Taylor, Bell: Map projection
Astronomical applications of
the quincuncial map projection
1a
1b
2a
2b
1 and 2: Different aspects of (left; 1a and 2a) a stereographic map of a hemisphere of the surface of the Earth and (right; 1b and 2b) its transformation into
a square using Schwarz’s integral with n = 4. (1): Normal aspect with central point l = 0°, φ = 90°. (2): Oblique aspect with central point l = 0°, φ = 45°. The
network of lines of latitude and longitude or the graticule uses a 10° interval.
A
n important part of the work associated
with publications of H M Nautical Almanac Office, in both paper and electronic
form, is the graphical representation of astronomical data, in particular astronomical pheA&G • October 2013 • Vol. 54 D B Taylor and S A Bell make
the case for C S Peirce’s littleused but practical quincuncial
astronomical map projection.
nomena. Both eclipse and occultation shadows
on the Earth’s surface are displayed and maps
of the night sky are also given. To produce these
maps, projections must be chosen of the Earth’s
surface for the shadows and of the celestial
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Taylor, Bell: Map projection
sphere for the night sky. Currently, for the former the orthographic projection is used and for
the latter the azimuthal equidistant projection.
For eclipses, in addition to maps of the shadows, it is helpful to give maps displaying umbral
limit lines or central limit lines covering a period
of years. These lines can be close to polar areas,
in mid-latitudes and, in some cases, cross equatorial regions. A map of the night sky can also
be used to show a comet track over a period of
time, such as the path of a topical bright comet,
for instance. Some comet tracks, such as C/2011
L4 PanSTARRS or C/2012 F6 Lemmon during
their 2013 apparitions, pass close to both the
south and north celestial poles of the sky. We
suggest that a suitable map projection for these
examples is the quincuncial map projection,
devised by Charles Sanders Peirce (1879).
The projection
Peirce called his map projection “quincuncial”
because it is made up of five parts. It has found
few uses, although one is a US Coast and Geodetic Survey world map of air routes by Stanley
(1946). The limited use that has been made of
this projection is due in part to the relative complexity of the formulae and also to the effort
required to produce the map before the advent
of electronic computers. Now the quincuncial
projection can be produced in a matter of a few
seconds of computing time.
This projection originated from the work
of H A Schwarz (1869), who showed that the
interior of the unit circle can be conformally
represented by the interior of a regular polygon
of n sides by means of an integral. Peirce was
the first to use this result in cartography for the
case n = 4. For this case, the transformation can
be expressed in terms of Jacobian elliptic functions for modulus 1/√2 = sin 45°. He mapped a
hemisphere conformally within a square, and
Guyou (1887) produced the transverse aspect
of the same projection. The word aspect is used
in cartography to indicate the appearance of the
network of parallels and meridians of the map
projection. An aspect of a map projection can
be subdivided using the terminology normal,
transverse or oblique aspect. A full description
and definition of these terms can be found in a
textbook on map projections, such as Maling
(1973). The network of parallels and meridians
of the map projection is known as the graticule.
A conformal map projection has no angular
deformation: that is, angles are preserved; an
important property for a map. Schwarz’s work
was extensively used by Adams (1925) and Lee
(1976) in their work on conformal map projections based on elliptic functions. A clear derivation of the formulae required to produce Peirce’s
quincuncial map projection is given by Lee.
Following Peirce’s paper, Pierpont (1896)
described an error he found in Peirce’s work
and also gave a fuller mathematical account of
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3: Quincuncial maps of the surface of the Earth with umbral limit lines for eclipses in the period 2012
to 2020; those in red refer to a total eclipse and those in black to an annular eclipse. The network of
lines of latitude and longitude or the graticule use a 10° interval.
the map projection. From these papers, the map
projection can be computed. However, the map
projection formulae, determining the distortion
of the map and computation of different aspects
of it, is best described in Lee’s work and this has
formed the basis of that part of Taylor (2013)
devoted to the quincuncial map.
We can represent the points in the circle with
radius one unit by the conformal stereographic
map projection. So, for example, a hemisphere
of the Earth can be placed conformally in a
square. In figures 1 and 2, by taking different
central points of the hemisphere we can draw
different aspects of the stereographic projection
and of the projection in a square. For each of the
central points the longitude l = 0° was taken and
for the latitude φ the values 90° (figure 1) and
45° (figure 2). The pole-centred figure 1 gives the
normal aspect, while figure 2 shows an example
of the oblique aspect. The transverse aspect is
given by φ = 0°. Here the central meridian connects in the projection in a square through the
mid-points on two opposite sides. The projections in the squares are obtained by first rotating the stereographic map by 45°, applying the
Schwarz mapping and then rotating the transformed map by –45°. Two transverse projections
alongside each other, with each representing a
different hemisphere, is the projection of Guyou.
If no rotations are made, the transformed projection in the square of the stereographic map
will have the central meridian connecting opposite vertices. Starting with the transverse aspect
of the stereographic map of a hemisphere, we
obtain the projection in a square as given by
Adams (1925). The quincuncial projection is
obtained starting with the normal aspects of the
stereographic map projection. The transformed
projections are used to produce Peirce’s quincuncial map projection. Centring on each pole,
we can form such a map for each in turn. Using
each as the central square, we can form a larger
square from attaching the other in four separate
pieces thus representing the sphere. The central
square plus the four additional pieces make up
the five pieces of the map and hence the name
quincuncial given to it by Peirce. The two maps
can be connected by a common quadrant.
Astronomical applications
In figure 3, quincuncial maps of the world are
given with the umbral limit lines for eclipses
from 2012 to 2020 shown. Those in red refer
to a total eclipse and those in black to an annular eclipse. These limit lines correspond naturally near to a pole and hence the use of the two
quincuncial maps. If only one of them were used
some of these lines would be broken. The proA&G • October 2013 • Vol. 54
Taylor, Bell: Map projection
centred quincuncial map. After passing near to
the south pole its declination steadily increases
and eventually moves to an area of sky in the
region of the north celestial pole, which is best
displayed on the north-pole-centred quincuncial map. The comet track has been positioned
away from the area most distorted in the map.
This map shows the overall path the comet takes
during 2013; maps of the night sky covering a
smaller area of sky would be required for observing at a given location and time.
Conclusions
4: Quincuncial maps of the night sky with the Comet C/2012 F6 Lemmon track covering 27 December
2012 (marked by a green dot) to 1 January 2014 (marked by a blue dot). Points in the track at five-day
intervals are marked in red. The network of lines of Dec and RA or the graticule uses a 10° interval.
jection used here offers some advantages over
cylindrical projections, such as those used by
Espenak in his World Atlas of Solar Eclipse
Maps on http://eclipse.gsfc.nasa.gov, handling
both polar and equatorial tracks with comparable levels of realism. The positioning of the lines
of longitude of the graticule are as those used by
Peirce. From a study of the scale of the projection
there are “singular” (non-conformal) points at
each bend in the equator; the projection is everywhere else conformal. Near to these non-conformal points the distortion of the map projection
will be greatest. This is taken into consideration
in setting the origin of longitudes. Note that
with axes Ox and Oy with origin O at the north
pole of the north-pole-centred quincuncial map
then the axis Ox corresponds to 25°W.
A map of the night sky can also be made using
the quincuncial projection. The graticule is now
made up of lines of declination (Dec) and of right
ascension (RA). Figure 4 shows a map of the
night sky using this projection, as an observer
would see looking up at the stars with the naked
eye. The upper map has at its centre the north
celestial pole, and the lower, the south celestial
pole. Lines of equal RA increase about the north
celestial pole in a clockwise direction while
they increase in a counter­clockwise direction
about the south celestial pole. The dashed line
A&G • October 2013 • Vol. 54 is the ecliptic. The magenta circle with a cross
inside indicates the point where the RA = 0° line
crosses the celestial equator. The star positions
were obtained from Urban (2007). This is the
MICA astrometric catalogue for epoch J2000.0,
which contains stars recorded in the Hipparcos
and/or the Tycho-2 catalogues with respect to
the International Celestial Reference System
(ICRS) with a V magnitude less than 9.5. There
are 232 317 entries in this catalogue. The map
drawn here uses all stars in this catalogue less
than or equal to fourth magnitude. The stars
are represented by filled-in circles, the radius of
which depends on the visual magnitude: i.e. the
brighter the star, the larger the circle.
On this map we show the track of Comet
C/2012 F6 Lemmon in 2013. From orbital elements given for this comet in Minor Planet Circular No. 83143, an ephemeris is generated by
numerical integration. In the equations of motion
for the comet, perturbations from all planets are
included, with positions for the planets from the
Jet Propulsion Laboratory DE406 ephemeris. An
astrometric ephemeris is produced with the positions for epoch J2000.0 given in the ICRS. The
equations of motion were integrated using the
Runge–Kutta–Nystrom method, RKN12(10)
from Dormand et al. (1987). The comet track
shown is first plotted on the south-celestial-pole-
Some astronomical applications of the quincuncial map projection have been presented here.
Taylor (2013) gives examples of occultation
shadows drawn on this projection as well as
giving a similar map to figure 3 but now using
central limit lines. A graphical representation of
periodic comets returning to perihelion for any
given year is also discussed, where the comet
osculating elements used are those published in
the Astronomical Almanac. From the stereographic map (in any aspect) we have seen, using
the Schwarz integral, this can be transformed
into a square. Thus, for example, eclipse and
occultation shadows can be represented on a
square map. Such maps can be useful for publications using the printed page and also for
websites in which one could pack several maps
together compactly.
The transformed map into the square, which
is used in forming the quincuncial map, is conformal everywhere except at the corners. In fact,
conformal projections have been used as the
bases for navigation charts and topographical
maps. Peirce’s map projection, used to show the
night sky, can be easily set up so as to allow a
rotation, if required, about the poles. This flexibility enables any area of interest to be located
in the least distorted part of the map. ●
Don B Taylor and Steve A Bell, HM Nautical
Almanac Office, National Hydrographer’s
Division, UK Hydrographic Office.
References
Adams O S 1925 Elliptic Functions Applied to Conformal
World Maps US Coast and Geodetic Survey Special
Publication 112.
Dormand J R et al. 1987 IMA J. Num. Anal. 7 423.
Guyou E 1887 Annales Hydrographiques 2nd ser. 9 16.
Lee L P 1976 Cartographica Monograph 16; supp. 1 to
Canadian Cartographer 13.
Maling D H 1973 Coordinate Systems and Map Projections (George Philip and Son, London), 2nd ed. 1992
(Pergamon, Oxford).
Peirce C S 1879 American Journal of Mathematics 2(4)
394.
Pierpont J 1896 American Journal of Mathematics
18(2) 145.
Schwarz H A 1869 Journal für die reine und Angewandte Mathematik (Crelle’s) 70(2) 105.
Stanley A A 1946 Surveying and Mapping 6(1) 19.
Taylor D B 2013 HMNAO Technical Notes 75 http://
astro.ukho.gov.uk/nao/technotes/naotn75.pdf.
Urban S E 2007 private communication.
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