Exponentially scaling distances for rendering objects that are far away using an imprecise floating Z buffer.

DistanceScaling.hs

module Main where
import Graphics.Gnuplot.Simple

-- Calculates the scaling factor for an object at distance 'dist'. The object
-- will be at scale 1 until 'max'/2, and then will be exponentially scaled down
-- as it increases. As 'dist' approaches infinity, 'dist * factor max dist'
-- approaches 'max'. See http://goo.gl/xLzrv
factor max dist = if dist < max / 2
    then 1
    else (1 - 1 / 2**(f+1)) / f
    where f = dist / max

-- Calculates the visual angular diameter of an object from its distance and
-- actual diameter. From http://en.wikipedia.org/wiki/Angular_diameter
angularSize diam dist = 2 * atan (diam / 2 / dist)

-- Convenience functions.
newDist max dist = dist * factor max dist
newDiam max diam dist = diam * factor max dist
newAngularSize max diam dist = angularSize (newDiam max diam dist) dist

-- Some constants
maxDist = 200 :: Float
initDiam = 1 :: Float

-- Main plotting function: demonstrates that the angular diameters of the object
-- with and without scaling are the same. The angular diameter with scaling is 
-- plotted one unit above that without. See an output example at
-- http://i.imgur.com/OuFzphF.png
plotBoth = plot [
    1 + newAngularSize maxDist initDiam,
    angularSize initDiam
 ]

-- And other plots, in case you want them.
plotWith = plot [newAngularSize maxDist initDiam]
plotWithout = plot [angularSize initDiam]

-- Common Gnuplot stuff.
plot = plotFuncs [
    Title "Apparent angular diameter",
    XLabel "Distance",
    YLabel "Angular diameter",
    Key (Just ["off"])
 ] (linearScale 200 (1, 100))