Map  Projections  Summary

• Projections  specify  a  two-­‐dimensional  coordinate  system  from  a  3-­‐D  globe  • All  projections  cause  some  distortion  • Errors  are  controlled  by  choosing  the  proper  projection  type,  limiting  the  area  applied  • There  are  standard  projections  • Projections  differ  by  datum  –  know  your  parameters  

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Intersect at90o angles

Projections  are  like  political  parties,  they  distort  everything

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There  is  ALWAYS  Distortion

Projections  Cause  Shape    and  Distance  Distortion

Where  is  the  distortion  the  worstest?

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BC

We  Manage  Distortion  by:    -­‐  Placement  of  intersecting  lines  

 -­‐  Size  of  the  area  projected

Someone sent me some data for my study area, they don’t know the coordinate

System…how do I figure it out?

Know  the  Likely  Projections  in  Your  Area,  Check  

Coordinates  for  Features  or  Extent

Common  Coordinates  

Projected:  -­‐County  

-­‐State  Plane  

-­‐UTM  

Unprojected:  -­‐Geographic,  NAD27  

-­‐Geographic,  NAD83(1986)  

-­‐Geographic,  NAD83(96)  

-­‐WGS84(xxxx)  

-­‐ITRFxx  

County  Projections:  Minnesota  examplehttp://www.dot.state.mn.us/surveying/toolstech/mapproj.html

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State  Plane  Coordinate  System  Zones

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State  Plane  is  Common  for  multi-­‐county  and  single-­‐county  

data

UTM  is  common  for  statewide  data  

Forward  and  Inverse  Equations  for  Projections

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Forward:  from  Lat/Lon  to  projected  x,  y  (or  N,  E)  e.g.,  from    Geographic  Coordinates  latitude  =  44o  59’  29.67571”(N)    longitude  =  93o  11  09.99753(W)  to  UTM  Zone  15N  coordinates  Northing  (y)  =  4,982,031.512    EasEng  (x)  =  485,329.503  

Forward  and  Inverse  Equations  for  Projections

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Inverse:  From  projected  coordinates  (x,y)  coordinates  

Back  To  geographic  (lat/lon)  coordinates  

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http://www.linz.govt.nz/data/geodetic-­‐services/coordinate-­‐conversion/projection-­‐conversions/transverse-­‐mercator-­‐transformation-­‐formulae

Forward  and  Inverse  Formulas

x  =  f(f,  l,  a,  b,  e,  other  “key”  parameters)  y  =  g(  f,  l,  a,  b,  e,  other  “key”  parameters)  

f  =  q(x,y,  a,  b,  e,  other  “key”  parameters)  l  =  p(x,y,  a,  b,  e,  other  “key”  parameters)  

Simple  Projection:  Regular  (NOT  TRANSVERSE)  Mercator

λ = 30, φ = 45

Example:

X  =  R  (π/180(30-­‐  0))          =  6,378  (1.7320)  

       =  11,047  

Y  =  R  ln(tan(π/180(45+45/2)))          =  6,378  ln(1.73205)          =  6,378  *  0.5493  

       =  3,503

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Simple  Mercator  Projection  -­‐  Inverse

•Orthographic  -­‐  ray  source  at  infinity  •Stereographic  -­‐  ray  source  at  antipode  •Gnomonic  -­‐  ray  source  at  spherical  center  

•Azimuthal  -­‐  project  on  to  a  plane  •Conic  -­‐  project  on  to  a  cone  •Cylindrical  -­‐  project  on  to  a  cylinder  

•Conformal  -­‐  preserves  shape  in  some  limited  area  •Equal  Area  -­‐  “squares”  formed  by  meridians/parallels  maintain  their  relative  size  proportion  from  the  Earth  to  the  map  

 

Projection  Properties

Azimuthal  Orthographic  Projection

Azimuthal  Stereographic  Projection

Azimuthal  Stereographic  Projection

Conic  Orthographic  Projection

Conic  Stereographic  Projection

Map  Projections  vs.  Datum  Transformations

• A  map  projections  is  a  systematic  rendering  from  3-­‐D  to  2-­‐D  

• Datum  transformations  are  from  one  datum  to  another,  3-­‐D  to  3-­‐D  

• Changing  from  one  projection  to  another  may  require  both

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From  one  Projection  to  Another    

If  in  the  SAME  DATUM,  just  forward  and  inverse  projection  equations

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From  one  Projection  to  Another    If  in  the  DIFFERENT  DATUMs,  must  do  a  datum  transformation  between  forward  and  inverse  projection  equations