive Gaussian curvature of elliptical points and
regions of negative Gaussian curvature of hyperbolic points separated by curves of points
with zero Gaussian curvature of parabolic lines.
4.2.4 Metric and principal curvatures
Let V be a vector on the tangent plane T
p
making an angle θ with
v
1
. V can be written as
VV
(
ˆ
cos
ˆ
θ
. The curvature along vector V with angle θ from
v
1
is given by
=
v
θ
+
v
sin)
1
2
κ = κ
1
cos
2
θ + κ
2
sin
2
θ
This result also suggests that the surface curvature is a second-order tensor following the
usual rule of tensor transformation with respect to angle θ. In terms of tensor notation, the
surface curvature tensor
κ
is written as
κκ=κ
1
ˆˆ
vv
+
κ
vv
ˆˆ
11
222
Curvature along vector V is given by
ˆˆ
ˆ
(
ˆ
ˆ
)
ˆ
ˆ
(
ˆ
ˆ
)
ˆ
ˆ
ˆ
)
2
(
ˆ
ˆ
)
2
(
κ⋅ ⋅= ⋅
vv
)
κ
vv vv
⋅ +
κ
vv vv
⋅ ⋅=
κ
(
vv
⋅ + ⋅
κ
vv
11 1
2
22
1
1
22
2
2
=
κ
cos
θ κ
+
sin
θ
1
2
where
v
=
v
v
is the unit vector along direction vector V.
