1. When the Going Gets Rough Automated Design of Low-Distortion Projections
Michael L. Dennis, RLS, PE
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Esri Corporate Template v2.1
April 18, 2014
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2. The Problem Many users want a “low-distortion” map product
Want distances on map to equal true distances on “ground”
But it is impossible – no such thing as “ground” coordinates
However, distortion CAN be minimized
“Low distortion projection” (LDP) concepts & objectives
Cover largest area with least distortion (optimization problem)
Want developable surface at topographic surface
Difficult in areas of variable elevation
Use existing CONFORMAL projection types
Linear distortion at a point same in all directions

3. Conformal map projections Use ones supported by wide range of software!
Cylindrical
Transverse Mercator (State Plane, UTM, USNG)
Oblique Mercator (State Plane for AK panhandle)
Conical
Lambert Conformal Conic (State Plane)
Azimuthal (“planar”)
Stereographic (all aspects)
Usually not good for large areas (> ~30 miles long)
Many others, but most global and usually not good for LDPs
E.g., “normal” Mercator

4. Linear map projection distortion
Difference in distance between pair of projected (map grid) coordinates and true horizontal “ground” distance
Can be positive or negative
Grid length greater than
ellipsoidal length
Projection
surface
To visualize this, we’re going to add ellipsoid (representing the earth’s surface) on the screen. Then, we’re going to slide a plane through this ellipsoid, which represents our projection surface. We’re going to place a light bulb at the center of the earth to see how features on the earth are represented on the map. (This is all done mathematically, of course. But we can picture it this way to understand it better). Notice how the length of the lines on the ellipsoid and the projection plane are different.
Grid length less than
ellipsoidal length
Ellipsoid

5. Horizontal distortion due to Earth curvature
Maximum projection
zone width
Maximum linear horizontal distortion
Parts per
million
Feet per mile
Ratio
(absolute value)
16 miles
±1 ppm
±0.005 ft/mi
1 : 1,000,000
35 miles
±5 ppm
±0.025 ft/mi
1 : 200,000
50 miles
±10 ppm
±0.05 ft/mi
1 : 100,000
71 miles
±20 ppm
±0.1 ft/mi
1 : 50,000
158 miles*
±100 ppm
±0.5 ft/mi
1 : 10,000
317 miles**
±400 ppm
±2.1 ft/mi
1 : 2,500

6. Low Distortion Projections (LDPs)
Conventional approach
Place projection axis near center of design area
Scale projection to “representative” ellipsoid height
Low distortion coverage
Ground
surface
Projection surface
To account for topographic height, we move the projection surface to “ground”.
We place the Local projection axis near the center of the project area (e.g., central meridian for TM).
Low distortion height limits
Local projection axis
Representative ellipsoid height
Ellipsoid surface

7. Horizontal distortion due to height above ellipsoid
Height below (–) and above (+) projection surface
Maximum linear horizontal distortion
Parts per
million
Feet per mile
Ratio
(absolute value)
–100 ft, +100 ft
±4.8 ppm
±0.03 ft/mi
~1 : 209,000
–400 ft, +400 ft
±19 ppm
±0.1 ft/mi
~1 : 52,000
–1000 ft, ft
±48 ppm
±0.3 ft/mi
~1 : 21,000
+2000 ft
–96 ppm
–0.5 ft/mi
~1 : 10,500
+2800 ft*
–134 ppm
–0.7 ft/mi
~1 : 7500
+14,400 ft**
–688 ppm
–3.6 ft/mi
~1 : 1500

8. Minimizing linear distortion
Methods for reducing linear distortion
Scale existing projection (e.g., State Plane) “to ground”
Scale ellipsoid “to ground” and use as basis for custom projection
Design custom projection “at ground”
All do same thing:
Put projection developable surface at topographic surface
But not all perform the same:
Certain methods better at minimize distortion

9. Lambert Conformal Conic projection Secant
State Plane –
Lambert Conformal Conic projection
Secant
Ellipsoid
Topographic surface
Standard parallel
Design location
Central parallel
Standard parallel

10. Lambert Conformal Conic projection Scaled to “ground”
State Plane –
Lambert Conformal Conic projection
Scaled to “ground”
Ellipsoid
Topographic surface
Design location
Central parallel

11. Low Distortion Projection – Lambert Conformal Conic projection
Best-fit to topographic surface
Ellipsoid
Topographic surface
Central parallel
Design location

14. General design approach for sloping topography
Choose projection type
Slope east-west  Transverse Mercator
Slope north-south  Lambert Conformal Conic
Slope oblique direction  Oblique Mercator (or Stereographic)
Choose several design points distributed throughout area
Place projection axis at centroid of design points
Scale projection using mean height of design points
Change projection axis location
Minimize distortion range and standard deviation
Change projection scale
To get mean distortion near zero or near center of range

22. LDP design challenge (Part 1) Areas with slope in oblique direction
Very difficult to solve by trial & error
Must change (at least) two variables simultaneously
Oblique Mercator: Projection axis latitude, longitude, and orientation
Oblique Stereographic: Projection origin latitude and longitude
Can automate design process
Use least squares to define “planar” surface through design points
Planar surface orientation determines projection axis location
Minimizes distortion at design points (for areas < ~25 miles long)

31. Best-fit plane slopes up 0.68°at azimuth of 29.7°

34. LDP design challenge (Part 2) Oblique slope PLUS Earth curvature
Even more difficult to solve by trial & error
Good news: Still can automate design process!
Guarantees distortion minimized at design points
But still need to check behavior throughout design area!

35. geo.ldpdesign.com

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40. Finalize LDP design Don’t rely only on statistics – avoid surprises!
Check distortion throughout project area
Check distortion at critical/important locations
Keep design SIMPLE and CLEAN!
Projection scale to no more than 6 decimal places
Latitude and longitude origins to nearest arc-minute
False northings and eastings as whole numbers
Grid coordinates different from existing (e.g., State Plane, UTM)
Define linear unit and geometric datum
E.g., international foot, North American Datum of 1983

41. Conclusions Minimizing linear distortion at topographic surface
Minimized when developable surface at topographic surface
No single design height for large areas
For areas of sloping topography
Offset projection axis from area centroid
Use existing projections (no special projection required)
For “small” areas (< ~25 miles long)
Can best-fit planar surface to topography
Gives projection axis offset and orientation
For “large” areas (> ~25 miles long)
Better results if simultaneously also account for Earth curvature
Presentation available at