Geography Reference
In-Depth Information
23
Map Projections of Alternative Structures: Torus,
Hyperboloid, Paraboloid, Onion Shape and Others
Up to now, we treated various mappings of the
ellipsoid
and the
sphere,
for instance of type
conformal, equidistant, or equal areal
or
perspective and geodetic. Here we focus on alternative
geometric structures like
(i) torus
2
with boundary.
Example
:
pneu
(ii) hyperboloid
T
2
.
Example
: cooling tower
H
2
.
Example
: parabolic mirror
(iv) onion shape figure
(iii) paraboloid
P
2
.
Example
:churchtower
Z
and many others.
Project surveying
often extends to the
design of space-time structures,
their
materialization
and
realization
by means of
measurement techniques
up to the
control of time-varying structures.
Spatial structures like
light tent roofs, bridges, dams, pneus, carosseries, cooling towers, church
towers,
and
parabolic mirrors
are designed by computers and usually represented in
orthogonal
or centralized projections.
“Better” projective representations like a
conformal mapping
,oran
equiareal mapping, or an equidistant mapping as the classical geodetic projections are barely
applied, probably due to missing knowledge. Here we aim by four work-out examples to document
elegance and power of these geodetic mappings.
Full of ideas from
Mapping Projections
is our next topic:
(v) clothoid
1
.
Example
: height-speed-railway
C
1
, minimum distance mapping.
Example
: height-speed-railway
We design a map projection for the
clothoid
(vi) clothoid
C
1
as well as the minimum distance mapping for this
clothoid. Applications are the designs of
High-Speed-Railway Tracks.
We begin with the definition
of the clothoid: A planar curve is called a
clothoid
C
1
if its
curvature
is positive
proportional to
its arc length s,
in particular
κ
(
s
)=
s/a
2
. The curvature radius
r
(
s
):=1
/κ
(
s
)aswellasitsarc
length
s
are positive constants subject to
rs
=
a
2
.
At first
we derive the differential equation which generates the clothoid
C
1
.The
initial value
problem
of such a differential equation is solved in terms of the
Fresnel integrals
by a power
series expansion.
Secondly
we succeed to solve the
Fresnel
integrals by a power series expansion
of the azimuth functions
sinα
(
s
)
, cosα
(
s
) relative to the initial curvature
x
0
of the
clothoid.
In
this way, the coordinate functions
x − x
0
=
f
(
α
0
,κ
0
,s− s
0
)
and y − y
0
=
g
(
α
0
,κ
0
,s− s
0
)are
derived, namely for (
x
,
y
) as conformal coordinates of
Gauß-Kruger
or
UTM
type.
Thirdly,
we
take advantage of
univariate series inversion
in order to derive the
clothoid functions y
C
−
y
0
=
h
(
x
x
0
,α
0
,κ
0
). As special cases the straight line and the circle are included.
Fourthly,
we present
case studies for the local representation of the
clothoid
for various degrees of approximations.
Finally, we introduce the
minimum distance mapping
of a point
close to the clothoid.
−
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