


                               Maps and Legends


                            7    TECHNICAL DETAILS

              It  has been said that the making of a good map requires
         a  compromise between all sorts of distortions so that in the
         end  none  is too objectionable.  Some maps keep area true at
         the expense of shape while others keep local directions exact
         but alter dimensions.  The projections available in "Maps and
         Legends:  The Cartographer" cover both of these kinds of maps
         and   more.    By   studying   the  differences  between  the
         projections  that you can make with this program you will see
         how these compromises can be handled to produce maps that are
         both effective and pleasing to view.


              In  this  section we present some background information
         on  each  of  the projections available in Maps and Legends.
         For  those  who  are  mathematically  inclined  we  have also
         provided the formulae that describe each projection.


                            Cylindrical Projection

              The   cylindrical   projection  provided  in  "Maps  and
         Legends"  is  the  rectangular even-spaced projection.  It is
         one  of  the simplest projections to express mathematically.
         As  its name implies, this projection results from projecting
         surface  features  outward  onto  a cylinder which is wrapped
         about  the  earth and aligned north and south.  Parallels and
         meridians  form straight horizontal and vertical lines, and a
         given change in latitude or longitude corresponds to the same
         horizontal or vertical distance anywhere on the map.


              The  formulae  which  relate  latitude  and longitude to
         horizontal (x) and vertical (y) position on the map are:


             x = a * (longitude) - b
             y = c * (latitude) - d

              In  these  expressions (and those which follow) a, b, c,
         and   d   are   constants  which  are  determined  from  your
         specification  of  the map coordinates to be displayed on the
         screen.  With the "Cylindrical" projection option of Maps and
         Legends  you specify the region to be mapped in both latitude
         and  longitude.  As a result, you control the relative scales
         in these two directions (a and c).  You can choose any region
         of  the  globe to appear within the screen.  You can also use
         this feature to distort your map by stretching it in latitude
         or  longitude.  Some interesting effects can be produced this
         way.   The  disadvantage  of this method is that you must use
         care  if you want your map to show the same distance scale in
         both   dimensions.     Mercator   Projection:   The  Mercator
         Projection  is  named  for  its  developer,  the  great Dutch
         cartographer  Gerard  Mercator  (1512  -  1594).  In Maps and



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         Legends  this  projection  is  available as one of the "World
         Map" projections.


              The  Mercator  projection  is  closely  related  to  the
         cylindrical  projection,  but  it differs in a very important
         way.   In the Mercator projection a given angular distance in
         latitude  corresponds to a greater distance on the map closer
         to  either  pole.  As a result, areas nearer the poles appear
         larger  than  they  really  are.  The great advantage of this
         projection  is  that  straight  lines  on  the  map show true
         compass  direction.   Thus  to  sail a ship from one point to
         another, one had only to sail to the compass course indicated
         by a straight line on Mercator's map.  The path that resulted
         may  not  be  the shortest path, but it would get you to your
         destination.   This  is  why  Mercator's  projection  was  so
         significant  to  the world navigators of his time.  The great
         disadvantage  of  the  Mercator  projection  is the distorted
         picture  that  it  presents  of the relative areas of various
         regions  of the world.  Accustomed to the Mercator projection
         from  maps  studied  in  school,  few people can believe that
         South  America  is really more than nine times larger in area
         than Greenland.


              Mathematically,  the  Mercator  projection  is expressed
         as:


             x = a * (longitude) - b
             y = c - a * log{ tan[(pi/2 - latitude) / 2] }

              The  constants  a,  b,  and c are determined in Maps and
         Legends  to  produce  horizontal  and  vertical scales in the
         proper ratio and to center the map at the longitude which you
         enter  in  the  dialog.   Because  of  the  way  the Mercator
         projection  is  constructed,  a finite map can never show the
         poles.   In  Maps and Legends the overall scale is defined to
         show  the  major  land areas of the world well.  If the scale
         were  decreased  to display Antarctica, for example, the rest
         of the world would appear quite small.


                              Conic Projections:

              To understand how a conic projection is defined, imagine
         the  earth  inside an ice cream cone (for southern hemisphere
         projections),  or  wearing  a  dunce  cap  (for  maps  in the
         northern  hemisphere).   Features  are projected onto the map
         from  the  center  of  the earth, and the map is displayed by
         cutting  the cone and unrolling it.  The conic projection was
         first published by Ptolemy in the 2nd century A.D.


              Conic  projections are at their best for showing regions



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         of the world that span a wide range of longitudes in the mid-
         latitudes,  like  the  continental US.  A conic projection is
         exact   on   the  parallel  (or  parallels)  where  the  cone
         intersects  the  earth.  (The cone can be made a bit smaller,
         to  cut  the earth along two parallels.)  These parallels are
         called  the  standard  parallels of the map.  The latitude of
         the standard parallels determines the angle of the cone.


              Conic  projections  encompass  a  large  family  of  map
         projections,  some  of which also go by other names.  If, for
         example, the standard parallel of the conic projection is the
         equator,  the  cone  becomes  a  cylinder  and  a cylindrical
         projection results.  If the standard parallel is at the pole,
         the cone flattens into a sheet and the projection is the same
         as  the azimuthal equidistant projection described below.  If
         you  request  a  conic  projection covering a large longitude
         range  near  a  pole, the cone on which the map is projected,
         when  unrolled,  will  not  fill  the display.  This is not a
         problem  with  the  program,  it's  just  the  way  the conic
         projection works.


              Maps  drawn  with  the  conic  projection  can  be quite
         accurate  in  scale.   However,  the distortions in direction
         produced by the conic projection are responsible for the fact
         that  many  Americans  think Washington and Maine are farther
         north than Minnesota.


              In  this  program,  standard  parallels  are chosen that
         split  the center meridian 1/4 from the top and the bottom of
         the  region  that  you  select.  (Unless, of course, you have
         chosen  a  map  centered on the equator or the north or south
         pole).


              The formulae for this map projection are:


             x = (a + pi/2 - latitude) * cos( b * longitude)
             y = (a + pi/2 - latitude) * sin( b * longitude)

              The  constant  b  is  the  sine of the angle of the cone
         (alpha),  and  a  is  determined  from  the  latitudes of the
         standard   parallels  and  distance  between  them.   If  the
         standard parallels are at latitudes Lt1 and Lt2 then:











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             b = sin(alpha) = cos(X) sin(d)/d
             a = d tan(X)/tan(d) - X

         where

             X = pi/2 - (Lt1 + Lt2)/2 and
             d = (Lt1 - Lt2)/2

         in  other  words, X is the average colatitude of the standard
         parallels,  and  d  is half the distance between the standard
         parallels.


                       Azimuthal Equidistant Projection

              The   azimuthal   equidistant  projection  is  the  only
         projection  in which every point is shown at both the correct
         distance  and the correct direction (azimuth) from the center
         point. All other distances and directions are distorted. This
         projection  is  used to determine great circle directions. If
         you  place  your  home town at the center of the map, you can
         easily  determine the azimuth for pointing shortwave antennae
         for  reception of distant radio stations. Straight lines away
         from  your  home  at  the center of the map will be the great
         circle  routes  flown  by  airplanes on international flights
         from an airport near your home.


              A point half way around the world from the center of the
         map is 20,000 kilometers away. To generate a map of the whole
         world,  enter  20000. Do not put any commas in the number. To
         generate  a  map  to a smaller great-circle distance, enter a
         number less than 20000.


              If  the  center  point  is at latitude Lt0 and longitude
         Lg0,  then  another point at Lt, Lg will appear on the map at
         an x,y position:


             x = A cos(Lt) sin(Lg-Lg0)
             y = A [sin(Lt) cos(Lg0) - cos(Lt) sin(Lt0) cos(Lg-Lg0)]

         where

             A = c / sin(c) and
             cos(c)    =   sin(Lt)   sin(Lt0)   +   cos(Lt)   cos(Lt0)
         cos(Lg-Lg0).

                            Perspective Projection

              Prospective  projections  are those which show the Earth
         exactly  as  it  appears when viewed from some point in space
         above the surface. This is one of the best projections to use
         for games and space war simulations.



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              If  you  enter  a very large number here you will get an
         orthographic projection. In practice a good height to use for
         an orthographic projection is 999999 miles.


              You  might enjoy putting in a low height to see a "birds
         eye" view of the area around your home town. A good number to
         use  is  250  kilometers.  That  is about the altitude of the
         Space  Shuttle  orbit. If you make a low altitude perspective
         projection  over  the  Unites States, it's fun to include the
         state  boundaries.  The view can seem very distorted from low
         altitudes.  Your  state  will  take  up  much of the field of
         view.


              If Lg is the longitude at the center of the map then


             x = r sin[Lg cos(b)]
             y = r cos[Lg sin(b)]

         where r = a [ cos(Lt) / cos(b) ] / { h cos(b) + cos(Lt - b)}

              This  is a lot like a conic projection except cos(b) = n
         is the sine of the angle of the cone.  b is the colatitude of
         the central parallel of the map.  The orthographic projection
         is  the  limit  where  h  goes  to  infinity.  Otherwise h is
         between 1 and a large number.


                            Orthographic Projection

              In  orthographic  projections  the  point  of view is at
         infinity.  Orthographic  projections  are  a  special case of
         perspective  projections.  The orthographic projection is the
         limit  where  h,  in  the  calculation  for  the  perspective
         projection, goes to infinity.


                         Other World Map Projections:

              We  have  included  a  selection  of different world map
         projections  in  Maps and Legends.  Some of these projections
         are useful, some are important in the history of cartography,
         others are included just because they are interesting to make
         and view.


           The Flamsteed, Sanson-Flamsteed, or Sinusoidal Projection

              This  projection  is  an  equal area projection in which
         parallels of latitude are straight and equally spaced. "Equal
         area"  means  just  that.   A  square mile, or hundred square
         mile, etc. region of the world will have the same area on the



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         map  no  matter  where that region is.  (Contrast this to the
         Mercator  projection.)   Equal  area,  however, does not mean
         constant  shape.   That  "square"  mile  will  not  be square
         everywhere on the map.


              In  the Sanson-Flamsteed projection all parallels are at
         their  true  distances  from  the  equator.  In addition, the
         scale  along  each  parallel  is  exact.   The  shapes of all
         meridians and the outline of the map follow a cosine curve so
         that:


             x = a (longitude) cos(latitude) + b
             y = a (latitude) + c

              Sanson  was  an  important early cartographer and he was
         the  first  to  publish this projection.  Flamsteed used this
         projection  in  his  famous  star atlas and it was due to him
         that this projection became popular.


                            The Foucault Projection

              The  Foucault  projection  is  closely  related  to  the
         Sanson-  Flamsteed  projection.  This  is  also an equal area
         projection,  named after the french mathematician who derived
         it.   It  is  interesting  to  compare  the  Foucault and the
         Sanson-Flamsteed projections.  Both projections have straight
         parallels  of  latitude, but in the Foucault projection, they
         are  not evenly spaced.  Notice how these two maps each bring
         out  different  aspects of the regions that they display, and
         how  they  differ  in  where  the  map distortions become too
         distracting.  The Foucault projection can be expressed as:


             x = a (longitude) cos(latitude) cos2(latitude/2)
             y = a pi tan(latitude/2) / 2

         where a is the map scale factor.

                           The Mollweide Projection

              One  of  the most beautiful world map projections is the
         Mollweide  projection.   It  is also an equal area projection
         with  straight  parallels of latitude.  Much of the appeal of
         the Mollweide projection is due to the way that it solves the
         polar  distortion  problems  that  are  so distracting in the
         Sanson-Flamsteed,  and  especially the Foucault, projections.
         In this projection, the outline of the map is an ellipse with
         a  ratio  of 2/1 in the lengths of the major and minor axes.
         The  Mollweide  projection  is  one  of the more difficult to
         calculate,  because  the  projection requires the solution of
         the transcendental equation:




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             2t + sin(2t) = pi sin(latitude)

         the  resulting  value  for  t  then  determines  the  x and y
         position on the map via:

             x = a (longitude) cos(t)
             y = a pi sin(t) / 2

              Maps  and  Legends doesn't solve this equation for every
         coordinate  point because that would take too long.  Instead,
         the equation is solved for a number of points in advance, and
         the  solution  for  any point of the map is looked up in this
         table.


                             The Werner Projection

              Certainly   the   most  remarkable  of  the  equal  area
         projections is the Werner projection.  In this projection the
         outline  of the world takes the shape of a heart.  It is this
         unusual  shape  which  justifies  its  presence  in  Maps and
         Legends. Here, finally, is an equal area projection where the
         parallels  are  neither straight nor parallel.  Johann Werner
         was  a young contemporary of Columbus.  While he knew nothing
         about the polar regions, he knew that the earth was round and
         he  had the mathematical genius to develop this map which has
         been called "the cartographer's valentine".


              Mathematically,  the  Werner  projection  resembles  the
         conic  projection,  except  that  n,  the  angle of the cone,
         depends on the latitude:


             x = a h sin[B (longitude)] - c
             y = -a h cos[B (longitude)] - d

         where B = sin(h)/h and the colatitude h = pi/2 - latitude.

                           The Polyconic Projection

              Finally  we  come  to  the  projection  which may be the
         ultimate map compromise.  The polyconic projection is neither
         equal  area  nor  does  it preserve directions, distances, or
         angles  exactly,  but  it  is  never  far  off,  either.  The
         polyconic  projection  was  developed  in  1820  by the first
         Superintendent  of  the  Coast  and  Geodetic  Survey  of the
         then-young  United  States,  Ferdinand Hassler.  At that time
         the  US  stretched mostly north and south along the Atlantic.
         Hassler   derived  his  projection  to  overcome  the  errors
         associated  with  the conic projection, when it is applied to
         regions that are larger north-south than east-west.





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              Conceptually,  the  polyconic  projection is a series of
         narrow  conic  projection  strips  cut  out  from  the region
         surrounding   the   conic's   standard  parallel  and  pasted
         together.    This   projection  is  ideal  for  a  series  of
         contiguous  maps of a large region.  In fact, the topological
         maps  of  the  US  Geological  Survey  are  all  based on the
         polyconic projection.


              The  formulae  which  express  the  polyconic projection
         are:


             x = (Lg) cos(Lt) / H
             y = (Lt) + (Lg)(Lg) sin(Lt) cos(Lt) / (2 H)

         where     H = 1 + {(Lg) sin(Lt) / 2}2









































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