We saw this cool Sol LeWitt wall at MASS MoCA. It did not escape our attention that it was basically an eikonal equation and that the weird junctures were caustic lines.

It was drawn with alternating colored marker lines appearing a cm away from the previous line. This is basically Huygens principal.

So I hacked together a demo in elm. Elm is a Haskell-ish language for the web.

So I made a quick rip and run elm program to do this. This is the output, which I could make more dynamic.

The algorithm is to turn a list of points into their connecting lines. Then move the line perpendicular to itself, then recompute the new intersection points. It’s somewhat reminiscent of Verlet integration. Lines coordinates are momentum-like and points are position like and we alternate them. This is a finite difference version of the geometric Huygen’s principle.

Alternative methods which might work better include the Fast Marching Method or just using the wave equation and then plotting iso surfaces.

I also had to resample the function to get only the maximum y value for each x value in order to duplicate the LeWitt effect.

These are the helper functions with lots of junk in there

module LineHelpers exposing (..) -- Maybe should just be doubles or nums import Debug fromJust : Maybe a -> a fromJust x = case x of Just y -> y Nothing -> Debug.crash "error: fromJust Nothing" toPointString : List (number, number) -> String toPointString xs = case xs of (x,y) :: ys -> (toString x) ++ "," ++ (toString y) ++ " " ++ (toPointString ys) _ -> "" crossProd : (number,number,number) -> (number,number,number) -> (number,number,number) crossProd (a,b,c) (d,e,f) = (b * f - c * e, c * d - a * f, a * e - b * d) type alias PointListH number = List (number,number,number) type alias LineListH number = List (number,number,number) -- gives the mapping function the list and the list shifted by 1 neighbormap f a = let a_ = fromJust (List.tail a) in List.map2 f a a_ crossNeighbor = neighbormap crossProd norm a b = sqrt (a * a + b * b) shiftLine delta (a,b,c) = (a,b, (norm a b) * delta + c) connectingLines = crossNeighbor shiftLines delta = List.map (shiftLine delta) intersections = crossNeighbor -- nearly striaght lines will find their intersection at infinity. -- maybe filter out a lower threshold on c -- keep first and last point last xs = let l = List.length xs in fromJust (List.head (List.drop (l - 1) xs)) timestep : Float -> List (Float,Float, Float) -> List (Float,Float, Float) timestep delta points = let firstpoint = fromJust (List.head points) lastpoint = last points connectlines = connectingLines points newlines = shiftLines delta connectlines newpoints = intersections newlines filterednewpoints = List.filter (\(a,b,c) -> (abs c) > 0.01) newpoints normpoints = List.map normalize filterednewpoints result = firstpoint :: (normpoints ++ [lastpoint]) resample = List.map (maxfunc (List.map dehomogenize result)) initx --result2 = removeoutoforder (-100000, 0,00) result in List.map homogenize (zip initx resample) homogenize (a,b) = (a,b,1) dehomogenize (a,b,c) = (a / c, b / c) normalize = dehomogenize >> homogenize zip = List.map2 (,) initx = List.map (toFloat >>((*) 4.5)) (List.range -200 400) --inity = List.map (\x -> x * x / 50) initx --inity = List.map (\x -> 300 + x * x / -50) initx --inity = List.map (\x -> 25 * sin (x / 20) + 250) initx inity = List.map (\x -> 25 * sin (x / 20) + 250 + 15 * sin (x/13)) initx initxy = zip initx inity initxyh = List.map homogenize initxy iterate n f x = if n == 0 then [] else (f x) :: iterate (n - 1) f (f x) paths = (List.map << List.map) dehomogenize (iterate 60 (timestep 5.0) initxyh) colors = List.concat (List.repeat (List.length paths) ["red", "blue", "yellow"] ) removeoutoforder prev xs = case xs of y :: ys -> if prev < y then (y :: removeoutoforder y ys) else removeoutoforder prev ys _ -> [] neighborzip a = let a_ = fromJust (List.tail a) in zip a a_ linearinterp x ((x1,y1), (x2,y2)) = (y1 * (x2 - x) + y2 * (x - x1)) / (x2 - x1) maxfunc : List (Float, Float) -> Float -> Float maxfunc points x = let pairs = neighborzip points filterfunc ((x1,y1), (x2,y2)) = (xor (x < x1) (x < x2)) candidates = List.filter filterfunc pairs yvals = List.map (linearinterp x) candidates in Maybe.withDefault 100 (List.maximum yvals)

And this is the svg main program.

import Html exposing (Html, button, div, text) import Html.Events exposing (onClick) import Svg exposing (..) import Svg.Attributes exposing (..) import LineHelpers exposing (..) roundRect : Html.Html msg roundRect = svg [ width "1000", height "1000",viewBox "-100 0 350 350" ] (List.reverse ([--[ rect [ x "10", y "10", width "100", height "100", rx "15", ry "15" ] [], -- polyline [ fill "none", stroke "red", points "20,100 40,60 70,80 100,20" ] [], polyline [ fill "none", stroke "black", strokeWidth "5.0", points (LineHelpers.toPointString LineHelpers.initxy) ] []] ++ (List.map2 (\path color -> polyline [ fill "none", stroke color, strokeWidth "3.0", points (LineHelpers.toPointString path)] []) LineHelpers.paths LineHelpers.colors))) main = Html.beginnerProgram { model = model, view = view, update = update } -- MODEL type alias Model = Int model : Model model = 0 -- UPDATE type Msg = Increment | Decrement update : Msg -> Model -> Model update msg model = case msg of Increment -> model + 1 Decrement -> model - 1 -- VIEW view : Model -> Html Msg view model = div [] [ button [ onClick Decrement ] [ Html.text "-" ] , div [] [ Html.text (toString model) ] , button [ onClick Increment ] [ Html.text "+" ], roundRect ]

notes on elm

elm is installed with npm

elm-repl

you import packages (including your own) with

import ThisPackage

and you check types by just writing them and hitting enter rather than :t

elm-live is a very handy thing. A live reloading server that watches for changes in your files.

elm-make myfile.elm

will generate the javascript and html

This is a good tutorial and a good snippet to get you going

Differences from Haskell:

elm isn’t lazy which is probably good.

The composition operator (.) is now <<

elm doesn’t have the multiline pattern match of haskell. You need to use case expressions. I miss them.

typeclass facilities are not emphasized.

The list type is List a rather than [a]

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