Nonlinear Projections - Transformations and Projections in Computer Graphics - page 201

the distance between consecutive circles shrinks for high latitudes. Thus, equatorial
regions are shown in more detail, while polar regions are shrunk in this type of pro-
jection. The figure demonstrates that this projection is limited to less than half the
sphere; it cannot include the equator. On the other hand, any great circle is displayed
in this projection as a straight segment. (A great circle is one whose center is at the
center of the sphere.) Great circles are important for navigation because a great circle
arc is the shortest distance between two points on the surface of a sphere. This is why
the gnomonic projection is commonly used in air navigation. This projection is neither
conformal nor equal-area.
Part (b) of the figure shows a stereographic projection. This is the case where
the center of projection is at the pole opposite the plane of projection. The circles of
latitude are uniformly spaced, which results in uniform distortions throughout.
When the center of projection is at infinity on the side of the sphere opposite that
of the projection plane, the lines of projections are parallel and the projection is referred
to as orthographic. Part (c) of the figure illustrates this type, and it is obvious that
the pole that's tangent to the plane of projection is shown in much detail and little
distortion, thereby making this projection ideal for mapping polar regions. Figure 4.50
illustrates the coordinate transformation for the orthographic projection. The polar
coordinates of the projected point are θ = longitude and r = R cos (latitude).

Projection

y North pole

plane

Point on

globe

Projected
point

R

r

φ

θ

North

Equator pole

r

x

Projected
point

Figure 4.50: Polar Coordinates in Orthographic Projection.

Cylindrical projections . Figure 4.51 shows the three main ways to project a sphere
on a cylinder tangent to it. Part (a) of the figure illustrates a perspective projection
from the center of the sphere. In part (b), points on the sphere are projected to the
cylinder in parallel, while the projection principle in part (c) is to project equal arc
lengths on the sphere to equal vertical segments on the cylinder.
In all three types of cylindrical projections, unrolling the cylinder results in equally
spaced longitudes on the map. However, in a perspective cylindrical projection, the
spaces between consecutive latitudes on the map increase as we move toward the pole
and approach infinity at the pole. Thus, it is impractical to extend this projection
beyond about latitude 80 ◦ . The simple projection depicted in Figure 4.51a projects a
pointatlatitude φ and longitude θ on the globe to Cartesian coordinates x = θ θ 0

(where θ 0 is the longitude at the center of the map) and y = R tan φ on the map. Such
a projection stretches the vertical dimensions of any regions between about latitude 30 ◦
and the poles, resulting in so much distortion that it is rarely used.

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