When the Going Gets Rough Automated Design of Low-Distortion Projections Michael L. Dennis, RLS, PE G67940_UC15_Tmplt_4x3_4-15 Esri Corporate Template v2.1 April 18, 2014 See http://arczone/resources/presentations.cfm for sample files and icons. Presentation available at www.geodetic.xyz

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

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

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

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

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

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, +1000 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

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

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

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

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

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

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)

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

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!

geo.ldpdesign.com

geo.ldpdesign.com

geo.ldpdesign.com

geo.ldpdesign.com

geo.ldpdesign.com

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

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 www.geodetic.xyz