Gnomonic projection

Source: R/gnomonic.R

Convert between geographic coordinates and the gnomonic projection. In this projection, geodesics (shortest paths) appear as straight lines, making it useful for navigation and great circle route planning.

gnomonic_fwd(x, lon0, lat0)

gnomonic_rev(x, y, lon0, lat0)

Arguments

x

For forward conversion: a two-column matrix or data frame of coordinates (longitude, latitude) in decimal degrees. For reverse conversion: numeric vector of x coordinates in meters.

lon0

Longitude of the projection center in decimal degrees.

lat0

Latitude of the projection center in decimal degrees.

y

Numeric vector of y coordinates in meters.

Value

Data frame with columns:

  • For forward conversion:

    • x: X coordinate in meters

    • y: Y coordinate in meters

    • azi: Azimuth of the geodesic at the center (degrees)

    • rk: Reciprocal of the azimuthal scale

    • lon, lat: Input coordinates (echoed)

  • For reverse conversion:

    • lon: Longitude in decimal degrees

    • lat: Latitude in decimal degrees

    • azi: Azimuth of the geodesic at the center (degrees)

    • rk: Reciprocal of the azimuthal scale

    • x, y: Input coordinates (echoed)

Details

The gnomonic projection has a unique property: all geodesics (great circles on a sphere, shortest paths on an ellipsoid) appear as straight lines. This makes it invaluable for:

  • Planning great circle routes in aviation and shipping

  • Seismic ray path analysis

  • Radio wave propagation studies

Limitations:

  • Can only show less than a hemisphere

  • Extreme distortion away from the center

  • Neither conformal nor equal-area

See also

azeq_fwd() for azimuthal equidistant projection

Examples

# Project cities relative to London
cities <- cbind(
  lon = c(-74, 139.7, 151.2, 2.3),
  lat = c(40.7, 35.7, -33.9, 48.9)
)
gnomonic_fwd(cities, lon0 = -0.1, lat0 = 51.5)
#>            x          y       azi          rk   lon   lat
#> 1 -7258985.5  2411279.2 -128.7635  0.64072159 -74.0  40.7
#> 2 46796001.7 75878096.6  156.2494  0.07150066 139.7  35.7
#> 3        NaN        NaN  139.2137 -0.88597093 151.2 -33.9
#> 4   176156.3  -286620.1  150.2696  0.99861359   2.3  48.9

# Great circle route appears as straight line
# London to NYC path
path <- geodesic_path(c(-0.1, 51.5), c(-74, 40.7), n = 10)
projected <- gnomonic_fwd(cbind(path$lon, path$lat), lon0 = -37, lat0 = 46)
# x and y should be approximately linear
plot(projected$x, projected$y, type = "l")