Calculating Distance in the Lambert Conformal Conic projection vs WGS84/GDA94

Lambert Conformal Conic 2 Standard Parallels

This projection is commonly used around the world. An example is the NSW Lambert projection.

The definition of the projection is described here.

Note that “[the] two standard parallels […] maintain exact scale, while the scale varies along the meridians” being reduced between the parallels and increased beyond them.

Use Case

I worked on a national (Australia) scale database which stored its data in a LCC projection. What was not understood by those who chose the projection was the distance/scale issue.

When accurate distances were required the application had to project the data into a geographic SRID (4283), compute the distance/length, and then throw away the projected data (a run-time penalty).

If the database had been defined to use a geographic SRID in the first place this additional processing could have been avoided.

Examples

The question of scale difference can be easily demonstrated using PostGIS.

Lines along parallels

Firstly we calculate distance along parallels including the two standard parallels for the NSW Lambert projection.

-- NSW Lambert
-- -30.75 is standard_parallel_1 (latitude)
-- -35.75 is standard_parallel_2 (latitude)
-- Length along parallel should be same.
select round(latitude,2) as latitude,
       round(ST_Length(line,true)::numeric,3) as llLength,
       round(ST_Length(ST_Transform(line,3308))::numeric,3) as lamLength,
       round(ABS(ST_Length(line,true)::numeric - ST_Length(ST_Transform(line,3308))::numeric),3) as diff
  from (select gs::decimal/100.0 as latitude, 
               ST_GeomFromEWKT('SRID=4283;LINESTRING(139.0 '||gs::decimal/100.0||',141.0 '||gs::decimal/100.0||')') as line
          from generate_series(-4075,-2575,100) as gs
       ) as f;

Note here that lines along the two standard parallels are, effectively, equal (as predicted) with parallels between and beyond the parallels being unequal.

Lines of random direction

-- Random linestrings
-- Projected Bounds: 8709235.6860, 4009911.3943, 9951732.0554, 5048642.3129
with data as (
  select 8709235.6860 as llx, 4009911.3943 as lly, 9951732.0554 as urx, 5048642.3129 as ury
)
select round(g.lamLength::numeric,3) as LamLength,
       round(g.llLength::numeric,3) as LLLength,
       round(ABS(g.lamLength-g.llLength)::numeric,3) as diff,  
       ST_Transform(g.line,4283)::geography as line
  from (select ST_Length(f.line) as lamLength,
               ST_Length(ST_Transform(f.line,4283)::geography,true) as llLength,
		       line
          from (select gs as id,
                       ST_SetSrid(
                          ST_MakeLine(ST_MakePoint(spdba.random_between(llx,urx),
                                                   spdba.random_between(lly,ury)),
                                      ST_MakePoint(spdba.random_between(llx,urx),
                                                   spdba.random_between(lly,ury))
                          ),3308) as line
                  from data,
                       generate_series(1,10,1) as gs
	           ) as f
         ) as g;

Note that the difference in distance between random linestrings is pronounced. Whether this issue affects your use of a Lambert Conformal Conic projection in your application depends on your needs. If, for example, accurate distance is important Lambert Conformation Conic may not be suitable.

If I have misrepresented the distance/scale issue for Lambert Conformal Conic projections please contact me.