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In the polar case, a plane of projection touches the globe at one of
the pole. In Figure A, a northern gnomonic polar projection is made. Thus,
the map of the earth created by projecting points on the surface of the
earth onto this plane from the centre of the globe.
The globe can be represented by a circle of 1 unit is created by using
a point (x, y) drawn a constant distance of 1 unit away from the point
B, the light source as seen in Figure B.
Focus on a quadrant of the circle bounded by a horizontal line segment
BC and a vertical line segment AB such that both segments are 90°
away from each other. As a representation of the earth, this would make
segment BC the radius of the equator and segment AC the semi-axis of the
polar axis touching the North Pole.
The plane of projection is represented by a straight line drawn from
point A to point G such that the segment AG is parallel to BC (or the
equator). As a result, the plane of projection is tangential to the globe
at the North Pole.
Projections of latitudes are drawn by extending a straight line from
the light source to the projection plane at the same angular distance
from the equator possessed by the latitude. These points are called projection
points. For example, point G, F, E, D, and A in Figure B are all projection
points. The same procedure is done for the supplementary quadrant to produce
the picture in Figure C.
With the light source at the centre and rays perpendicular to the tangential
point on the North Pole, that makes the projection of latitudes concentric
circles with their centres at the North Pole. As a result, the radii of
the circular latitudes are equivalent to the segment created from the
tangential point to the corresponding projection points (refer to Figure
D). The radii of such concentric circles used to draw the projected parallels
is equivalent to the radius of the circle * tan (90° - angular distance
of latitude). The proof is as follows:
Refering to Figure C, for an arbitrary latitude MN, its projected radius
is equal to FA because AB (which represents the polar axis) is tangential
to the projection plane GL therefore, they intersect at 90°. Based
on circle geometry, AB will bisect FK since it is perpendicular to GL
and goes through the centre of the circle. If <FBC represents the
angular distance of latitude MN away from the equator CH, then <ABF=90°
- <FBC because <ABF+ <FBC = 90°. In other words, <ABF
is the co-latitude of <FBC. Therefore, the tangent relationship tan
(<ABF)=FA/AB. With AB being the radius of the circle with FA being
the radius of the latitude MN and <ABF being the co-latitude, the
radius of the latitude MN is equal to the radius of the circle * tan
(co-latitude).
The meridian, on the other hand, would appear as straight lines on the
plane of projection. In fact, this is the case for all great circles.
This is because the meridians' plane all passes through the poles and
the centre of the globe, which is also the centre of projection. Therefore,
when seen from the centre of the globe would appear as straight lines.
Another way of visualizing this is using the relationship between meridians
and parallels. Based on the fact that the two lines intersect at 90°,
this implies that the meridians are perpendicular to tangential lines
on the circular latitudes. From circle geometry, one knows that a line
that is perpendicular to a tangent must go through the centre of the circle,
therefore making the meridians all equal to the radius of the concentric
circles in the gnomonic projection.
However, this is not the case for the equator. The equator cannot be
represented on the projection because the equator is parallel to the plane.
Thus there cannot exists any projection points and it becomes impossible
to get a projection of the entire hemisphere.
In conclusion, a map with gnomonic projection in the polar case will
have concentric circles drawn from common centres on the North Pole. The
centre point will have latitude of 90° as the radius of the circles
increase with a decrease in latitude. And the radius of the circles becomes
the meridians. Therefore, the separation of the meridians by a constant
angular distance implies a division of the latitudes into equal arcs and
thus producing the map as seen in Figure E.
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