Civil Engineering Reference
In-Depth Information
=
∂
∂
p
=
∂
∂
φ
=
∂
∂
p
=
∂
∂
φ
u
=
φ
and
v
=
φ
u
v
uu
vv
The unit normal to the surface S at
p
is given by
uv
uv
×
×
n
=
Hence, (
u
,
v
,
n
) represents a local base at
p
, as shown in Figure 4.8.
As
u
and
v
are basis vectors of T
p
, every vector V on T
p
can be written as
V
=+ ∈
αβ αβ
uv
with
,
and the first fundamental form Φ
1
is given by
Φ
1
= V ⋅ V = Eα
2
+ 2Fαβ + Gβ
2
where
E =
u
⋅
u
, F =
u
⋅
v
, G =
v
⋅
v
α
β
EF
FG
then VV
T
Let
Λ
=
andM
=
⋅ =
Λ
M
Λ
.
Matrix M, which measures the length of vector V, is symmetric and positive-definite
called the tangent plane metric at
p
. By means of metric M, the length of a line segment
created on the parametric space can be evaluated. As shown in Figure 4.8, a curve Γ on S is
defined by
γ: (u(t), v(t)) → S
t ∈ [a,b]
v
S
n
Γ
u
p
φ
v
γ
Ω
I: t [a,b]
C
ξ
u
Figure 4.8
Curved surface S produced by parametric mapping
ϕ
.
