Weather routing

From Rosetta Code
(Redirected from Weather Routing)
Weather routing is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The weather routing problem has the following parts:

  • a predicted surface wind direction and speed, at increments of longitude, latitude, and time
  • an expected surface current direction and speed, at increments of longitude, latitude, and time
  • 'polar data' describing maximum speed of a sailboat at points of sail for a given speed of wind over water
  • regions for sailing (the open ocean) and not (the land, shallows, restricted areas, etc.)
  • a starting location and time, and a destination

Given the above information and a specific path, progress and arrival time are determined. The weather routing problem, conversely, is to determine the path which results in the earliest arrival time.


Go[edit]

Translation of: Julia

This runs in only 37 seconds which is surprisingly quick compared to Julia. However, I've just noticed that I'm using an out of date version of Julia (1.0.4) so hopefully the latest version will be able to close the gap.

package main

import (
    "fmt"
    "io/ioutil"
    "log"
    "math"
    "strconv"
    "strings"
)

type matrixF = [][]float64
type pred = func(float64) bool

/*
   Structure that represents a polar CSV file's data.
   Note 0 degrees is directly into the wind, 180 degrees is directly downwind.
*/
type SailingPolar struct {
    winds   []float64 // vector of windspeeds
    degrees []float64 // vector of angles in degrees of direction relative to the wind
    speeds  matrixF   // matrix of sailing speeds indexed by wind velocity and angle of boat to wind
}

/*
   Structure that represents wind and surface current direction and velocity for a given position.
   Angles in degrees, velocities in knots.
*/
type SurfaceParameters struct{ windDeg, windKts, currentDeg, currentKts float64 }

// Checks for fatal errors.
func check(err error) {
    if err != nil {
        log.Fatal(err)
    }
}

// Reads a sailing polar CSV file and returns a SailingPolar struct containing the file data.
// A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma.
// The first line of file contains labels for the wind velocities that make up columns, and
// the first entry of each row makes up a column of angle of sailing direction from wind in degrees.
func getPolarData(fileName string) *SailingPolar {
    content, err := ioutil.ReadFile(fileName)
    check(err)
    lines := strings.Split(string(content), "\n")
    line0 := strings.TrimSpace(lines[0])
    header := strings.Split(line0, ";")
    var winds, degrees []float64
    var speeds matrixF
    for _, col := range header[1:] {
        t, err := strconv.ParseFloat(col, 64)
        check(err)
        winds = append(winds, t)
    }
    for _, line := range lines[1:] {
        line = strings.TrimSpace(line)
        if line == "" {
            break // ignore final blank line if there is one
        }
        cols := strings.Split(line, ";")
        f, err := strconv.ParseFloat(cols[0], 64)
        check(err)
        degrees = append(degrees, f)
        var temp []float64
        for _, col := range cols[1:] {
            t, err := strconv.ParseFloat(col, 64)
            check(err)
            temp = append(temp, t)
        }
        speeds = append(speeds, temp)
    }
    return &SailingPolar{winds, degrees, speeds}
}

const R = 6372800.0 // Earth's approximate radius in meters

/* various helper methods which work with degrees rather than radians. */

// Converts degrees to radians.
func deg2Rad(deg float64) float64 { return math.Mod(deg*math.Pi/180+2*math.Pi, 2*math.Pi) }

// Converts radians to degrees.
func rad2Deg(rad float64) float64 { return math.Mod(rad*180/math.Pi+360, 360) }

// Trig functions.
func sind(d float64) float64     { return math.Sin(deg2Rad(d)) }
func cosd(d float64) float64     { return math.Cos(deg2Rad(d)) }
func asind(d float64) float64    { return rad2Deg(math.Asin(d)) }
func atand(x, y float64) float64 { return rad2Deg(math.Atan2(x, y)) }

// Calculates the haversine function for two points on the Earth's surface.
// Given two latitude, longitude pairs in degrees for a point on the Earth,
// get distance in meters and the initial direction of travel in degrees for
// movement from point 1 to point 2.
func haversine(lat1, lon1, lat2, lon2 float64) (float64, float64) {
    dlat := lat2 - lat1
    dlon := lon2 - lon1
    a := math.Pow(sind(dlat/2), 2) + cosd(lat1)*cosd(lat2)*math.Pow(sind(dlon/2), 2)
    c := 2 * asind(math.Sqrt(a))
    theta := atand(sind(dlon)*cosd(lat2), cosd(lat1)*sind(lat2)-sind(lat1)*cosd(lat2)*cosd(dlon))
    theta = math.Mod(theta+360, 360)
    return R * c * 0.5399565, theta
}

// Returns the index of the first element of 'a' for which 'p' returns true or -1 otherwise.
func findFirst(a []float64, p pred) int {
    for i, e := range a {
        if p(e) {
            return i
        }
    }
    return -1
}

// Returns the index of the last element of 'a' for which 'p' returns true or -1 otherwise.
func findLast(a []float64, p pred) int {
    for i := len(a) - 1; i >= 0; i-- {
        if p(a[i]) {
            return i
        }
    }
    return -1
}

// Calculate the expected sailing speed in a specified direction in knots,
// given sailing polar data, a desired point of sail in degrees, and wind speed in knots.
func boatSpeed(sp *SailingPolar, pointOfSail, windSpeed float64) float64 {
    winds := sp.winds
    degrees := sp.degrees
    speeds := sp.speeds
    udeg := findLast(degrees, func(t float64) bool { return t <= pointOfSail })
    odeg := findFirst(degrees, func(t float64) bool { return t >= pointOfSail })
    uvel := findLast(winds, func(t float64) bool { return t <= windSpeed })
    ovel := findFirst(winds, func(t float64) bool { return t >= windSpeed })
    if udeg == -1 || odeg == -1 || uvel == -1 || ovel == -1 {
        return -1
    }
    var frac float64
    switch {
    case odeg == udeg && uvel == ovel:
        frac = 1
    case odeg == udeg:
        frac = (windSpeed - winds[uvel]) / (winds[ovel] - winds[uvel])
    case uvel == ovel:
        frac = (pointOfSail - degrees[udeg]) / (degrees[odeg] - degrees[udeg])
    default:
        frac = ((pointOfSail-degrees[udeg])/(degrees[odeg]-degrees[udeg]) +
            (windSpeed-winds[uvel])/(winds[ovel]-winds[uvel])) / 2
    }
    return speeds[udeg][uvel] + frac*(speeds[odeg][ovel]-speeds[udeg][uvel])
}

// Calculates the expected net boat speed in a desired direction versus the wind ('azimuth').
// This is generally different from the actual boat speed in its actual direction.
// Directions are in degrees ('pointos' is point of sail the ship direction from the wind),
// and velocity of wind ('ws') is in knots.
func sailingSpeed(sp *SailingPolar, azimuth, pointos, ws float64) float64 {
    return boatSpeed(sp, pointos, ws) * cosd(math.Abs(pointos-azimuth))
}

// Calculates the net direction and velocity of a sailing ship.
// Arguments are sailing polar data, direction of travel in degrees from north, wind direction in
// degrees from north, wind velocity in knots, surface current direction in degrees, and
// current velocity in knots.
func bestVectorSpeed(sp *SailingPolar, dirTravel, dirWind, windSpeed, dirCur, velCur float64) (float64, float64) {
    azimuth := math.Mod(dirTravel-dirWind, 360)
    if azimuth < 0 {
        azimuth += 360
    }
    if azimuth > 180 {
        azimuth = 360 - azimuth
    }
    vmg := boatSpeed(sp, azimuth, windSpeed)
    other := -1.0
    idx := -1
    for i, d := range sp.degrees {
        ss := sailingSpeed(sp, azimuth, d, windSpeed)
        if ss > other {
            other = ss
            idx = i
        }
    }
    if other > vmg {
        azimuth = sp.degrees[idx]
        vmg = other
    }
    dirChosen := deg2Rad(dirWind + azimuth)
    wx := vmg * math.Sin(dirChosen)
    wy := vmg * math.Cos(dirChosen)
    curX := velCur * math.Sin(deg2Rad(dirCur))
    curY := velCur * math.Cos(deg2Rad(dirCur))
    return rad2Deg(math.Atan2(wy+curY, wx+curX)), math.Sqrt(math.Pow(wx+curX, 2) + math.Pow(wy+curY, 2))
}

// Calculates the trip time in minutes from (lat1, lon1) to the destination (lat2, lon2).
// Uses the data in SurfaceParameters for wind and current velocity and direction.
func sailSegmentTime(sp *SailingPolar, p SurfaceParameters, lat1, lon1, lat2, lon2 float64) float64 {
    distance, dir := haversine(lat1, lon1, lat2, lon2)
    _, vel := bestVectorSpeed(sp, dir, p.windDeg, p.windKts, p.currentDeg, p.currentKts)
    // minutes/s * m / (knots * (m/s / knot)) = minutes
    return (1.0 / 60.0) * distance / (vel * 1.94384)
}

/* Structure that represents a point in 2-D space. */
type Point2 struct{ x, y int }

func (p Point2) add(p2 Point2) Point2  { return Point2{p.x + p2.x, p.y + p2.y} }
func (p Point2) equals(p2 Point2) bool { return p.x == p2.x && p.y == p2.y }
func (p Point2) String() string        { return fmt.Sprintf("[%d, %d]", p.x, p.y) }

/*
   Structure that consists of a tuple of latitude and longitude in degrees.
   NB: This uses latitude (often considered to be y) first then longitude (often considered to be x).
   This latitude, then longitude ordering is as per ISO 6709 (en.wikipedia.org/wiki/ISO_6709).
*/
type Position struct{ lat, lon float64 }

/*  Structure that represents a Position with the SurfaceParameters of wind and current at the Position. */
type GridPoint struct {
    pt Position
    sp SurfaceParameters
}
type MatrixG = [][]*GridPoint

/*
   Type alias for a matrix of GridPoints, each Position point with their SurfaceParameters.
   A Vector of TimeSlice can give the surface characteristics for an ocean region over time.
*/
type TimeSlice = MatrixG

/* Structure that represents a routing problem. */
type RoutingProblem struct {
    timeInterval    float64     // the minutes duration for each TimeSlice
    timeFrame       []TimeSlice // a vector of sequential timeslices for the ocean region
    obstacleIndices []Point2    // the Cartesian indices in each TimeSlice for
    // obstacles, such as land or shoals, where the ship may not go
    startIndex        int    // the TimeSlice position for time of starting
    start             Point2 // starting location on grid of GridPoints
    finish            Point2 // destination / finish location on grid of GridPoints
    allowRepeatVisits bool   // whether the vessel may overlap its prior path, usually false
}

/* Structure that represents a timed path. */
type TimedPath struct {
    duration float64  // minutes total to travel the path
    path     []Point2 // vector of Cartesian indices of points in grid for path to travel
}

func (t TimedPath) String() string           { return fmt.Sprintf("%g %v", t.duration, t.path) }
func (t TimedPath) equals(t2 TimedPath) bool { return t.String() == t2.String() }

func findMin(a []float64) (float64, int) {
    min := a[0]
    idx := 0
    for i, e := range a[1:] {
        if e < min {
            min = e
            idx = i + 1
        }
    }
    return min, idx
}

var ntuples = [][2]int{{-1, -1}, {-1, 0}, {-1, 1}, {0, -1}, {0, 1}, {1, -1}, {1, 0}, {1, 1}}
var neighbors = make([]Point2, len(ntuples))

func init() {
    for i := 0; i < len(ntuples); i++ {
        neighbors[i] = Point2{ntuples[i][0], ntuples[i][1]}
    }
}

func contains(points []Point2, point Point2) bool {
    for _, p := range points {
        if p.equals(point) {
            return true
        }
    }
    return false
}

// Returns a list of points surrounding 'p' which are not otherwise excluded.
func surround(p Point2, mat TimeSlice, excluded []Point2) []Point2 {
    xmax := len(mat)
    ymax := len(mat[0])
    var res []Point2
    for _, x := range neighbors {
        q := x.add(p)
        if (0 <= q.x && q.x < xmax) && (0 <= q.y && q.y < ymax) && !contains(excluded, q) {
            res = append(res, q)
        }
    }
    return res
}

// Get the route (as a TimedPath) that minimizes time from start to finish for a given
// RoutingProblem (sea parameters) given sailing polar data (ship parameters).
func minimumTimeRoute(rp *RoutingProblem, sp *SailingPolar, verbose bool) *TimedPath {
    timedPaths := []*TimedPath{&TimedPath{0, []Point2{rp.start}}}
    completed := false
    minPath := &TimedPath{1000, []Point2{}}
    for i := 0; i < 1000; i++ {
        var newPaths []*TimedPath
        if verbose {
            fmt.Printf("Checking %d paths of length %d\n", len(timedPaths), len(timedPaths[0].path))
        }
        for _, tpath := range timedPaths {
            le := len(tpath.path)
            if tpath.path[le-1] == rp.finish {
                completed = true
                newPaths = append(newPaths, tpath)
            } else {
                p1 := tpath.path[le-1]
                num := int(math.Round(tpath.duration))
                den := int(math.Round(rp.timeInterval))
                slice := rp.timeFrame[(num/den)%len(rp.timeFrame)]
                for _, p2 := range surround(p1, slice, rp.obstacleIndices) {
                    if !rp.allowRepeatVisits && contains(tpath.path, p2) {
                        continue
                    }
                    gp1 := slice[p1.x][p1.y]
                    gp2 := slice[p2.x][p2.y]
                    lat1 := gp1.pt.lat
                    lon1 := gp1.pt.lon
                    lat2 := gp2.pt.lat
                    lon2 := gp2.pt.lon
                    t := sailSegmentTime(sp, gp1.sp, lat1, lon1, lat2, lon2)
                    path := make([]Point2, len(tpath.path))
                    copy(path, tpath.path)
                    path = append(path, p2)
                    newPaths = append(newPaths, &TimedPath{tpath.duration + t, path})
                }
            }
        }
        set := make(map[string]*TimedPath)
        for _, np := range newPaths {
            set[np.String()] = np
        }
        timedPaths = timedPaths[:0]
        for k := range set {
            timedPaths = append(timedPaths, set[k])
        }
        if completed {
            var durations []float64
            for _, x := range timedPaths {
                durations = append(durations, x.duration)
            }
            minDur, _ := findMin(durations)
            var finished []*TimedPath
            for _, x := range timedPaths {
                le := len(x.path)
                if x.path[le-1] == rp.finish {
                    finished = append(finished, x)
                }
            }
            durations = durations[:0]
            for _, x := range finished {
                durations = append(durations, x.duration)
            }
            minFinDur, idx := findMin(durations)
            if verbose {
                fmt.Printf("Current finished minimum: %v, others %v\n", finished[idx], minDur)
            }
            if minDur == minFinDur {
                minPath = finished[idx]
                break
            }
        }
    }
    return minPath
}

/*
   The data is selected so the best time path is slightly longer than the
   shortest length path. The forbidden regions are x, representing land or reef.
   The allowed sailing points are . and start and finish are S and F.

   x  .  .  F  .  .  x  .  x
   .  .  .  .  .  .  .  x  x
   x  .  .  x  x  x  .  .  .
   .  .  x  x  x  x  .  x  x
   x  .  .  .  x  x  .  x  .
   x  .  .  .  x  x  .  x  .
   .  .  .  .  x  .  .  x  .
   x  .  .  .  .  .  .  x  .
   .  .  .  S  .  .  .  .  .
*/

// These need to be changed to 0-based for Go.
var ftuples = [][2]int{
    {1, 8}, {2, 1}, {2, 8}, {3, 5}, {3, 8}, {4, 1}, {4, 5}, {4, 6}, {4, 8}, {5, 1},
    {5, 5}, {5, 6}, {5, 8}, {6, 3}, {6, 4}, {6, 5}, {6, 6}, {6, 8}, {6, 9}, {7, 1},
    {7, 4}, {7, 5}, {7, 6}, {8, 8}, {8, 9}, {9, 1}, {9, 7}, {9, 9},
}

var forbidden = make([]Point2, len(ftuples))

func init() {
    for i := 0; i < len(ftuples); i++ {
        forbidden[i] = Point2{ftuples[i][0] - 1, ftuples[i][1] - 1}
    }
}

// Create regional wind patterns on the map.
func surfaceByLongitude(lon float64) SurfaceParameters {
    switch {
    case lon < -155.03:
        return SurfaceParameters{-5, 8, 150, 0.5}
    case lon < -155.99:
        return SurfaceParameters{-90, 20, 150, 0.4}
    default:
        return SurfaceParameters{180, 25, 150, 0.3}
    }
}

// Vary wind speeds over time.
func mutateTimeSlices(slices []TimeSlice) {
    i := 1
    for _, slice := range slices {
        for j := 0; j < len(slice); j++ {
            for k := 0; k < len(slice[j]); k++ {
                x := slice[j][k]
                x.sp = SurfaceParameters{x.sp.windDeg, x.sp.windKts * (1 + 0.002*float64(i)),
                    x.sp.currentDeg, x.sp.currentKts}
            }
        }
        i++
    }
}

func main() {
    startPos := Point2{0, 3} // 0-based
    endPos := Point2{8, 3}   // ditto
    slices := make([]MatrixG, 200)
    for s := 0; s < 200; s++ {
        gpoints := make([][]*GridPoint, 9)
        for i := 0; i < 9; i++ {
            gpoints[i] = make([]*GridPoint, 9)
            for j := 0; j < 9; j++ {
                pt := Position{19.78 - 1.0/60.0 + float64(i)/60, -155.0 - 5.0/60.0 + float64(j)/60}
                gpoints[i][j] = &GridPoint{pt, surfaceByLongitude(pt.lon)}
            }
        }
        slices[s] = gpoints
    }
    mutateTimeSlices(slices)
    routeProb := &RoutingProblem{10, slices, forbidden, 0, startPos, endPos, false}
    fileName := "polar.csv"
    sp := getPolarData(fileName)
    tp := minimumTimeRoute(routeProb, sp, false)
    fmt.Println("The route taking the least time found was:\n", tp.path, "\nwhich has duration",
        int(tp.duration/60), "hours,", int(math.Round(math.Mod(tp.duration, 60))), "minutes.")
}
Output:
The route taking the least time found was:
 [[0, 3] [0, 4] [1, 5] [2, 6] [3, 6] [4, 6] [5, 6] [6, 6] [7, 5] [7, 4] [8, 3]] 
which has duration 4 hours, 44 minutes.

Julia[edit]

Brute force optimization search, practical for shorter path lengths, but would require a better algorithm for paths over twice this size.

module SailingPolars

using DelimitedFiles

export SailingPolar, SurfaceParameters, getpolardata, deg2rad, rad2deg, cartesian2polar
export polar2cartesian, haversine, inverse_haversine, boatspeed, bestvectorspeed
export sailingspeed, sailsegmenttime

"""
    Structure to represent a polar CSV file's data.

Contains a matrix, speeds, of sailing speeds indexed by wind velocity and angle of boat to wind
winds is a list of wind speeds
degrees is a list of angles in degrees of direction relative to the wind
Note 0.0 degrees is directly into the wind, 180 degrees is directly downwind.
"""
struct SailingPolar
    winds::Vector{Float32}
    degrees::Vector{Float32}
    speeds::Matrix{Float32} # speeds[wind direction degrees, windspeed knots]
end

"""
    struct SurfaceParameters

Structure that represents wind and surface current direction and velocity for a given position
Angles in degrees, velocities in knots
"""
struct SurfaceParameters
    winddeg::Float32
    windkts::Float32
    currentdeg::Float32
    currentkts::Float32
end

"""
function getpolardata(filename)

Read a sailing polar CSV file and return a SailingPolar containing the file data.

A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma.
The first line of file contains labels for the wind velocities that make up columns, and
the first entry of each row makes up a column of angle of sailing direction from wind in degrees
"""
function getpolardata(filename)
    datacells, headercells = readdlm(filename, ';', header=true)
    winds = map(x -> parse(Float32, x), headercells[2:end])
    degrees = datacells[:, 1]
    speeds = datacells[:, 2:end]
    return SailingPolar(winds, degrees, speeds)
end


const R = 6372800  # Earth's approximate radius in meters

"""
    deg2rad(deg)

Convert degrees to radians
"""
deg2rad(deg) = (deg * π / 180.0 + 2π) % 2π

"""
    rad2deg(rad)

Convert radians to degrees
"""
rad2deg(rad) = (rad * (180.0 / π) + 360.0) % 360.0

"""
    cartesian2polard(x, y)

Convert x, y coordinates to polar coordinates with angle in degrees
"""
cartesian2polard(x, y) = sqrt(x * x + y * y), atand(x, y)

"""
    polard2cartesian(r, deg)

Convert polar coordinates in degrees to cartesian x, y coordinates
"""
polard2cartesian(r, deg) = r .* sincosd(deg)

"""
    function haversine(lat1, lon1, lat2, lon2)

Calculate the haversine function for two points on the Earth's surface.

Given two latitude, longitude pairs in degrees for a point on the Earth,
get distance in meters and the initial direction of travel in degrees for
movement from point 1 to point 2.
"""
function haversine(lat1, lon1, lat2, lon2)
    dlat = lat2 - lat1
    dlon = lon2 - lon1
    a = sind(dlat / 2)^2 + cosd(lat1) * cosd(lat2) * sind(dlon / 2)^2
    c = 2.0 * asind(sqrt(a))
    theta = atand(sind(dlon) * cosd(lat2),
        cosd(lat1) * sind(lat2) - sind(lat1) * cosd(lat2) * cosd(dlon))
    theta = (theta + 360) % 360
    return R * c * 0.5399565, theta
end

"""
    function inverse_haversine(lat1, lon1, distance, direction)

Calculate an inverse haversine function.

Takes the point of origin in degrees (latitude, longitude), distance in meters, and
initial direction in degrees, and returns the latitude and longitude of the endpoint
in degrees after traveling the specified distance.
"""
function inverse_haversine(lat1, lon1, distance, direction)
    lat2 = asind(sind(lat1) * cos(distance / R) + cosd(lat1) * sin(distance / R) * cosd(direction))
    lon2 = lon1 + atand(sind(direction) * sin(distance / R) * cosd(lat1),
                       cos(distance / R) - sind(lat1) * sind(lat2))
    return lat2, lon2
end

"""
    function boatspeed(sp::SailingPolar, pointofsail, windspeed)

Calculate the expected sailing speed in a specified direction in knots,
given sailing polar data, a desired point of sail in degrees, and wind speed in knots
"""
function boatspeed(sp::SailingPolar, pointofsail, windspeed)
    winds, degrees, speeds = sp.winds, sp.degrees, sp.speeds
    udeg = findlast(t -> t <= pointofsail, degrees)
    odeg = findfirst(t -> t >= pointofsail, degrees)
    uvel = findlast(t -> t <= windspeed, winds)
    ovel = findfirst(t -> t >= windspeed, winds)
    if any(t -> t == nothing, [udeg, odeg, uvel, ovel])
        return -1.0
    end
    frac = (odeg == udeg && uvel == ovel) ? 1.0 :
            (odeg == udeg) ? (windspeed - winds[uvel]) / (winds[ovel] - winds[uvel]) :
            (uvel == ovel) ? (pointofsail - degrees[udeg]) / (degrees[odeg] - degrees[udeg]) :
            ((pointofsail - degrees[udeg]) / (degrees[odeg] - degrees[udeg]) +
            (windspeed - winds[uvel]) / (winds[ovel] - winds[uvel])) / 2
    return speeds[udeg, uvel] + frac * (speeds[odeg, ovel] - speeds[udeg, uvel])
end


"""
    sailingspeed(sp::SailingPolar, azimuth, pointos, ws)

Calculate the expected net boat speed in a desired direction versus the wind (azimuth).
This is generally different from the actual boat speed in its actual direction.
Directions are in degrees (pointos is point of sail, the ship direction from wind),
and velocity of wind (ws) is in knots.
"""
sailingspeed(sp, azimuth, pointos, ws) = boatspeed(sp, pointos, ws) * cosd(abs(pointos - azimuth))


"""
    function bestvectorspeed(sp::SailingPolar, dirtravel, dirwind, windspeed, dircur, velcur)

Calculate the net direction and velocity of a sailing ship.

Arguments are sailing polar data, direction of travel in degrees from north, wind direction in
degrees from north, wind velocity in knots, surface current direction in degrees, and
current velocity in knots.
"""
function bestvectorspeed(sp::SailingPolar, dirtravel, dirwind, windspeed, dircur, velcur)
    azimuth = (dirtravel - dirwind) % 360.0
    azimuth = azimuth < 0 ? azimuth + 360.0 : azimuth
    azimuth = azimuth > 180.0 ? 360.0 - azimuth : azimuth
    VMG = boatspeed(sp, azimuth, windspeed)
    other, idx = findmax([sailingspeed(sp, azimuth, x, windspeed) for x in sp.degrees])
    if other > VMG
        azimuth = sp.degrees[idx]
        VMG = other
    end
    dirchosen = deg2rad(dirwind + azimuth)
    wx, wy = VMG * sin(dirchosen), VMG * cos(dirchosen)
    curx, cury = velcur * sin(deg2rad(dircur)), velcur * cos(deg2rad(dircur))
    return rad2deg(atan(wy + cury, wx + curx)), sqrt((wx + curx)^2 + (wy + cury)^2)
end

"""
    function sailsegmenttime(sp::SailingPolar, p::SurfaceParameters, lat1, lon1, lat2, lon2)

Calculate the trip time in minutes from (lat1, lon1) to the destination (lat2, lon2).
Uses the data in SurfaceParameters for wind and current velocity and direction.
"""
function sailsegmenttime(sp::SailingPolar, p::SurfaceParameters, lat1, lon1, lat2, lon2)
    distance, dir = haversine(lat1, lon1, lat2, lon2)
    dir2, vel = bestvectorspeed(sp, dir, p.winddeg, p.windkts, p.currentdeg, p.currentkts)
    # minutes/s * m / (knots * (m/s / knot)) = minutes
    return (1 / 60) * distance / (vel * 1.94384)
end


end # module


module SailingNavigation

export Position, lat, lon, GridPoint, TimeSlice, TimedPath, closestpoint, surround
export RoutingProblem, minimumtimeroute

using GeometryTypes
using ..SailingPolars

# NB: This uses latitude (often considered to be y) first then longitude (often considered to be x).
# This latitude, then longitude ordering is as per ISO 6709 (en.wikipedia.org/wiki/ISO_6709)

# Position is a Float32 2-tuple of latitude and longitude in degrees
Position = Point2f0

# latitude from Position
lat(p::Position) = p[1]

# longitude from Position
lon(p::Position) = p[2]

# A GridPoint is a Position with the SurfaceParameters of wind and current at the Position
mutable struct GridPoint
    pt::Position
    sp::SurfaceParameters
end

"""
    TimeSlice

A TimeSlice is a matrix of GridPoints, each Position point with their SurfaceParameters
A Vector of TimeSlice can give the surface characteristics for an ocean region over time.
"""
TimeSlice = Matrix{GridPoint}

"""
    mutable struct RoutingProblem

timeinterval: the minutes duration for each TimeSlice
timeframe: a vector of sequential timeslices for the ocean region
obstacleindices: the Cartesian indices in each timeslice for
    obstacles, such as land or shoals, where the ship may not go
startindex: the timeslice position for time of starting
start: starting location on grid of GridPoints
finish: destination / finish location on grid of GridPoints
allowrepeatvisits: whether the vessel may overlap its prior path, usually false
"""
mutable struct RoutingProblem
    timeinterval::Float64 # minutes between timeframe slices
    timeframe::Vector{TimeSlice}
    obstacleindices::Vector{Point2{Int}}
    startindex::Int
    start::Point2{Int}
    finish::Point2{Int}
    allowrepeatvisits::Bool
end

"""
    mutable struct TimedPath

duration: minutes total to travel the path
path: vector of Cartesian indices of points in grid for path to travel
"""
mutable struct TimedPath
    duration::Float64
    path::Vector{Point2{Int}}
end

"""
    closestpoint(p, mat)

Get the closest GridPoint in matrix mat to a given position p.
p: Cartesian indices of a Position (latitude, longitude in degrees) in grid of GridPoints
mat: matrix of Gridpoints
"""
closestpoint(p, mat) = findmin(gp -> haversine(p[1], p[2], gp.pt[1], gp.pt[2])[1], mat)[2]

function surround(p, mat, excluded)
    neighbors = Point2{Int}[(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)]
    (xmax, ymax) = size(mat)
    return filter(q -> 1 <= q[1] <= xmax && 1 <= q[2] <= ymax && !(q in excluded),
        [x + p for x in neighbors])
end

"""
    function minimumtimeroute(rp::RoutingProblem, sp::SailingPolar, verbose=false)

Get the route (as a TimedPath) that minimizes time from start to finish for a given
RoutingProblem (sea parameters) given sailing polar data (ship parameters).
"""
function minimumtimeroute(rp::RoutingProblem, sp::SailingPolar, verbose=false)
    timedpaths = [TimedPath(0.0, [rp.start])]
    completed, mintime, minpath = false, 1000.0, TimedPath(1000.0, [])
    for i in 1:1000
        newpaths = TimedPath[]
        verbose && println("Checking ", length(timedpaths), " paths of length ",
            length(timedpaths[1].path) - 1)
        for tpath in timedpaths
            if tpath.path[end] == rp.finish
                completed = true
                push!(newpaths, tpath)
            else
                p1 = tpath.path[end]
                slice = rp.timeframe[div(Int(round(tpath.duration)),
                                     Int(round(rp.timeinterval))) %
                                     length(rp.timeframe) + 1]
                for p2 in surround([p1[1], p1[2]], slice, rp.obstacleindices)
                    !rp.allowrepeatvisits && p2 in tpath.path && continue
                    gp1, gp2 = slice[p1[1], p1[2]], slice[p2[1], p2[2]]
                    lat1, lon1, lat2, lon2 = gp1.pt[1], gp1.pt[2], gp2.pt[1], gp2.pt[2]
                    t = sailsegmenttime(sp, gp1.sp, lat1, lon1, lat2, lon2)
                    path = deepcopy(tpath.path)
                    push!(path, p2)
                    push!(newpaths, TimedPath(tpath.duration + t, path))
                end
            end
        end
        timedpaths = unique(newpaths)
        if completed
            mindur = minimum(map(x -> x.duration, timedpaths))
            finished = filter(x -> x.path[end] == rp.finish, timedpaths)
            minfindur, idx = findmin(map(x -> x.duration, finished))
            verbose && println("Current finished minimum: ", finished[idx], ", others $mindur")
            if mindur == minfindur
                minpath = finished[idx]
                break
            end
        end
    end
    return minpath
end

end # module

using GeometryTypes
using .SailingNavigation, .SailingPolars

#=
The data is selected so the best time path is slightly longer than the
shortest length path. The forbidden regions are x, representing land or reef.
The allowed sailing points are . and start and finish are S and F.

x  .  .  F  .  .  x  .  x
.  .  .  .  .  .  .  x  x
x  .  .  x  x  x  .  .  .
.  .  x  x  x  x  .  x  x
x  .  .  .  x  x  .  x  .
x  .  .  .  x  x  .  x  .
.  .  .  .  x  .  .  x  .
x  .  .  .  .  .  .  x  .
.  .  .  S  .  .  .  .  .
=#
const forbidden = Point2{Int}.([
    [1, 8], [2, 1], [2, 8], [3, 5], [3, 8], [4, 1], [4, 5], [4, 6], [4, 8], [5, 1],
    [5, 5], [5, 6], [5, 8], [6, 3], [6, 4], [6, 5], [6, 6], [6, 8], [6, 9], [7, 1],
    [7, 4], [7, 5], [7, 6], [8, 8], [8, 9], [9, 1], [9, 7], [9, 9],
])

# Create regional wind patterns on the map.
function surfacebylongitude(lon)
    return lon < -155.03 ? SurfaceParameters(-5.0, 8, 150, 0.5) :
           lon < -155.99 ? SurfaceParameters(-90.0, 20, 150, 0.4) :
                           SurfaceParameters(180.0, 25, 150, 0.3)
end

# Vary wind speeds over time.
function mutatetimeslices!(slices)
    for (i, slice) in enumerate(slices), x in slice
        x.sp = SurfaceParameters(x.sp.winddeg, x.sp.windkts * (1 + 0.002 * i),
            x.sp.currentdeg, x.sp.currentkts)
    end
end


const startpos = Point2{Int}(1, 4)
const endpos = Point2{Int}(9, 4)
const pmat  = [Position(19.78 - 1/60 + i/60, -155.0 - 5/60 + j/60) for i in 0:8, j in 0:8]
const gpoints = map(pt -> GridPoint(pt, surfacebylongitude(lon(pt))), pmat)
const slices = [deepcopy(gpoints) for _ in 1:200]
mutatetimeslices!(slices)

const routeprob = RoutingProblem(10.0, slices, forbidden, 1, startpos, endpos, false)
const filename = "polar.csv"
const sp = getpolardata(filename)
const tp = minimumtimeroute(routeprob, sp)

println("The route taking the least time found was:\n    ", tp.path,
    "\nwhich has duration $(div(tp.duration, 60)) hours, $(rem(tp.duration, 60)) minutes.")

The polar CSV file used for this solution, named polar.csv, is as follows. Note that this is a very detailed polar, chosen to stress the testing of the code. Most polar files are far smaller, with fewer choices for angle and wind speed.

TWA\TWS;0;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18;19;20;21;22;23;24;25;26;27;28;29;30;35;40;60;70
40;0;0.53;0.54;0.49;0.4;0.31;0.21;0.16;0.11;0.08;0.05;0.03;0.02;0.01;0;0;0;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.05;-0.1;-0.11
41;0;0.61;0.62;0.56;0.47;0.36;0.25;0.19;0.14;0.1;0.07;0.04;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;0;0;-0.04;-0.09;-0.1
44;0;0.89;0.91;0.82;0.69;0.56;0.42;0.33;0.24;0.18;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;-0.02;-0.06;-0.06
45;0;0.99;1.02;0.92;0.78;0.64;0.49;0.39;0.29;0.22;0.15;0.1;0.07;0.04;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.05;-0.05
46;0;1.11;1.14;1.02;0.87;0.73;0.57;0.45;0.35;0.26;0.18;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.04;-0.05
47;0;1.23;1.25;1.14;0.97;0.82;0.66;0.53;0.41;0.31;0.22;0.15;0.1;0.07;0.04;0.02;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.01;-0.03;-0.04
48;0;1.37;1.37;1.26;1.08;0.93;0.76;0.61;0.48;0.36;0.26;0.19;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.02;-0.03
49;0;1.5;1.5;1.39;1.2;1.05;0.87;0.71;0.56;0.42;0.31;0.22;0.15;0.1;0.07;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.02;-0.02
50;0;1.65;1.64;1.52;1.33;1.18;1;0.81;0.65;0.49;0.37;0.26;0.19;0.13;0.08;0.05;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;-0.01;-0.02
51;0;1.79;1.77;1.67;1.46;1.32;1.13;0.92;0.74;0.57;0.43;0.31;0.22;0.15;0.1;0.07;0.05;0.03;0.02;0.02;0.01;0.01;0;0;0;0;0;0;0;0;-0.01;-0.02
53;0;2.1;2.07;1.99;1.76;1.62;1.4;1.14;0.95;0.74;0.57;0.43;0.31;0.22;0.16;0.1;0.08;0.06;0.04;0.03;0.02;0.01;0.01;0.01;0;0;0;0;0;0;-0.01;-0.01
54;0;2.26;2.22;2.16;1.92;1.78;1.55;1.28;1.06;0.84;0.65;0.5;0.37;0.27;0.19;0.13;0.1;0.07;0.06;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;-0.01
55;0;2.43;2.39;2.34;2.09;1.95;1.7;1.42;1.18;0.95;0.74;0.57;0.43;0.32;0.23;0.16;0.12;0.09;0.07;0.05;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;-0.01
60;0;3.29;3.33;3.33;3.08;2.93;2.64;2.29;1.98;1.66;1.36;1.1;0.88;0.68;0.53;0.39;0.32;0.26;0.21;0.17;0.13;0.1;0.07;0.05;0.04;0.03;0.02;0.01;0;0;0;0
70;0;5.2;5.53;5.74;5.59;5.5;5.22;4.84;4.46;3.94;3.51;3.08;2.65;2.26;1.9;1.55;1.38;1.22;1.06;0.92;0.78;0.66;0.55;0.46;0.37;0.3;0.24;0.18;0.03;0;0;0
80;0;6.8;7.43;7.97;8.02;8.23;8.34;8.2;7.9;7.37;6.91;6.43;5.9;5.32;4.72;4.12;3.83;3.55;3.25;2.96;2.67;2.4;2.13;1.88;1.65;1.43;1.22;1.04;0.37;0.09;0.01;0
90;0;7.59;8.5;9.4;9.73;10.4;11.16;11.53;11.56;11.3;11.05;10.77;10.44;9.83;9.07;8.34;8;7.65;7.27;6.88;6.46;6.04;5.61;5.15;4.74;4.33;3.88;3.51;1.72;0.67;0.12;0.03
100;0;7.34;8.25;9.16;9.86;10.5;11.95;12.79;13.5;14.02;14.4;14.37;14.5;14.4;13.92;13.52;13.19;12.79;12.51;12.1;11.66;11.22;10.77;10.26;9.72;9.2;8.58;8.01;4.87;2.51;0.7;0.23
110;0;7.09;7.97;8.84;9.74;10.09;11.85;12.75;13.84;14.99;16.02;16.33;17.1;17.83;17.99;18.32;18.14;17.81;17.84;17.6;17.3;17.05;16.83;16.53;16.03;15.59;15.03;14.37;10.26;6.41;2.32;0.86
120;0;6.59;7.42;8.3;9.1;9.56;10.83;11.6;13.1;13.87;14.66;15.75;16.67;17.63;18.43;19.62;20.17;20.6;21.12;21.55;21.75;21.91;22.07;21.9;21.58;21.29;20.92;20.29;16.47;12.03;5.49;2.26
129;0;6.14;6.93;7.83;8.52;9.09;9.89;10.57;12.42;12.87;13.43;15.23;16.16;17.08;18.07;19.48;20.35;21.22;21.93;22.85;23.44;23.98;24.55;24.59;24.55;24.51;24.46;24;21.56;17.75;9.64;4.25
130;0;6.07;6.87;7.76;8.44;9.02;9.8;10.48;12.29;12.73;13.27;15.08;16.03;16.97;17.96;19.36;20.25;21.15;21.88;22.82;23.44;24.03;24.6;24.66;24.68;24.67;24.64;24.24;22;18.33;10.11;4.5
135;0;5.72;6.57;7.36;8.02;8.65;9.38;10.11;11.52;11.97;12.55;13.85;15.31;16.31;17.33;18.54;19.48;20.35;21.28;22.3;23.08;24.09;24.63;24.69;24.78;24.79;24.91;24.82;23.74;20.98;12.39;5.78
136;0;5.66;6.5;7.28;7.93;8.57;9.3;10.04;11.34;11.82;12.42;13.62;15.06;16.17;17.2;18.35;19.29;20.15;21.12;22.15;22.96;24.07;24.6;24.67;24.76;24.75;24.85;24.81;23.98;21.45;12.8;6.03
139;0;5.42;6.31;6.92;7.67;8.34;9.08;9.86;10.86;11.32;12.03;12.99;14.3;15.73;16.76;17.76;18.71;19.53;20.6;21.66;22.54;23.92;24.44;24.53;24.64;24.58;24.65;24.67;24.47;22.68;13.79;6.73
140;0;5.35;6.22;6.79;7.59;8.26;9;9.8;10.72;11.16;11.89;12.79;14.06;15.5;16.62;17.57;18.51;19.32;20.43;21.49;22.4;23.84;24.36;24.46;24.58;24.51;24.57;24.61;24.56;23.02;14.08;6.96
141;0;5.29;6.12;6.67;7.48;8.18;8.93;9.74;10.57;11.02;11.77;12.62;13.82;15.26;16.47;17.38;18.32;19.04;20.28;21.31;22.07;23.53;24;24.21;24.29;24.43;24.48;24.55;24.61;23.33;14.31;7.18
142;0;5.23;6.02;6.57;7.39;8.1;8.86;9.67;10.43;10.88;11.64;12.45;13.59;15.03;16.24;17.14;18.06;18.77;19.98;21.01;21.75;23.18;23.65;23.86;23.95;24.34;24.39;24.48;24.61;23.61;14.54;7.4
143;0;5.16;5.93;6.45;7.3;8;8.78;9.54;10.27;10.75;11.5;12.28;13.36;14.8;16.01;16.9;17.81;18.5;19.69;20.72;21.43;22.84;23.31;23.52;23.61;24.05;24.27;24.41;24.57;23.85;14.8;7.6
144;0;5.09;5.83;6.33;7.23;7.92;8.66;9.41;10.13;10.62;11.39;12.13;13.13;14.57;15.78;16.65;17.56;18.24;19.41;20.43;21.12;22.5;22.97;23.19;23.28;23.73;24.08;24.33;24.49;24.04;15;7.8
145;0;5.02;5.73;6.23;7.15;7.85;8.55;9.28;9.98;10.51;11.27;11.98;12.92;14.35;15.56;16.42;17.31;17.97;19.13;20.14;20.82;22.17;22.64;22.87;22.96;23.42;23.81;24.23;24.41;24.19;15.14;7.98
146;0;4.96;5.64;6.12;7.07;7.77;8.43;9.15;9.84;10.38;11.16;11.83;12.71;14.12;15.35;16.19;17.07;17.72;18.86;19.86;20.51;21.84;22.31;22.56;22.65;23.12;23.48;23.94;24.33;24.3;15.3;8.16
148;0;4.82;5.45;5.91;6.9;7.59;8.21;8.89;9.55;10.14;10.89;11.55;12.29;13.7;14.92;15.74;16.6;17.23;18.32;19.3;19.91;21.2;21.67;21.95;22.05;22.53;22.87;23.38;24.13;24.39;15.52;8.46
149;0;4.76;5.35;5.81;6.78;7.49;8.09;8.78;9.42;10.01;10.76;11.41;12.1;13.48;14.71;15.52;16.36;16.98;18.06;19.03;19.63;20.89;21.37;21.67;21.77;22.26;22.61;23.12;23.98;24.37;15.57;8.58
150;0;4.69;5.26;5.7;6.67;7.37;7.96;8.64;9.26;9.86;10.6;11.24;11.89;13.26;14.48;15.29;16.11;16.73;17.79;18.74;19.33;20.55;21.04;21.37;21.48;21.98;22.34;22.83;23.69;24.11;15.6;8.67
155;0;4.33;4.74;5.16;6.16;6.79;7.33;7.96;8.51;9.15;9.81;10.4;10.85;12.14;13.37;14.1;14.87;15.48;16.42;17.3;17.8;18.88;19.39;19.86;20;20.54;20.97;21.45;22.25;22.77;15.38;8.89
160;0;4.09;4.41;4.83;5.77;6.39;6.94;7.55;8.04;8.67;9.28;9.83;10.24;11.46;12.69;13.39;14.11;14.73;15.6;16.41;16.87;17.85;18.4;18.97;19.15;19.72;20.2;20.65;21.35;21.84;14.95;8.74
162;0;4;4.29;4.69;5.62;6.23;6.77;7.38;7.86;8.48;9.07;9.6;10;11.18;12.42;13.1;13.81;14.43;15.27;16.06;16.5;17.43;18;18.62;18.81;19.39;19.89;20.33;20.99;21.48;14.76;8.61
168;0;3.74;3.93;4.35;5.15;5.75;6.31;6.93;7.34;7.92;8.45;8.95;9.35;10.44;11.68;12.32;12.99;13.63;14.39;15.11;15.5;16.31;16.94;17.7;17.92;18.53;19.08;19.49;19.99;20.46;14.34;8.3
170;0;3.69;3.85;4.27;5.04;5.65;6.22;6.82;7.23;7.8;8.31;8.8;9.22;10.27;11.51;12.15;12.81;13.45;14.19;14.9;15.28;16.06;16.7;17.51;17.73;18.34;18.91;19.31;19.77;20.22;14.24;8.24
174;0;3.57;3.69;4.11;4.83;5.43;6.01;6.62;7;7.55;8.03;8.5;8.93;9.95;11.19;11.81;12.45;13.11;13.81;14.48;14.84;15.57;16.24;17.11;17.35;17.98;18.57;18.95;19.33;19.77;14.03;8.13
180;0;3.51;3.6;4.03;4.71;5.31;5.91;6.51;6.88;7.41;7.88;8.33;8.79;9.78;11.02;11.63;12.26;12.93;13.61;14.26;14.61;15.31;15.99;16.9;17.15;17.79;18.39;18.77;19.09;19.52;13.87;8.07
Output:
The route taking the least time found was:
    Point{2,Int64}[[1, 4], [1, 5], [2, 6], [3, 7], [4, 7], [5, 7], [6, 7], [7, 7], [8, 6], [8, 5], [9, 4]]
which has duration 4.0 hours, 43.697879668707344 minutes.

Nim[edit]

Translation of: Go

The Go version runs in about 44 seconds on my computer. This Nim version, compiled in release mode (which includes runtime checks), runs in 22 seconds (18 seconds if link time optimization is activated). When compiled without checks (“danger” mode), it runs in 15.5 seconds.

import hashes, math, parsecsv, sequtils, sets, strutils, sugar

type
  MatrixF = seq[seq[float]]
  Pred = float -> bool

  # Structure that represents a polar CSV file's data.
  # Note 0 degrees is directly into the wind, 180 degrees is directly downwind.
  SailingPolar = object
    winds: seq[float]     # Vector of windspeeds.
    degrees: seq[float]   # Vector of angles in degrees of direction relative to the wind.
    speeds: MatrixF       # Matrix of sailing speeds indexed by wind velocity and angle of boat to wind.

  # Structure that represents wind and surface current direction and velocity for a given position.
  # Angles in degrees, velocities in knots.
  SurfaceParameters = tuple[windDeg, windKts, currentDeg, currentKts: float]


proc getPolarData(filename: string): SailingPolar =
  ## Reads a sailing polar CSV file and returns a SailingPolar struct containing the file data.
  ## A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma.
  ## The first line of file contains labels for the wind velocities that make up columns, and
  ## the first entry of each row makes up a column of angle of sailing direction from wind in degrees.
  var parser: CsvParser
  parser.open(filename, separator = ';')
  parser.readHeaderRow()
  for col in 1..parser.headers.high:
    result.winds.add parser.headers[col].parseFloat()
  while parser.readRow():
    if parser.row.len == 0: break # Ignore final blank line if there is one.
    result.degrees.add parser.row[0].parseFloat()
    result.speeds.add @[]
    for col in 1..parser.row.high:
      result.speeds[^1].add parser.row[col].parseFloat()


const R = 6372800.0 # Earth's approximate radius in meters.

template sind(d: float): float = sin(degToRad(d))
template cosd(d: float): float = cos(degToRad(d))
template asind(x: float): float = radToDeg(arcsin(x))
template atand(x, y: float): float = radToDeg(arctan2(x, y))


func haversine(lat1, long1, lat2, long2: float): (float, float) =
  ## Calculates the Haversine function for two points on the Earth's surface.
  ## Given two latitude, longitude pairs in degrees for a point on the Earth,
  ## get distance in meters and the initial direction of travel in degrees for
  ## movement from point 1 to point 2.
  let dlat = lat2 - lat1
  let dlong = long2 - long1
  let a = sind(dlat/2)^2 + cosd(lat1) * cosd(lat2) * sind(dlong/2)^2
  let c = 2 * asind(sqrt(a))
  var theta = atand(sind(dlong) * cosd(lat2),
                    cosd(lat1) * sind(lat2) - sind(lat1) * cosd(lat2) * cosd(dlong))
  theta = (theta + 360) mod 360
  result = (R * c * 0.5399565, theta)


func findFirst(a: seq[float]; p: Pred): int =
  ## Returns the index of the first element of 'a' for which 'p' returns true or -1 otherwise.
  for i in 0..a.high:
    if p(a[i]): return i
  result = -1


func findLast(a: seq[float]; p: Pred): int =
  ## Returns the index of the last element of 'a' for which 'p' returns true or -1 otherwise.
  for i in countdown(a.high, 0):
    if p(a[i]): return i
  result = -1


func boatSpeed(sp: SailingPolar; pointOfSail, windSpeed: float): float =
  ## Calculate the expected sailing speed in a specified direction in knots,
  ## given sailing polar data, a desired point of sail in degrees, and wind speed in knots.
  let
    udeg = sp.degrees.findLast(t => t <= pointOfSail)
    odeg = sp.degrees.findFirst(t => t >= pointOfSail)
    uvel = sp.winds.findLast(t => t <= windSpeed)
    ovel = sp.winds.findFirst(t => t >= windSpeed)
  if udeg == -1 or odeg == -1 or uvel == -1 or ovel == -1: return -1
  let frac = if odeg == udeg and uvel == ovel:
               1.0
             elif odeg == udeg:
               (windSpeed - sp.winds[uvel]) / (sp.winds[ovel] - sp.winds[uvel])
             elif uvel == ovel:
               (pointOfSail - sp.degrees[udeg]) / (sp.degrees[odeg] - sp.degrees[udeg])
             else:
               ((pointOfSail - sp.degrees[udeg]) / (sp.degrees[odeg] - sp.degrees[udeg]) +
               (windSpeed - sp.winds[uvel]) / (sp.winds[ovel] - sp.winds[uvel])) / 2
  result = sp.speeds[udeg][uvel] + frac * (sp.speeds[odeg][ovel] - sp.speeds[udeg][uvel])


func sailingSpeed(sp: SailingPolar; azimuth, pointos, ws: float): float =
  ## Calculates the expected net boat speed in a desired direction versus the wind ('azimuth').
  ## This is generally different from the actual boat speed in its actual direction.
  ## Directions are in degrees ('pointos' is point of sail the ship direction from the wind),
  ## and velocity of wind ('ws') is in knots.
  sp.boatSpeed(pointos, ws) * cosd(abs(pointos - azimuth))


func bestVectorSpeed(sp: SailingPolar;
                     dirTravel, dirWind, windSpeed, dirCur, velCur: float): (float, float) =
  ## Calculates the net direction and velocity of a sailing ship.
  ## Arguments are sailing polar data, direction of travel in degrees from north, wind direction in
  ## degrees from north, wind velocity in knots, surface current direction in degrees, and
  ## current velocity in knots.
  var azimuth = (dirTravel - dirWind) mod 360
  if azimuth < 0: azimuth += 360
  if azimuth > 180: azimuth = 360 - azimuth

  var vmg = sp.boatSpeed(azimuth, windSpeed)
  var other = -1.0
  var idx = -1
  for i, d in sp.degrees:
    let ss = sp.sailingSpeed(azimuth, d, windSpeed)
    if ss > other:
      other = ss
      idx = i
  if other > vmg:
    azimuth = sp.degrees[idx]
    vmg = other

  let
    dirChosen = dirWind + azimuth
    wx = vmg * sind(dirChosen)
    wy = vmg * cosd(dirChosen)
    curX = velCur * sind(dirCur)
    curY = velCur * cosd(dirCur)
  result = (atand(wy + curY, wx + curX), sqrt((wx + curX)^2 + (wy + curY)^2))


func sailSegmentTime(sp: SailingPolar; p: SurfaceParameters;
                     lat1, long1, lat2, long2: float): float =
  ## Calculates the trip time in minutes from (lat1, long1) to the destination (lat2, long2).
  ## Uses the data in SurfaceParameters for wind and current velocity and direction.
  let (distance, dir) = haversine(lat1, long1, lat2, long2)
  let (_, vel) = sp.bestVectorSpeed(dir, p.windDeg, p.windKts, p.currentDeg, p.currentKts)
  ## minutes/s * m / (knots * (m/s / knot)) = minutes
  result = (1 / 60) * distance / (vel * 1.94384)


# Structure that represents a point in 2-D space.
type Point2 = tuple[x, y: int]

func `+`(p1, p2: Point2): Point2 = (p1.x + p2.x, p1.y + p2.y)

func `$`(p: Point2): string = "($1, $2)".format(p.x, p.y)


type

  # Tuple that consists of a tuple of latitude and longitude in degrees.
  # NB: This uses latitude (often considered to be y) first then longitude (often considered to be x).
  # This latitude, then longitude ordering is as per ISO 6709 (en.wikipedia.org/wiki/ISO_6709).
  Position = tuple[lat, long: float]

  # Tuple that represents a Position with the SurfaceParameters of wind and current at the Position.
  GridPoint = tuple[pt: Position; sp: SurfaceParameters]

  MatrixG = seq[seq[GridPoint]]

  # Type alias for a matrix of GridPoints, each Position point with their SurfaceParameters.
  # A vector of TimeSlice can give the surface characteristics for an ocean region over time.
  TimeSlice = MatrixG

  # Structure that represents a routing problem.
  RoutingProblem = object
    timeInterval: float           # The minutes duration for each TimeSlice.
    timeFrame: seq[TimeSlice]     # A vector of sequential timeslices for the ocean region.
    obstacleIndices: seq[Point2]  # The cartesian indices in each TimeSlice for obstacles
                                  # such as land or shoals, where the ship may not go.
    startIndex: int               # The TimeSlice position for time of starting.
    start: Point2                 # Starting location on grid of GridPoints.
    finish: Point2                # Destination / finish location on grid of GridPoints.
    allowRepeatVisits: bool       # Whether the vessel may overlap its prior path, usually false.

  # Structure that represents a timed path.
  TimedPath = object
    duration: float     # Minutes total to travel the path.
    path: seq[Point2]   # Vector of cartesian indices of points in grid for path to travel.


func hash(t: TimedPath): Hash =
  ## Hash function to allow building a set of TimedPath values.
  result = t.duration.hash !& t.path.hash
  result = !$result


const Neighbors: seq[Point2] = @[(-1, -1), (-1,  0), (-1, 1), (0, -1),
                                 ( 0,  1), ( 1, -1), ( 1, 0), (1,  1)]

func surround(p: Point2; mat: TimeSlice; excluded: openArray[Point2]): seq[Point2] =
  ## Returns a list of points surrounding 'p' which are not otherwise excluded.
  let xmax = mat.len
  let ymax = mat[0].len
  for x in Neighbors:
    let q = p + x
    if q.x >= 0 and q.x < xmax and q.y >= 0 and q.y < ymax and q notin excluded:
      result.add q


proc minimumTimeRoute(rp: RoutingProblem; sp: SailingPolar; verbose: bool): TimedPath =
  ## Get the route (as a TimedPath) that minimizes time from start to finish for a given
  ## RoutingProblem (sea parameters) given sailing polar data (ship parameters).

  var timedPaths = @[TimedPath(duration: 0, path: @[rp.start])]
  var completed = false
  result = TimedPath(duration: 1000)
  for _ in 1..1000:

    var newPaths: seq[TimedPath]
    if verbose:
      echo "Checking $1 paths of length $2".format(timedPaths.len, timedPaths[0].path.len)
    for tpath in timedPaths:
      if tpath.path[^1] == rp.finish:
        completed = true
        newPaths.add tpath
      else:
        let p1 = tpath.path[^1]
        let num = tpath.duration.toInt
        let den = rp.timeInterval.toInt
        let slice = rp.timeFrame[num div den mod rp.timeFrame.len]
        for p2 in p1.surround(slice, rp.obstacleIndices):
          if not rp.allowRepeatVisits and p2 in tpath.path:
            continue
          let gp1 = slice[p1.x][p1.y]
          let gp2 = slice[p2.x][p2.y]
          let (lat1, long1) = gp1.pt
          let (lat2, long2) = gp2.pt
          let t = sp.sailSegmentTime(gp1.sp, lat1, long1, lat2, long2)
          let path = tpath.path & p2
          newPaths.add TimedPath(duration: tpath.duration + t, path: path)

    timedPaths = newPaths.toHashSet().toSeq()
    if completed:
      var durations = collect(newSeq, for t in timedPaths: t.duration)
      let minDur = min(durations)
      let finished = collect(newSeq):
                       for t in timedPaths:
                         if t.path[^1] == rp.finish: t
      durations = collect(newSeq, for f in finished: f.duration)
      let idx = minIndex(durations)
      let minFinDur = durations[idx]
      if verbose:
        echo "Current finished minimum: $1, others $2".format(finished[idx], minDur)
      if minDur == minFinDur:
        result = finished[idx]
        break


#[ The data is selected so the best time path is slightly longer than the
   shortest length path. The forbidden regions are x, representing land or reef.
   The allowed sailing points are . and start and finish are S and F.

   x  .  .  F  .  .  x  .  x
   .  .  .  .  .  .  .  x  x
   x  .  .  x  x  x  .  .  .
   .  .  x  x  x  x  .  x  x
   x  .  .  .  x  x  .  x  .
   x  .  .  .  x  x  .  x  .
   .  .  .  .  x  .  .  x  .
   x  .  .  .  .  .  .  x  .
   .  .  .  S  .  .  .  .  .
]#


const Forbidden: seq[Point2] = @[(0, 7), (1, 0), (1, 7), (2, 4), (2, 7), (3, 0), (3, 4),
                                 (3, 5), (3, 7), (4, 0), (4, 4), (4, 5), (4, 7), (5, 2),
                                 (5, 3), (5, 4), (5, 5), (5, 7), (5, 8), (6, 0), (6, 3),
                                 (6, 4), (6, 5), (7, 7), (7, 8), (8, 0), (8, 6), (8, 8)]


func surfaceByLongitude(long: float): SurfaceParameters =
  ## Create regional wind patterns on the map.
  if long < -155.03:
    (-5.0, 8.0, 150.0, 0.5)
  elif long < -155.99:
    (-90.0, 20.0, 150.0, 0.4)
  else:
    (180.0, 25.0, 150.0, 0.3)


func mutateTimeSlices(slices: var seq[TimeSlice]) =
  var i = 1
  for slice in slices.mitems:
    for j in 0..slice.high:
      for x in slice[j].mitems:
        x.sp = (x.sp.windDeg, x.sp.windKts * (1 + 0.002 * float64(i)),
                x.sp.currentDeg, x.sp.currentKts)
    inc i


let startPos: Point2 = (0, 3)
let endPos: Point2 = (8, 3)
var slices = newSeq[MatrixG](200)

for s in 0..slices.high:
  var gpoints = newSeq[seq[GridPoint]](9)
  for i in 0..<9:
    gpoints[i].setLen(9)
    for j in 0..<9:
      let pt: Position = (19.78 - 1/60 + i/60, -155.0 - 5/60 + j/60)
      gpoints[i][j] = (pt, surfaceByLongitude(pt.long))
  slices[s] = move(gpoints)

slices.mutateTimeSlices()
let routeProb = RoutingProblem(timeInterval: 10, timeFrame: slices,
                               obstacleIndices: Forbidden, startIndex: 0,
                               start: startPos, finish: endPos, allowRepeatVisits: false)
let fileName = "polar.csv"
let sp = getPolarData(fileName)
let tp = routeProb.minimumTimeRoute(sp, false)
echo "The route taking the least time found was:"
echo tp.path
echo "which has duration ", int(tp.duration / 60), " hours ", toInt(tp.duration mod 60), " minutes."
Output:
The route taking the least time found was:
@[(0, 3), (0, 4), (1, 5), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (7, 5), (7, 4), (8, 3)]
which has duration 4 hours 44 minutes.

Phix[edit]

Translation of: Go
--
-- demo\rosetta\Weather_Routing.exw
-- ================================
--
with javascript_semantics
constant polar_csv = """
TWA\TWS;0;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18;19;20;21;22;23;24;25;26;27;28;29;30;35;40;60;70
40;0;0.53;0.54;0.49;0.4;0.31;0.21;0.16;0.11;0.08;0.05;0.03;0.02;0.01;0;0;0;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.05;-0.1;-0.11
41;0;0.61;0.62;0.56;0.47;0.36;0.25;0.19;0.14;0.1;0.07;0.04;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;0;0;-0.04;-0.09;-0.1
44;0;0.89;0.91;0.82;0.69;0.56;0.42;0.33;0.24;0.18;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;-0.02;-0.06;-0.06
45;0;0.99;1.02;0.92;0.78;0.64;0.49;0.39;0.29;0.22;0.15;0.1;0.07;0.04;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.05;-0.05
46;0;1.11;1.14;1.02;0.87;0.73;0.57;0.45;0.35;0.26;0.18;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.04;-0.05
47;0;1.23;1.25;1.14;0.97;0.82;0.66;0.53;0.41;0.31;0.22;0.15;0.1;0.07;0.04;0.02;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.01;-0.03;-0.04
48;0;1.37;1.37;1.26;1.08;0.93;0.76;0.61;0.48;0.36;0.26;0.19;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.02;-0.03
49;0;1.5;1.5;1.39;1.2;1.05;0.87;0.71;0.56;0.42;0.31;0.22;0.15;0.1;0.07;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.02;-0.02
50;0;1.65;1.64;1.52;1.33;1.18;1;0.81;0.65;0.49;0.37;0.26;0.19;0.13;0.08;0.05;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;-0.01;-0.02
51;0;1.79;1.77;1.67;1.46;1.32;1.13;0.92;0.74;0.57;0.43;0.31;0.22;0.15;0.1;0.07;0.05;0.03;0.02;0.02;0.01;0.01;0;0;0;0;0;0;0;0;-0.01;-0.02
53;0;2.1;2.07;1.99;1.76;1.62;1.4;1.14;0.95;0.74;0.57;0.43;0.31;0.22;0.16;0.1;0.08;0.06;0.04;0.03;0.02;0.01;0.01;0.01;0;0;0;0;0;0;-0.01;-0.01
54;0;2.26;2.22;2.16;1.92;1.78;1.55;1.28;1.06;0.84;0.65;0.5;0.37;0.27;0.19;0.13;0.1;0.07;0.06;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;-0.01
55;0;2.43;2.39;2.34;2.09;1.95;1.7;1.42;1.18;0.95;0.74;0.57;0.43;0.32;0.23;0.16;0.12;0.09;0.07;0.05;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;-0.01
60;0;3.29;3.33;3.33;3.08;2.93;2.64;2.29;1.98;1.66;1.36;1.1;0.88;0.68;0.53;0.39;0.32;0.26;0.21;0.17;0.13;0.1;0.07;0.05;0.04;0.03;0.02;0.01;0;0;0;0
70;0;5.2;5.53;5.74;5.59;5.5;5.22;4.84;4.46;3.94;3.51;3.08;2.65;2.26;1.9;1.55;1.38;1.22;1.06;0.92;0.78;0.66;0.55;0.46;0.37;0.3;0.24;0.18;0.03;0;0;0
80;0;6.8;7.43;7.97;8.02;8.23;8.34;8.2;7.9;7.37;6.91;6.43;5.9;5.32;4.72;4.12;3.83;3.55;3.25;2.96;2.67;2.4;2.13;1.88;1.65;1.43;1.22;1.04;0.37;0.09;0.01;0
90;0;7.59;8.5;9.4;9.73;10.4;11.16;11.53;11.56;11.3;11.05;10.77;10.44;9.83;9.07;8.34;8;7.65;7.27;6.88;6.46;6.04;5.61;5.15;4.74;4.33;3.88;3.51;1.72;0.67;0.12;0.03
100;0;7.34;8.25;9.16;9.86;10.5;11.95;12.79;13.5;14.02;14.4;14.37;14.5;14.4;13.92;13.52;13.19;12.79;12.51;12.1;11.66;11.22;10.77;10.26;9.72;9.2;8.58;8.01;4.87;2.51;0.7;0.23
110;0;7.09;7.97;8.84;9.74;10.09;11.85;12.75;13.84;14.99;16.02;16.33;17.1;17.83;17.99;18.32;18.14;17.81;17.84;17.6;17.3;17.05;16.83;16.53;16.03;15.59;15.03;14.37;10.26;6.41;2.32;0.86
120;0;6.59;7.42;8.3;9.1;9.56;10.83;11.6;13.1;13.87;14.66;15.75;16.67;17.63;18.43;19.62;20.17;20.6;21.12;21.55;21.75;21.91;22.07;21.9;21.58;21.29;20.92;20.29;16.47;12.03;5.49;2.26
129;0;6.14;6.93;7.83;8.52;9.09;9.89;10.57;12.42;12.87;13.43;15.23;16.16;17.08;18.07;19.48;20.35;21.22;21.93;22.85;23.44;23.98;24.55;24.59;24.55;24.51;24.46;24;21.56;17.75;9.64;4.25
130;0;6.07;6.87;7.76;8.44;9.02;9.8;10.48;12.29;12.73;13.27;15.08;16.03;16.97;17.96;19.36;20.25;21.15;21.88;22.82;23.44;24.03;24.6;24.66;24.68;24.67;24.64;24.24;22;18.33;10.11;4.5
135;0;5.72;6.57;7.36;8.02;8.65;9.38;10.11;11.52;11.97;12.55;13.85;15.31;16.31;17.33;18.54;19.48;20.35;21.28;22.3;23.08;24.09;24.63;24.69;24.78;24.79;24.91;24.82;23.74;20.98;12.39;5.78
136;0;5.66;6.5;7.28;7.93;8.57;9.3;10.04;11.34;11.82;12.42;13.62;15.06;16.17;17.2;18.35;19.29;20.15;21.12;22.15;22.96;24.07;24.6;24.67;24.76;24.75;24.85;24.81;23.98;21.45;12.8;6.03
139;0;5.42;6.31;6.92;7.67;8.34;9.08;9.86;10.86;11.32;12.03;12.99;14.3;15.73;16.76;17.76;18.71;19.53;20.6;21.66;22.54;23.92;24.44;24.53;24.64;24.58;24.65;24.67;24.47;22.68;13.79;6.73
140;0;5.35;6.22;6.79;7.59;8.26;9;9.8;10.72;11.16;11.89;12.79;14.06;15.5;16.62;17.57;18.51;19.32;20.43;21.49;22.4;23.84;24.36;24.46;24.58;24.51;24.57;24.61;24.56;23.02;14.08;6.96
141;0;5.29;6.12;6.67;7.48;8.18;8.93;9.74;10.57;11.02;11.77;12.62;13.82;15.26;16.47;17.38;18.32;19.04;20.28;21.31;22.07;23.53;24;24.21;24.29;24.43;24.48;24.55;24.61;23.33;14.31;7.18
142;0;5.23;6.02;6.57;7.39;8.1;8.86;9.67;10.43;10.88;11.64;12.45;13.59;15.03;16.24;17.14;18.06;18.77;19.98;21.01;21.75;23.18;23.65;23.86;23.95;24.34;24.39;24.48;24.61;23.61;14.54;7.4
143;0;5.16;5.93;6.45;7.3;8;8.78;9.54;10.27;10.75;11.5;12.28;13.36;14.8;16.01;16.9;17.81;18.5;19.69;20.72;21.43;22.84;23.31;23.52;23.61;24.05;24.27;24.41;24.57;23.85;14.8;7.6
144;0;5.09;5.83;6.33;7.23;7.92;8.66;9.41;10.13;10.62;11.39;12.13;13.13;14.57;15.78;16.65;17.56;18.24;19.41;20.43;21.12;22.5;22.97;23.19;23.28;23.73;24.08;24.33;24.49;24.04;15;7.8
145;0;5.02;5.73;6.23;7.15;7.85;8.55;9.28;9.98;10.51;11.27;11.98;12.92;14.35;15.56;16.42;17.31;17.97;19.13;20.14;20.82;22.17;22.64;22.87;22.96;23.42;23.81;24.23;24.41;24.19;15.14;7.98
146;0;4.96;5.64;6.12;7.07;7.77;8.43;9.15;9.84;10.38;11.16;11.83;12.71;14.12;15.35;16.19;17.07;17.72;18.86;19.86;20.51;21.84;22.31;22.56;22.65;23.12;23.48;23.94;24.33;24.3;15.3;8.16
148;0;4.82;5.45;5.91;6.9;7.59;8.21;8.89;9.55;10.14;10.89;11.55;12.29;13.7;14.92;15.74;16.6;17.23;18.32;19.3;19.91;21.2;21.67;21.95;22.05;22.53;22.87;23.38;24.13;24.39;15.52;8.46
149;0;4.76;5.35;5.81;6.78;7.49;8.09;8.78;9.42;10.01;10.76;11.41;12.1;13.48;14.71;15.52;16.36;16.98;18.06;19.03;19.63;20.89;21.37;21.67;21.77;22.26;22.61;23.12;23.98;24.37;15.57;8.58
150;0;4.69;5.26;5.7;6.67;7.37;7.96;8.64;9.26;9.86;10.6;11.24;11.89;13.26;14.48;15.29;16.11;16.73;17.79;18.74;19.33;20.55;21.04;21.37;21.48;21.98;22.34;22.83;23.69;24.11;15.6;8.67
155;0;4.33;4.74;5.16;6.16;6.79;7.33;7.96;8.51;9.15;9.81;10.4;10.85;12.14;13.37;14.1;14.87;15.48;16.42;17.3;17.8;18.88;19.39;19.86;20;20.54;20.97;21.45;22.25;22.77;15.38;8.89
160;0;4.09;4.41;4.83;5.77;6.39;6.94;7.55;8.04;8.67;9.28;9.83;10.24;11.46;12.69;13.39;14.11;14.73;15.6;16.41;16.87;17.85;18.4;18.97;19.15;19.72;20.2;20.65;21.35;21.84;14.95;8.74
162;0;4;4.29;4.69;5.62;6.23;6.77;7.38;7.86;8.48;9.07;9.6;10;11.18;12.42;13.1;13.81;14.43;15.27;16.06;16.5;17.43;18;18.62;18.81;19.39;19.89;20.33;20.99;21.48;14.76;8.61
168;0;3.74;3.93;4.35;5.15;5.75;6.31;6.93;7.34;7.92;8.45;8.95;9.35;10.44;11.68;12.32;12.99;13.63;14.39;15.11;15.5;16.31;16.94;17.7;17.92;18.53;19.08;19.49;19.99;20.46;14.34;8.3
170;0;3.69;3.85;4.27;5.04;5.65;6.22;6.82;7.23;7.8;8.31;8.8;9.22;10.27;11.51;12.15;12.81;13.45;14.19;14.9;15.28;16.06;16.7;17.51;17.73;18.34;18.91;19.31;19.77;20.22;14.24;8.24
174;0;3.57;3.69;4.11;4.83;5.43;6.01;6.62;7;7.55;8.03;8.5;8.93;9.95;11.19;11.81;12.45;13.11;13.81;14.48;14.84;15.57;16.24;17.11;17.35;17.98;18.57;18.95;19.33;19.77;14.03;8.13
180;0;3.51;3.6;4.03;4.71;5.31;5.91;6.51;6.88;7.41;7.88;8.33;8.79;9.78;11.02;11.63;12.26;12.93;13.61;14.26;14.61;15.31;15.99;16.9;17.15;17.79;18.39;18.77;19.09;19.52;13.87;8.07
"""

function to_numbers(sequence s)
    for i=1 to length(s) do
        s[i] = to_number(s[i])
    end for
    return s
end function

function getpolardata(string s)
--
-- A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma.
-- The first line of the file contains labels for the wind velocities that make up columns, and
-- the first entry of each row makes up a column of angle of sailing direction from wind in degrees
--
    sequence lines = split_any(s,"\r\n"),
             winds = to_numbers(split(lines[1],";")[2..$]),
             degrees = {}, speeds = {}
    for i=2 to length(lines) do
        sequence l = to_numbers(split(lines[i],";"))
        if length(l)!=length(winds)+1 then ?9/0 end if
        degrees = append(degrees,l[1])
        speeds = append(speeds,l[2..$])
    end for
    return {winds, degrees, speeds}
end function

--
-- winds is a list of wind speeds
-- degrees is a list of angles in degrees of direction relative to the wind
--  (note 0 degrees is directly into the wind, 180 degrees is directly downwind)
-- each speeds[i] is an array of length(winds) for each degrees[i]
--
constant {winds, degrees, speeds} = getpolardata(polar_csv) 
--constant {winds, degrees, speeds} = getpolardata(get_text("polar.csv")) -- alt
constant R = 6372800  -- Earth's approximate radius in meters
constant timeinterval = 10  -- the minutes duration for each TimeSlice

function deg2rad(atom deg) return remainder(deg*PI/180+2*PI,2*PI) end function
function rad2deg(atom rad) return remainder (rad*(180/PI)+360,360) end function
function sind(atom deg) return sin(deg2rad(deg)) end function
function cosd(atom deg) return cos(deg2rad(deg)) end function
function asind(atom deg) return rad2deg(arcsin(deg)) end function
function atand(atom x,y) return rad2deg(atan2(x,y)) end function

function haversine(atom lat1, lon1, lat2, lon2)
--
-- Calculate the haversine function for two points on the Earth's surface.
-- 
-- Given two latitude, longitude pairs in degrees for a point on the Earth,
-- get distance in meters and the initial direction of travel in degrees for
-- movement from point 1 to point 2.
--
    atom dlat = lat2 - lat1,
         dlon = lon2 - lon1,
         a = power(sind(dlat/2),2) + cosd(lat1)*cosd(lat2)*power(sind(dlon/2),2),
         c = 2.0 * asind(sqrt(a)),
         theta = atand(sind(dlon)*cosd(lat2),
                       cosd(lat1)*sind(lat2) - sind(lat1)*cosd(lat2)*cosd(dlon))
    theta = remainder(theta+360, 360)
    return {R*c*0.5399565, theta}
end function

function find_range(atom v, sequence s)
-- Returns the indexes of s of the first >=v, and the last <=v
    for i=1 to length(s) do
        if s[i]>=v then
            for j=length(s) to 1 by -1 do
                if s[j]<=v then return {i,j} end if
            end for
            exit
        end if
    end for
    return {-1,-1}
end function

function boatspeed(atom pointofsail, windspeed)
--
-- Calculate the expected sailing speed in a specified direction in knots,
-- given a desired point of sail in degrees, and wind speed in knots (for
-- the previously loaded sailing polar data)
--
    integer {ld,ud} = find_range(pointofsail, degrees),
            {lv,uv} = find_range(windspeed, winds)
    if find(-1,{ld, ud, lv, uv}) then return -1 end if
    atom wu = winds[uv],
         wl = winds[lv],
         du = degrees[ud],
         dl = degrees[ld],
         f
    if ld==ud then
        f = iff(uv==lv ? 1 :
            (wu-windspeed)/(wu-wl))
    elsif uv==lv then
        f = (du-pointofsail)/(du-dl)
    else
        f = ((du-pointofsail)/(du-dl)+
             (wu-windspeed)/(wu-wl))/2
    end if
    atom su = speeds[ud,uv],
         sl = speeds[ld,lv],
--       res = su + f*(sl-su)       -- (original)
         res = su - f*(su-sl)       -- (equivalent)
--       res = sl + (1-f)*(su-sl)   -- (also equivalent)
    return res
end function

function sailingspeed(atom azimuth, pointofsail, ws)
--
-- Calculate the expected net boat speed in a desired direction versus the wind (azimuth).
-- This is generally different from the actual boat speed in its actual direction.
-- Directions are in degrees (pointofsail is the ship direction from wind),
-- and velocity of wind (ws) is in knots.
--
    return boatspeed(pointofsail, ws) * cosd(abs(pointofsail-azimuth))
end function

function bestvectorspeed(atom dirtravel, sequence surfaceparameters)
--
-- Calculate the net direction and velocity of a sailing ship.
--
    atom {winddirection,
          windvelocity,
          currentdirection,
          currentvelocity} = surfaceparameters,
         azimuth = remainder(dirtravel-winddirection,360)
    if azimuth<0 then azimuth += 360 end if
    if azimuth>180 then azimuth = 360-azimuth end if
    atom vmg = boatspeed(azimuth, windvelocity),
         other = -1
    integer idx = -1
    for i=1 to length(degrees) do
        atom ss = sailingspeed(azimuth, degrees[i], windvelocity)
        if ss>other then {other,idx} = {ss,i} end if
    end for
    if other>vmg then
        azimuth = degrees[idx]
        vmg = other
    end if
    atom dirchosen = deg2rad(winddirection + azimuth),
         dircurrent = deg2rad(currentdirection),
         wx = vmg * sin(dirchosen),
         wy = vmg * cos(dirchosen),
         curx = currentvelocity * sin(dircurrent),
         cury = currentvelocity * cos(dircurrent)
    return sqrt(power(wx+curx,2) + power(wy+cury,2))
end function

function sailsegmenttime(sequence surfaceparameters, atom lat1, lon1, lat2, lon2)
--
-- Calculate the trip time in minutes from (lat1, lon1) to the destination (lat2, lon2).
-- Uses the data in surfaceparameters for wind and current velocity and direction.
--
    atom {distance, direction} = haversine(lat1, lon1, lat2, lon2),
         velocity = bestvectorspeed(direction, surfaceparameters)
    -- minutes/s * m / (knots * (m/s / knot)) = minutes
    atom res = (1/60) * distance / (velocity * 1.94384)
    return res
end function

--
--  The data is selected so the best time path is slightly longer than the
--  shortest length path. The forbidden regions are x, representing land or reef.
--  The allowed sailing points are . and start and finish are S and F.
--   
constant chart = split("""
...S.....
x......x.
....x..x.
x...xx.x.
x...xx.x.
..xxxx.xx
x..xxx...
.......xx
x..F..x.x""",'\n')

function minimum_time_route(sequence timeframe, start, finish)
--
-- Get the fastest route from start to finish for some detailed sea/ship parameters.
-- timeframe is a massive 200 * 9x9 * {pt,surfaceparameters}
-- note that polar data (ie winds, degrees, speeds) is static here, for simplicity.
--
    atom t0 = time(),
         mintime = 1000.0
    integer xmax = length(chart[1]),
            ymax = length(chart),
            {py,px} = start
    sequence todo = {start},
             costs = repeat(repeat(-1,xmax),ymax),  -- (lowest durations)
             paths = repeat(repeat(0,xmax),ymax),   -- (single backlinks)
             minpath = {}
    costs[py,px] = 0
    while length(todo) do
        {py,px} = todo[1]
        todo = todo[2..$]
        atom duration = costs[py,px]
        integer sdx = remainder(floor(round(duration)/timeinterval),length(timeframe))+1
        sequence s = timeframe[sdx]
        for nx=px-1 to px+1 do
            for ny=py-1 to py+1 do
                if (nx!=px or ny!=py) 
                and nx>=1 and nx<=xmax
                and ny>=1 and ny<=ymax 
                and chart[ny,nx]!='x' then
                    sequence gp1 = s[py,px],    -- {pt,surfaceparameters}
                             gp2 = s[ny,nx]     --          ""
                    atom {lat1, lon1} = gp1[1],
                         {lat2, lon2} = gp2[1]
                    sequence surfaceparameters = gp1[2]
                    atom nt = duration + sailsegmenttime(surfaceparameters, lat1, lon1, lat2, lon2)
                    if costs[ny,nx]=-1 or nt<costs[ny,nx] then
                        -- a larger (than 9x9) simulation might benefit from not
                        -- putting any already-too-long routes back on the todo
                        -- list and/or processing todo lowest duration first.
                        costs[ny,nx] = nt
                        paths[ny,nx] = {py,px}
                        if not find({ny,nx},todo) then
                            todo = append(todo,{ny,nx})
                        end if
                    elsif nt==costs[ny,nx] then
                        -- (Should multiple same-time routes exist, we could store 
                        --  multiple back-links and whip up a simple [recursive]
                        --  routine to rebuild them all. Or just ignore them.)
                        ?9/0 
                    end if
                end if
            end for
        end for
        s = {} -- (simplify debugging)
    end while
    timeframe = {} -- (simplify debugging)
    {py,px} = finish
    mintime = costs[py,px]
    minpath = {finish}
    while true do
        object pyx = paths[py,px]
        if pyx=0 then exit end if
        minpath = prepend(minpath,pyx)
        paths[py,px] = 0 -- (be safe, why not)
        {py,px} = pyx
    end while
    if minpath[1]!=start then ?9/0 end if
    return {minpath,elapsed(mintime*60),elapsed(time()-t0)}
end function
 
function surfacebylongitude(atom lon)
-- Create regional wind patterns on the map.
    sequence surfaceparameters = iff(lon < -155.03 ? { -5.0,  8, 150, 0.5} :
                                 iff(lon < -155.99 ? {-90.0, 20, 150, 0.4} :
                                                     {180.0, 25, 150, 0.3}))
    return surfaceparameters
end function

sequence slices = repeat(null,200)

procedure mutatetimeslices()
-- Vary wind speeds over time.
    for i=1 to length(slices) do
        sequence s = deep_copy(slices[i])
        for j=1 to length(s) do
            sequence sj = s[j]
            s[j] = 0
            for k=1 to length(sj) do
                atom windvelocity = sj[k][2][2]
                windvelocity *= 1 + 0.002*i
                sj[k][2][2] = windvelocity
            end for
            s[j] = sj
        end for
        slices[i] = s
    end for
end procedure
 
for s=1 to length(slices) do
    sequence gpoints = repeat(null,9)
    for i=1 to 9 do
        atom lat = 19.78 - 2/60 + i/60
        gpoints[i] = repeat(null,9)
        for j=1 to 9 do
            atom lon = -155.0 - 6/60 + j/60
            gpoints[i][j] = {{lat,lon}, surfacebylongitude(lon)}
        end for
    end for
    slices[s] = gpoints
end for
mutatetimeslices()
constant fmt = """
The route taking the least time found was:
    %v
which has duration %s [route found in %s]
"""
printf(1,fmt,minimum_time_route(slices,{1,4},{9,4}))

{} = wait_key()
Output:
The route taking the least time found was:
    {{1,4},{1,5},{2,6},{3,7},{4,7},{5,7},{6,7},{7,7},{8,6},{8,5},{9,4}}
which has duration 4 hours, 43 minutes and 41s [route found in 0.0s]

Wren[edit]

Translation of: Julia

A reasonably faithful translation though I haven't bothered to split the code up into modules (which would mean separate files in Wren) and have dispensed altogether with four functions which aren't actually called.

As Wren uses 0-based indexing the points in the minimum path have coordinates one less than those in the Julia results.

As you'd expect, this takes many times longer than Julia to run (about 24.5 minutes versus 3 minutes 20 seconds) but gets there in the end :)

import "io" for File

/*
    Class that represents a polar CSV file's data.
    Contains a matrix, 'speeds', of sailing speeds indexed by wind velocity and angle of boat to wind.
    'winds' is a list of wind speeds.
    'degrees' is a list of angles in degrees of direction relative to the wind.
    Note 0 degrees is directly into the wind, 180 degrees is directly downwind.
*/
class SailingPolar {
    construct new(winds, degrees, speeds) {
        _winds = winds
        _degrees = degrees
        _speeds = speeds // speeds[wind direction degrees, windspeed knots]
    }
    winds { _winds }
    degrees { _degrees }
    speeds {_speeds }
}

/*
    Class that represents wind and surface current direction and velocity for a given position.
    Angles in degrees, velocities in knots.
*/
class SurfaceParameters {
    construct new(windDeg, windKts, currentDeg, currentKts) {
        _windDeg = windDeg
        _windKts = windKts
        _currentDeg = currentDeg
        _currentKts = currentKts
    }
    windDeg { _windDeg }
    windKts { _windKts }
    currentDeg { _currentDeg }
    currentKts { _currentKts }
}

// Reads a sailing polar CSV file and returns a SailingPolar object containing the file data.
// A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma.
// The first line of file contains labels for the wind velocities that make up columns, and
// the first entry of each row makes up a column of angle of sailing direction from wind in degrees.
var getPolarData = Fn.new { |fileName|
    var lines = File.read(fileName).split("\n")
    var header = lines[0].trim().split(";")
    var winds = header[1..-1].map { |x| Num.fromString(x) }.toList
    var degrees = []
    var speeds = []
    for (line in lines[1..-1]) {
        line = line.trim()
        if (line == "") break // ignore final blank line if there is one
        var cols = line.split(";")
        degrees.add(Num.fromString(cols[0]))
        speeds.add(cols[1..-1].map{ |x| Num.fromString(x) }.toList)
    }
    return SailingPolar.new(winds, degrees, speeds)
}

var R = 6372800  // Earth's approximate radius in meters

/*  Class containing various helper methods which work with degrees rather than radians. */
class D {
    // Converts degrees to radians.
    static deg2Rad(deg) { (deg*Num.pi/180 + 2*Num.pi) % (2*Num.pi) }

    // Converts radians to degrees.
    static rad2Deg(rad) { (rad*180/Num.pi + 360) % 360 }

    // Trig functions.
    static sin(d) { deg2Rad(d).sin }
    static cos(d) { deg2Rad(d).cos }
    static asin(d) { rad2Deg(d.asin) }
    static atan(x, y) { rad2Deg(x.atan(y)) }
}

// Calculates the haversine function for two points on the Earth's surface. 
// Given two latitude, longitude pairs in degrees for a point on the Earth,
// get distance in meters and the initial direction of travel in degrees for
// movement from point 1 to point 2.
var haversine = Fn.new { |lat1, lon1, lat2, lon2|
    var dlat = lat2 - lat1
    var dlon = lon2 - lon1
    var a = D.sin(dlat/2).pow(2) + D.cos(lat1) * D.cos(lat2) * (D.sin(dlon/2).pow(2))
    var c = 2 * D.asin(a.sqrt)
    var theta = D.atan(D.sin(dlon) * D.cos(lat2), 
          D.cos(lat1)*D.sin(lat2) - D.sin(lat1) * D.cos(lat2) * D.cos(dlon))
    theta = (theta + 360) % 360
    return [R * c * 0.5399565, theta]
}

// Returns the index of the first element of 'a' for which 'pred' returns true or -1 otherwise.
var findFirst = Fn.new { |a, pred|
    for (i in 0...a.count) if (pred.call(a[i])) return i
    return -1
}

// Returns the index of the last element of 'a' for which 'pred' returns true or -1 otherwise.
var findLast = Fn.new { |a, pred|
    for (i in a.count-1..0) if (pred.call(a[i])) return i
    return -1
}

// Calculate the expected sailing speed in a specified direction in knots,
// given sailing polar data, a desired point of sail in degrees, and wind speed in knots.
var boatSpeed = Fn.new { |sp, pointOfSail, windSpeed|
    var winds = sp.winds
    var degrees = sp.degrees
    var speeds = sp.speeds
    var udeg = findLast.call(degrees)  { |t| t <= pointOfSail }
    var odeg = findFirst.call(degrees) { |t| t >= pointOfSail }
    var uvel = findLast.call(winds)    { |t| t <= windSpeed }
    var ovel = findFirst.call(winds)   { |t| t >= windSpeed }
    if ([udeg, odeg, uvel, ovel].any { |t| t == -1 }) return -1
    var frac = (odeg == udeg && uvel == ovel) ? 1 :
               (odeg == udeg) ? (windSpeed - winds[uvel])/(winds[ovel] - winds[uvel]) :
               (uvel == ovel) ? (pointOfSail - degrees[udeg])/(degrees[odeg] - degrees[udeg]) :
               ((pointOfSail - degrees[udeg])/(degrees[odeg] - degrees[udeg]) +
               (windSpeed - winds[uvel])/(winds[ovel] - winds[uvel]))/2
    return speeds[udeg][uvel] + frac * (speeds[odeg][ovel] - speeds[udeg][uvel])
}

// Calculates the expected net boat speed in a desired direction versus the wind ('azimuth').
// This is generally different from the actual boat speed in its actual direction.
// Directions are in degrees ('pointos' is point of sail the ship direction from the wind),
// and velocity of wind ('ws') is in knots.
var sailingSpeed = Fn.new { |sp, azimuth, pointos, ws|
    return boatSpeed.call(sp, pointos, ws) * D.cos((pointos - azimuth).abs)
}

// Calculates the net direction and velocity of a sailing ship.
// Arguments are sailing polar data, direction of travel in degrees from north, wind direction in
// degrees from north, wind velocity in knots, surface current direction in degrees, and
// current velocity in knots.
var bestVectorSpeed = Fn.new { |sp, dirTravel, dirWind, windSpeed, dirCur, velCur|
    var azimuth = (dirTravel - dirWind) % 360
    azimuth = (azimuth < 0) ? azimuth + 360 : azimuth
    azimuth = (azimuth > 180) ? 360 - azimuth : azimuth
    var VMG = boatSpeed.call(sp, azimuth, windSpeed)
    var other = -1
    var idx = -1
    for (i in 0...sp.degrees.count) {
        var ss = sailingSpeed.call(sp, azimuth, sp.degrees[i], windSpeed)
        if (ss > other) {
             other = ss
             idx = i
        }
    }
    if (other > VMG) {
        azimuth = sp.degrees[idx]
        VMG = other
    }
    var dirChosen = D.deg2Rad(dirWind + azimuth)
    var wx = VMG * (dirChosen.sin)
    var wy = VMG * (dirChosen.cos)
    var curX = velCur * (D.deg2Rad(dirCur).sin)
    var curY = velCur * (D.deg2Rad(dirCur).cos)
    return [D.rad2Deg((wy + curY).atan(wx + curX)), ((wx + curX).pow(2) + (wy + curY).pow(2)).sqrt]
}

// Calculates the trip time in minutes from (lat1, lon1) to the destination (lat2, lon2).
// Uses the data in SurfaceParameters for wind and current velocity and direction.
var sailSegmentTime = Fn.new { |sp, p, lat1, lon1, lat2, lon2|
    var h = haversine.call(lat1, lon1, lat2, lon2)
    var distance = h[0]
    var dir = h[1]
    var vel = bestVectorSpeed.call(sp, dir, p.windDeg, p.windKts, p.currentDeg, p.currentKts)[1]
    // minutes/s * m / (knots * (m/s / knot)) = minutes
    return (1 / 60) * distance / (vel * 1.94384)
}

/* Class that represents a point in 2-D space. Need value type semantics for comparisons etc. */
class Point2 {
    construct new(x, y) {
        _x = x
        _y = y
    }
    x { _x }
    y { _y }

    + (other) { Point2.new(x + other.x, y + other.y) }

    == (other) { x == other.x && y == other.y }
    != (other) { !(this == other) }

    toString { "[%(_x), %(_y)]" }
}

/*
    Class that consists of a tuple of latitude and longitude in degrees.
    NB: This uses latitude (often considered to be y) first then longitude (often considered to be x).
    This latitude, then longitude ordering is as per ISO 6709 (en.wikipedia.org/wiki/ISO_6709).
*/
class Position {
    construct new(lat, lon) {
        _lat = lat
        _lon = lon
    }
    lat { _lat }
    lon { _lon }
}

/*  Class that represents a Position with the SurfaceParameters of wind and current at the Position. */
class GridPoint {
    construct new(pt, sp) {
        _pt = pt
        _sp = sp
    }
    pt { _pt }
    pt=(value) { _pt = value }
    sp { _sp }
    sp=(value) { _sp = value }
}

/*
    Class that consists of a matrix of GridPoints, each Position point with their SurfaceParameters.
    A Vector of TimeSlice can give the surface characteristics for an ocean region over time.
*/
class TimeSlice {
    construct new(gridPoints) {
      _gridPoints = gridpoints
    }
    gridPoints { _gridPoints }
}

/*
    Class that represents a routing problem and requiring the following parameters:
    * timeinterval: the minutes duration for each TimeSlice
    * timeframe: a vector of sequential timeslices for the ocean region
    * obstacleindices: the Cartesian indices in each timeslice for
        obstacles, such as land or shoals, where the ship may not go
    * startindex: the timeslice position for time of starting
    * start: starting location on grid of GridPoints
    * finish: destination / finish location on grid of GridPoints
    * allowrepeatvisits: whether the vessel may overlap its prior path, usually false.
*/
class RoutingProblem {
    construct new(timeInterval, timeFrame, obstacleIndices, startIndex, start, finish, allowRepeatVisits) {
        _timeInterval = timeInterval // minutes between timeFrame slices
        _timeFrame = timeFrame
        _obstacleIndices = obstacleIndices
        _startIndex = startIndex
        _start = start
        _finish = finish
        _allowRepeatVisits = allowRepeatVisits
    }

    timeInterval { _timeInterval }
    timeFrame  { _timeFrame }
    obstacleIndices { _obstacleIndices }
    startIndex { _startIndex }
    start { _start }
    finish { _finish }
    allowRepeatVisits { _allowRepeatVisits }
}

/*
    Class that represents a timed path and requires the following parameters:
    * duration: minutes total to travel the path
    * path: vector of Cartesian indices of points in grid for path to travel.
*/
class TimedPath {
    construct new(duration, path) {
        _duration = duration
        _path = path
    }
    duration { _duration }
    path { _path }

    toString { "%(_duration) %(_path)" }

    == (other) { this.toString == other.toString }
    != (other) { this.toString != other.toString }
}

var findMin = Fn.new { |a|
    var min = a[0]
    var idx = 0
    for (i in 1...a.count) {
        if (a[i] < min) {
            min = a[i]
            idx = i
        }
    }
    return [min, idx]
}

var ntuples = [ [-1, -1], [-1, 0], [-1, 1], [0, -1], [0, 1], [1, -1], [1, 0], [1, 1] ]
var neighbors = List.filled(ntuples.count, null)
(0...ntuples.count).each { |i| neighbors[i] = Point2.new(ntuples[i][0], ntuples[i][1]) }

// Returns a list of points surrounding 'p' which are not otherwise excluded.
var surround = Fn.new { |p, mat, excluded|
    var xmax = mat.count
    var ymax = mat[0].count
    return neighbors.map { |x| x + p }.where { |q|
        return (0 <= q.x && q.x < xmax) && (0 <= q.y && q.y < ymax) && !excluded.contains(q)
    }.toList
}

// Get the route (as a TimedPath) that minimizes time from start to finish for a given
// RoutingProblem (sea parameters) given sailing polar data (ship parameters).
var minimumTimeRoute = Fn.new { |rp, sp, verbose|
    var timedPaths = [TimedPath.new(0, [rp.start])]
    var completed = false
    var minPath = TimedPath.new(1000, [])
    for (i in 0...1000) {
        var newPaths = []
        verbose && System.print("Checking %(timedPaths.count) paths of length %(timedPaths[0].path.count)")
        for (tpath in timedPaths) {
            if (tpath.path[-1] == rp.finish) {
                completed = true
                newPaths.add(tpath)
            } else {
                var p1 = tpath.path[-1]
                var num = tpath.duration.round
                var den = rp.timeInterval.round
                var slice = rp.timeFrame[(num/den).truncate % rp.timeFrame.count]
                for (p2 in surround.call(p1, slice, rp.obstacleIndices)) {
                    if (rp.allowRepeatVisits || !tpath.path.contains(p2)) {
                        var gp1 = slice[p1.x][p1.y]
                        var gp2 = slice[p2.x][p2.y]
                        var lat1 = gp1.pt.lat
                        var lon1 = gp1.pt.lon
                        var lat2 = gp2.pt.lat
                        var lon2 = gp2.pt.lon
                        var t = sailSegmentTime.call(sp, gp1.sp, lat1, lon1, lat2, lon2)
                        var path = tpath.path.toList
                        path.add(p2)
                        newPaths.add(TimedPath.new(tpath.duration + t, path))
                    }
                }
            }
        }
        var set = {}
        for (np in newPaths) set[np.toString] = np
        timedPaths = set.values.toList
        if (completed) {
            var minDur = findMin.call(timedPaths.map { |x| x.duration }.toList)[0]
            var finished = timedPaths.where { |x| x.path[-1] == rp.finish }.toList
            var mi = findMin.call(finished.map { |x| x.duration }.toList)
            var minFinDur = mi[0]
            var idx = mi[1]
            if (verbose) {
                System.print("Current finished minimum: %(finished[idx]), others %(minDur)")
            }
            if (minDur == minFinDur) {
                minPath = finished[idx]
                break
            }
        }
    }
    return minPath
}

/*
    The data is selected so the best time path is slightly longer than the
    shortest length path. The forbidden regions are x, representing land or reef.
    The allowed sailing points are . and start and finish are S and F.

    x  .  .  F  .  .  x  .  x
    .  .  .  .  .  .  .  x  x
    x  .  .  x  x  x  .  .  .
    .  .  x  x  x  x  .  x  x
    x  .  .  .  x  x  .  x  .
    x  .  .  .  x  x  .  x  .
    .  .  .  .  x  .  .  x  .
    x  .  .  .  .  .  .  x  .
    .  .  .  S  .  .  .  .  .
*/

// These need to be changed to 0-based for Wren.
var ftuples = [
    [1, 8], [2, 1], [2, 8], [3, 5], [3, 8], [4, 1], [4, 5], [4, 6], [4, 8], [5, 1],
    [5, 5], [5, 6], [5, 8], [6, 3], [6, 4], [6, 5], [6, 6], [6, 8], [6, 9], [7, 1],
    [7, 4], [7, 5], [7, 6], [8, 8], [8, 9], [9, 1], [9, 7], [9, 9]
]

var forbidden = List.filled(ftuples.count, null)
(0...ftuples.count).each { |i| forbidden[i] = Point2.new(ftuples[i][0]-1, ftuples[i][1]-1) }

// Create regional wind patterns on the map.
var surfaceByLongitude = Fn.new { |lon|
    return (lon < -155.03) ? SurfaceParameters.new(-5, 8, 150, 0.5) :
           (lon < -155.99) ? SurfaceParameters.new(-90, 20, 150, 0.4) :
                             SurfaceParameters.new(180, 25, 150, 0.3)
}

// Vary wind speeds over time.
var mutateTimeSlices = Fn.new { |slices|
    var i = 1
    for (slice in slices) {
        for (j in 0...slice.count) {
            for (k in 0...slice[j].count) {
                var x = slice[j][k]
                x.sp = SurfaceParameters.new(x.sp.windDeg, x.sp.windKts * (1 + 0.002 * i),
                    x.sp.currentDeg, x.sp.currentKts)
             }
        }
        i = i + 1
    }
}

var startPos = Point2.new(0, 3)  // 0-based
var endPos = Point2.new(8, 3)    // ditto
var slices = List.filled(200, null)
for (s in 0...200) {
    var gpoints = List.filled(9, null)
    for (i in 0..8) {
        gpoints[i] = List.filled(9, null)
        for (j in 0..8) {
            var pt = Position.new(19.78 - 1/60 + i/60, -155.0 - 5/60 + j/60)
            gpoints[i][j] = GridPoint.new(pt, surfaceByLongitude.call(pt.lon))
        }
    }
    slices[s] = gpoints
}
mutateTimeSlices.call(slices)
var routeProb = RoutingProblem.new(10, slices, forbidden, 0, startPos, endPos, false)
var fileName = "polar.csv"
var sp = getPolarData.call(fileName)
var tp = minimumTimeRoute.call(routeProb, sp, false)
System.print("The route taking the least time found was:\n    %(tp.path) \nwhich has duration " +
   "%((tp.duration/60).truncate) hours, %((tp.duration%60).round) minutes.")
Output:
The route taking the least time found was:
    [[0, 3], [0, 4], [1, 5], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6], [7, 5], [7, 4], [8, 3]] 
which has duration 4 hours, 44 minutes.