Geoscience Reference
In-Depth Information
How can F
1
and F
2
be determined? Projection problems have different solutions
based on different demands. We turn these demands into certain mathematical
conditions and allow them to be expressed in the projection formula to get the
specific mathematical relationship. Next we will discuss the requirements of pro-
jections for controlling mapping.
6.1.3 Conformal Projection and Conformality
Projection means the planar representation of quantities on the ellipsoid, which will
inevitably create distortions. A projection distortion refers to the alteration of angle,
distance, or area after projection. Cones and cylinders are developable surfaces, on
which quantities will not be distorted if represented on a plane. An ellipsoid and a
sphere are undevelopable surfaces that crumple or fold when unrolled and flattened
by force. Projection distortions are no doubt unfavorable, but they can be allocated
and controlled reasonably by determining F
1
and F
2
in (
6.1
).
There are three kinds of projection distortions, i.e., angle distortion, distance
distortion, and area distortion. They can be controlled according to specific needs.
A distortion of some kind can be zero, such as when the projection is equiangular,
equivalent (equal-area), or equidistant; distortions of all kinds can also coexist but
have to be kept within a proper degree. It is obviously impossible to eliminate all
distortions at the same time since the ellipsoid is an undevelopable surface and a
projection will invariably introduce distortions.
For large-scale mapping, if figures on a map can be maintained conformable to
the original on the ellipsoid within a certain area, i.e., angles remain undistorted
after projection, then on such maps terrains and land features will be completely
conformable to the real features, which will bring great convenience in use. A
projection in which all angles at any point are preserved is known as the equiangular
or conformal projection.
As shown in Fig.
6.1
, a small midpoint polygon OABCDE on the ellipsoid is
conformally projected onto the plane as O
0
A
0
B
0
C
0
D
0
E
0
. Each line segment on the
ellipsoid in Fig.
6.1
is a differential line segment (called an arc element), which is
considered a straight line and remains the same after being projected onto the plane.
According to the definition of conformal projection, the internal angles of every
triangle are not altered after projection. Triangles are correctly represented after
conformal projection, so their corresponding sides are proportional, thus:
O
0
A
0
OA
ﾼ
O
0
B
0
OB
ﾼ
O
0
C
0
OC
ﾼ
O
0
D
0
OD
ﾼ
O
0
E
0
OE
ﾼ
m
ﾼ
constant
where m is the scale factor. Consequently, for certain points in conformal projec-
tion, the scale factor m is independent of direction. However, this property of
conformal projection is conditional and is only valid within a small area. It is
impossible that a map projection can be achieved in which large areas are rendered
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