Civil Engineering Reference
In-Depth Information
The length of curve Γ is given by
b
b
∫
∫
L
()
Γ=
γ
()
t dt
=
γ
()
t
⋅
γ
()
t dt
a
a
From the interval I = [a,b], a curve C can be created by the mapping ξ: I → Ω, such that
γ = ϕ ∘ ξ ⇒ γ′ ⋅ γ′ = γ′
T
γ′ = (ϕ′ ∘ ξ′)
T
(ϕ′ ∘ ξ′)
T
= ξ′
T
ϕ′
T
ϕ′ξ′
However,
∂
∂
∂
∂
φ
∂
∂
∂
∂
p
u
u
u
v
u
v
uuuv
uvvv
= M
⋅
⋅
T
φ
=
=
=
φφ
=
=
uv
φ
p
⋅
⋅
v
v
b
=
∫
T
T
T
γγ ξξ
=
MorL
(Γ
ξξ
Mdt
a
In case C is a line segment from point A to point B, we have
1
∫
T
ξ
()
tAtABt
=+ ∈
,
[ ,],
01
ξ
=
AB andL
( )
Γ
=
AB MA tABABdt
(
+
)
0
Geometric interpretation of M
: Let A be a point on Ω and M(ϕ(A)) be the metric at point
A. For arbitrary real value ε, the locus of point B on Ω such that
T
2
AB MAAB
(( ))
φ
=
ε
is an ellipse. In other words, an ellipse on Ω will map to a circle on T
p
of S and vice versa. In
general, an ellipse on Ω will map to an ellipse of different size and orientation on the tangent
plane of S, and by controlling the shape and size of the elements on Ω, meshes of various
characteristics on S can be created.
4.2.3 Principal curvatures
Principal curvatures (the maximum and minimum curvatures at a point) are of fundamen-
tal importance to the curved surface under consideration, which can also be used to define
natural characteristic lines on the surface. Let V = α
u
+ β
v
be a vector on the tangent plane
T
p
of S at
p
and
n
be the unit normal at
p
. Set
2
2
2
=⋅
∂
∂
φ
=⋅
∂
∂∂
φ
=⋅
∂
∂
φ
L
n
,
M
n
,
N
n
u
2
uv
v
2
The second fundamental form, which is the projection of
n
onto the second derivatives of
ϕ, is given by
Φ
2
= Lα
2
+ 2Mαβ + Nβ
2
