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Stereographic projections are created by projecting from a point at one
end of the diameter in a circle to a projection plane which sit tangential
to the other end of the diameter. This situation is clearly depicted in
Figure A where the light source comes from point A with white rays radiating
out of it. Geometrically, the light source sits on the left side of the
diameter AB. In terms of geography, the light source sits on the equator
of the earth opposite to the tangential point that the projection plane
CI makes with the equator. Such characteristics make the projections on
this plane a equatorial projection.
When viewed in this perspective, parallels are seen as the horizontal
arcs in Figure D, which is equivalent to the red arc CD in Figure B. The
derivation of such arcs is as follows:
In general, the projection of latitudes in the stereographic equatorial
projection is created by rotating a line of length CE on point E. So
long as this line touches the white circle (which represents the earth),
a path of points is drawn at the end of CE to create a projected parallel
on the plane. With EG representing the polar axis and AB representing
the equator, one knows that the two lines intersect perpendicularly.
As such, <CEO has the relationship of equivalence with the angular
distance of the latitude CD or <AOC. This is because <AOC+<COE=90°
therefore <COE is the co-latitude of <AOC. Since CO is the radius
of the circle and touches the tangential plane CE, <ECO is a right
angle. And if all the angles in a triangle add up to 180°, then
the following situation would occur in triangle CEO, <OCE + <CEO
+ <EOC = 180°. Because <ECO = 90°, this implies that <CEO+<EOC=90°.
With <AOC+<COE=90°, this means that <CEO = <AOC. Therefore,
the following relationship can be obtained:
- sin(<CEO) = CO/EO
- CO = circle's radius
- <CEO = the angular distance of the latitude away from the equator
- EO = CO/sin (<CEO)
- EO = circle's radius / sin (angular distance of latitude).
Therefore the point that a line is rotated from to draw the parallels
is circle's radius * cosec (angular distance of latitude) units away
from the centre of the projected circle and always on the polar axis.
For parallels in the southern hemisphere of the earth, the point of
rotation lies on the southern polar axis as for parallels in the norther
hemisphere of the earth, the point of roation lies on the northern
polar axis.
- tan (<CEO) = CO/CE
- CE = CO/tan (<CEO)
- CE = circle's radius / tan (angular distance of latitude)
In conclusion, the length of the line being rotated is equal to the
circle's radius * cot (angular distance of latitude).
The meridians are constructed by a similar manner. The meridians are
the vertical arcs depicted in Figure D, which is equivalent to the blue
arc CD in Figure C. The derivation of such arcs is as follows:
In general, the projection of longitudes in the stereographic equatorial
projection is created by rotating a line of length CE on point E. So
long as this line touches the white circle (which represents the earth),
a path of points is drawn at the end of CE to create a projected meridian
on the plane. With CD representing the polar axis and AB representing
the equator, one knows that the two lines intersect perpendicularly.
In addition, a line CF is drawn such that it is tangential to the blue
arc at point C, therefore <FCE = 90°. Using the interior angle
sum theorem, triangle FOC has <CFO, <FOC, and <OCF adding up
to 180°. With <FOC=90°, this implies that <CFO + <OCF
= 90°. The three angles <FCE, <CEF and <EFC also all add
up to 180° in triangle FCE. Similar to triangle FOC, triangle FCE
has one 90° angle (<FCE), therefore, <CFE + <FEC = 90°.
The following conclusion can be deduced:
- <CFE + <FEC = 90°
- <CFO + <OCF = 90°
- <CFE = <CFO
- <FEC = <FCO
This implies that <FEC is equal to the angular distance of meridian
away from the central meridian. This is because any ray that is tangent
to the meridian at the poles have the same angular distance away from
the central meridian. As such the distance CE and EO can be derived
at as follows
- tan (<FEC) = CO/EO
- CO = radius of circle
- EO = circle's radius / tan (angular distance of meridian)
- EO = circle's radius * cot (angular distance of meridian)
- sin (<FEC) = CO/CE
- CE = circle's radius / sin (angular distance of meridian)
- CE = circle's radius * cosec (angular distance of meridian)
This means that the point that a line is rotated from to draw the
meridians is equivalent to the circle's radius * cosec (angular distance
of longitude) units away from the centre of the projected circle and
always on the equator. For meridians in the eastern hemisphere of
the earth, the point of rotation lies on the eastern equator as for
meridians in the western hemisphere of the earth, the point of roation
lies on the western equator. The length of such a line is equal to
circle's radius * cot (angular distance of meridian).
Figure D gives an illustration of a stereographic equatorial projection.
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