Civil Engineering Reference
In-Depth Information
Given a point
p
(u,v) on S, a close neighbour, say,
q
on S at (Δu, Δv) from
p
, can be
expressed as a Taylor's expansion about
p
:
Δ
q
=+ +
p
() ()
Δ
u
u
v
v
+
Hence, given a point
p
on S and the associated tangent vectors
u
and
v
, a irst-o
r
der
approximation of a point
q
=
q
(u + Δu, v + Δv) can be computed. However, the point
q
, in
general, will not lie on the surface S.
Now consider a point
r
on S whose spatial co-ordinates are known but not its parametric
co-ordinates. If a point
p
=
p
(u,v) on S sufficiently close to
r
is known, then a first-order
approximation of the increment of the parametric co-ordinates from
p
to
r
(Δu, Δv) can be
determined by solving the equation
∆
p
=
r
−
p
= (∆u)
u
+ (∆v)
v
However, a solution may not exist as there are more equations than the number of
unknowns (three equations with two unknowns). In fact, it is not possible to find Δu and
Δv because Δ
p
is not lying on the tangent plane at point
p
. A solution for this is to use the
projection of Δ
p
onto the tangent plane spanned by vectors
u
and
v
, i.e.
() ()
(4.1)
Δp
=
Projectionof
ΔΔ
p
=
u uv
+
Δ
v
Better approximation can be obtained by iteration with updated co-ordinates of point
p
,
i.e.
p
↦
p
(u + ∆u, v + ∆v)
Repeatedly solving Equation 4.1 with updated point
p
until Δ
p
is normal to the tangent
plane at
p
, or
Δp
is smaller than a specified tolerance
ε
, i.e.
Δp <ε
.
For mesh generation based on space-to-surface projection, we have to solve a slightly dif-
ferent problem: determine the point C on S, or more strictly speaking, find its parametric
co-ordinates (u,v) such that the size and the shape of the triangle formed with the base seg-
ment is preserved. This leads to the condition that the projected point C on S should lie also
on the plane normal to the segment AB at M and at a distance h from M.
The initial guess C
o
on S can be obtained from the closest point projection (Equation 4.1)
of a spatial point
r
on the tangent plane at M:
r
= M + h
e
2
An improvement of Ci
i
based on a first-order approximation is given by
C
i+1
=
p
(u
i+1
, v
i+1
) =
p
(u
i
+ ∆u, v
i
+ ∆v) ≈ C
i
+ (∆u)
u
+ (∆v)
v
Δu and Δv are to be solved based on the following conditions:
i.
e
1
⋅MC
i+1
= 0
ii. ‖MCi+1‖
i+1
‖ = h
The first condition ensures that Ci+1
i+1
is on the plane normal to segment AB, and the sec-
ond condition ensures that Ci+1
i+1
is of a distance h from M. Iteration is needed as there is a
