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# Orbital elements

Orbital elements 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.

When neglecting the influence of other objects, two celestial bodies orbit one another along a conic trajectory. In the orbital plane, the radial equation is thus:

r = L/(1 + e cos(angle))

L , e and angle are respectively called semi-latus rectum, eccentricity and true anomaly. The eccentricity and the true anomaly are two of the six so-called orbital elements often used to specify an orbit and the position of a point on this orbit.

The four other parameters are the semi-major axis, the longitude of the ascending node, the inclination and the argument of periapsis. An other parameter, called the gravitational parameter, along with dynamical considerations described further, also allows for the determination of the speed of the orbiting object.

The semi-major axis is half the distance between perihelion and aphelion. It is often noted a, and it's not too hard to see how it's related to the semi-latus-rectum:

a = L/(1 - e2)

The longitude of the ascending node, the inclination and the argument of the periapsis specify the orientation of the orbiting plane with respect to a reference plane defined with three arbitrarily chosen reference distant stars.

The gravitational parameter is the coefficent GM in Newton's gravitational force. It is sometimes noted µ and will be chosen as one here for the sake of simplicity:

µ = GM = 1

As mentioned, dynamical considerations allow for the determination of the speed. They result in the so-called vis-viva equation:

v2 = GM(2/r - 1/a)

This only gives the magnitude of the speed. The direction is easily determined since it's tangent to the conic.

Those parameters allow for the determination of both the position and the speed of the orbiting object in cartesian coordinates, those two vectors constituting the so-called orbital state vectors.

Task

Show how to perform this conversion from orbital elements to orbital state vectors in your programming language.

TODO: pick an example from a reputable source, and bring the algorithm description onto this site. (Restating those pages in concise a fashion comprehensible to the coders and readers of this site will be a good exercise.)

## 11l

Translation of: Python
`F mulAdd(v1, x1, v2, x2)   R v1 * x1 + v2 * x2 F rotate(i, j, alpha)   R [mulAdd(i, cos(alpha), j, sin(alpha)), mulAdd(i, -sin(alpha), j, cos(alpha))] F orbitalStateVectors(semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly)   V i = (1.0, 0.0, 0.0)   V j = (0.0, 1.0, 0.0)   V k = (0.0, 0.0, 1.0)    V p = rotate(i, j, longitudeOfAscendingNode)   i = p[0]   j = p[1]   p = rotate(j, k, inclination)   j = p[0]   p = rotate(i, j, argumentOfPeriapsis)   i = p[0]   j = p[1]    V l = I (eccentricity == 1.0) {2.0} E 1.0 - eccentricity * eccentricity   l *= semimajorAxis   V c = cos(trueAnomaly)   V s = sin(trueAnomaly)   V r = 1 / (1.0 + eccentricity * c)   V rprime = s * r * r / l   V position = mulAdd(i, c, j, s) * r   V speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c)   speed = normalize(speed) * sqrt(2.0 / r - 1.0 / semimajorAxis)    R [position, speed] V ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)print(‘Position : ’ps[0])print(‘Speed    : ’ps[1])`
Output:
```Position : (0.787295801, 0.45454549, 0)
Speed    : (-0.5477226, 0.948683274, 0)
```

## ALGOL W

Translation of: C
(which is a translation of Kotlin which is a translation of ...).
`begin    % compute orbital elements                                               %    % 3-element vector                                                       %    record Vector( real x, y, z );    % prints the components of the vector v                                  %    procedure writeOnVector( reference(Vector) value v ) ;        writeon( r_format := "A", r_w := 10, r_d := 6, s_w := 0, "( ", x(v), ", ", y(v), ", ", z(v), " )" );    % returns a vector whose elements are the sum of the elements of v & w   %    reference(Vector) procedure add( reference(Vector) value v, w ) ;        Vector( x(v) + x(w), y(v) + y(w), z(v) + z(w) );    % returns a vector whose elements are those of v multiplied by m         %    reference(Vector) procedure mul( reference(Vector) value v                                   ; real              value m                                   ) ;        Vector( x(v) * m, y(v) * m, z(v) * m );    % returns a vector whose elements are those of v divided by d            %    reference(Vector) procedure divVR( reference(Vector) value v                                     ; real              value d                                     ) ;        mul( v, 1 / d );    % returns the norm of the vector v                                       %    real procedure vabs( reference(Vector) value v ) ;        sqrt( ( x(v) * x(v) ) + y(v) * y(v) + z(v) * z(v) );    % returns the sum of v1 * x1 and v2 * x2                                 %    reference(Vector) procedure mulAdd( reference(Vector) value v1, v2                                      ; real              value x1, x2                                      ) ;        add( mul( v1, x1 ), mul( v2, x2 ) );    % sets ps to rotations of i and j by alpha                               %    procedure rotate( reference(Vector) value i, j                    ; real              value alpha                    ; reference(Vector) array ps ( * )                    ) ;    begin        ps( 0 ) := mulAdd( i, j,  cos( alpha ), sin( alpha ) );        ps( 1 ) := mulAdd( i, j, -sin( alpha ), cos( alpha ) )    end rotate ;    % sets position and speed vectors from the supplied elements             %    procedure orbitalStateVectors( real              value  semimajorAxis                                 ,                          eccentricity                                 ,                          inclination                                 ,                          longitudeOfAscendingNode                                 ,                          argumentOfPeriapsis                                 ,                          trueAnomaly                                 ; reference(Vector) result position                                 ,                          speed                                 ) ;    begin        reference(Vector) i, j, k;        reference(Vector) array qs ( 0 :: 1 );        real              L, c, s, r, rprime;        i        := Vector( 1.0, 0.0, 0.0 );        j        := Vector( 0.0, 1.0, 0.0 );        k        := Vector( 0.0, 0.0, 1.0 );        L        := 2.0;        rotate( i, j, longitudeOfAscendingNode, qs );        i        := qs( 0 );        j        := qs( 1 );        rotate( j, k, inclination, qs );        j        := qs( 0 );        rotate( i, j, argumentOfPeriapsis, qs );        i        := qs( 0 );        j        := qs( 1 );        if eccentricity not = 1 then L := 1 - eccentricity * eccentricity;        l        := L * semimajorAxis;        c        := cos( trueAnomaly );        s        := sin( trueAnomaly );        r        := L / ( 1.0 + eccentricity * c );        rprime   := s * r * r / L;        position := mulAdd( i, j, c, s );        position := mul( position, r ) ;        speed    := mulAdd( i, j, rprime * c - r * s, rprime * s + r * c );        speed    := divVR( speed, vabs( speed ) );        speed    := mul( speed, sqrt( 2 / r - 1 / semimajorAxis ) );    end orbitalStateVectors ;    % test the orbitalStateVectors routine                                   %    begin        real longitude;        reference(Vector) position, speed;        longitude := 355.0 / ( 113.0 * 6.0 );        orbitalStateVectors( 1.0, 0.1, 0.0, longitude, 0.0, 0.0, position, speed );        write( "Position : " ); writeOnVector( position );        write( "Speed    : " ); writeOnVector( speed    )    endend.`
Output:
```Position : (   0.779422,   0.450000,   0.000000 )
Speed    : (  -0.552770,   0.957427,   0.000000 )```

## C

Translation of: Kotlin
`#include <stdio.h>#include <math.h> typedef struct {    double x, y, z;} vector; vector add(vector v, vector w) {    return (vector){v.x + w.x, v.y + w.y, v.z + w.z};} vector mul(vector v, double m) {    return (vector){v.x * m, v.y * m, v.z * m};} vector div(vector v, double d) {    return mul(v, 1.0 / d);} double vabs(vector v) {    return sqrt(v.x * v.x + v.y * v.y + v.z * v.z);} vector mulAdd(vector v1, vector v2, double x1, double x2) {    return add(mul(v1, x1), mul(v2, x2)); } void vecAsStr(char buffer[], vector v) {    sprintf(buffer, "(%.17g, %.17g, %.17g)", v.x, v.y, v.z);} void rotate(vector i, vector j, double alpha, vector ps[]) {    ps[0] = mulAdd(i, j, cos(alpha), sin(alpha));    ps[1] = mulAdd(i, j, -sin(alpha), cos(alpha));} void orbitalStateVectors(    double semimajorAxis, double eccentricity, double inclination,    double longitudeOfAscendingNode, double argumentOfPeriapsis,    double trueAnomaly, vector ps[]) {     vector i = {1.0, 0.0, 0.0};    vector j = {0.0, 1.0, 0.0};    vector k = {0.0, 0.0, 1.0};    double l = 2.0, c, s, r, rprime;    vector qs[2];    rotate(i, j, longitudeOfAscendingNode, qs);    i = qs[0]; j = qs[1];    rotate(j, k, inclination, qs);    j = qs[0];    rotate(i, j, argumentOfPeriapsis, qs);    i = qs[0]; j = qs[1];    if (eccentricity != 1.0)  l = 1.0 - eccentricity * eccentricity;    l *= semimajorAxis;    c = cos(trueAnomaly);    s = sin(trueAnomaly);    r = l / (1.0 + eccentricity * c);    rprime = s * r * r / l;    ps[0] = mulAdd(i, j, c, s);    ps[0] = mul(ps[0], r);    ps[1] = mulAdd(i, j, rprime * c - r * s, rprime * s + r * c);    ps[1] = div(ps[1], vabs(ps[1]));    ps[1] = mul(ps[1], sqrt(2.0 / r - 1.0 / semimajorAxis));} int main() {    double longitude = 355.0 / (113.0 * 6.0);    vector ps[2];    char buffer[80];    orbitalStateVectors(1.0, 0.1, 0.0, longitude, 0.0, 0.0, ps);    vecAsStr(buffer, ps[0]);    printf("Position : %s\n", buffer);    vecAsStr(buffer, ps[1]);    printf("Speed    : %s\n", buffer);    return 0;}`
Output:
```Position : (0.77942284339867973, 0.45000003465368416, 0)
Speed    : (-0.55277084096044382, 0.95742708317976177, 0)
```

## C#

Translation of: D
`using System; namespace OrbitalElements {    class Vector {        public Vector(double x, double y, double z) {            X = x;            Y = y;            Z = z;        }         public double X { get; set; }        public double Y { get; set; }        public double Z { get; set; }         public double Abs() {            return Math.Sqrt(X * X + Y * Y + Z * Z);        }         public static Vector operator +(Vector lhs, Vector rhs) {            return new Vector(lhs.X + rhs.X, lhs.Y + rhs.Y, lhs.Z + rhs.Z);        }         public static Vector operator *(Vector self, double m) {            return new Vector(self.X * m, self.Y * m, self.Z * m);        }         public static Vector operator /(Vector self, double m) {            return new Vector(self.X / m, self.Y / m, self.Z / m);        }         public override string ToString() {            return string.Format("({0}, {1}, {2})", X, Y, Z);        }    }     class Program {        static Tuple<Vector, Vector> OrbitalStateVectors(            double semiMajorAxis,            double eccentricity,            double inclination,            double longitudeOfAscendingNode,            double argumentOfPeriapsis,            double trueAnomaly        ) {            Vector mulAdd(Vector v1, double x1, Vector v2, double x2) {                return v1 * x1 + v2 * x2;            }             Tuple<Vector, Vector> rotate(Vector iv, Vector jv, double alpha) {                return new Tuple<Vector, Vector>(                    mulAdd(iv, +Math.Cos(alpha), jv, Math.Sin(alpha)),                    mulAdd(iv, -Math.Sin(alpha), jv, Math.Cos(alpha))                );            }             var i = new Vector(1, 0, 0);            var j = new Vector(0, 1, 0);            var k = new Vector(0, 0, 1);             var p = rotate(i, j, longitudeOfAscendingNode);            i = p.Item1; j = p.Item2;            p = rotate(j, k, inclination);            j = p.Item1;            p = rotate(i, j, argumentOfPeriapsis);            i = p.Item1; j = p.Item2;             var l = semiMajorAxis * ((eccentricity == 1.0) ? 2.0 : (1.0 - eccentricity * eccentricity));            var c = Math.Cos(trueAnomaly);            var s = Math.Sin(trueAnomaly);            var r = l / (1.0 + eccentricity * c);            var rprime = s * r * r / l;            var position = mulAdd(i, c, j, s) * r;            var speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c);            speed /= speed.Abs();            speed *= Math.Sqrt(2.0 / r - 1.0 / semiMajorAxis);             return new Tuple<Vector, Vector>(position, speed);        }         static void Main(string[] args) {            var res = OrbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0);            Console.WriteLine("Position : {0}", res.Item1);            Console.WriteLine("Speed    : {0}", res.Item2);        }    }}`
Output:
```Position : (0.77942284339868, 0.450000034653684, 0)
Speed    : (-0.552770840960444, 0.957427083179762, 0)```

## C++

Translation of: C#
`#include <iostream>#include <tuple> class Vector {private:    double _x, _y, _z; public:    Vector(double x, double y, double z) : _x(x), _y(y), _z(z) {        // empty    }     double getX() {        return _x;    }     double getY() {        return _y;    }     double getZ() {        return _z;    }     double abs() {        return sqrt(_x * _x + _y * _y + _z * _z);    }     Vector operator+(const Vector& rhs) const {        return Vector(_x + rhs._x, _y + rhs._y, _z + rhs._z);    }     Vector operator*(double m) const {        return Vector(_x * m, _y * m, _z * m);    }     Vector operator/(double m) const {        return Vector(_x / m, _y / m, _z / m);    }     friend std::ostream& operator<<(std::ostream& os, const Vector& v);}; std::ostream& operator<<(std::ostream& os, const Vector& v) {    return os << '(' << v._x << ", " << v._y << ", " << v._z << ')';} std::pair<Vector, Vector> orbitalStateVectors(    double semiMajorAxis,    double eccentricity,    double inclination,    double longitudeOfAscendingNode,    double argumentOfPeriapsis,    double trueAnomaly) {    auto mulAdd = [](const Vector& v1, double x1, const Vector& v2, double x2) {        return v1 * x1 + v2 * x2;    };     auto rotate = [mulAdd](const Vector& iv, const Vector& jv, double alpha) {        return std::make_pair(            mulAdd(iv, +cos(alpha), jv, sin(alpha)),            mulAdd(iv, -sin(alpha), jv, cos(alpha))        );    };     Vector i(1, 0, 0);    Vector j(0, 1, 0);    Vector k(0, 0, 1);     auto p = rotate(i, j, longitudeOfAscendingNode);    i = p.first; j = p.second;    p = rotate(j, k, inclination);    j = p.first;    p = rotate(i, j, argumentOfPeriapsis);    i = p.first; j = p.second;     auto l = semiMajorAxis * ((eccentricity == 1.0) ? 2.0 : (1.0 - eccentricity * eccentricity));    auto c = cos(trueAnomaly);    auto s = sin(trueAnomaly);    auto r = l / (1.0 + eccentricity * c);;    auto rprime = s * r * r / l;    auto position = mulAdd(i, c, j, s) * r;    auto speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c);    speed = speed / speed.abs();    speed = speed * sqrt(2.0 / r - 1.0 / semiMajorAxis);     return std::make_pair(position, speed);} int main() {    auto res = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0);    std::cout << "Position : " << res.first << '\n';    std::cout << "Speed    : " << res.second << '\n';     return 0;}`
Output:
```Position : (0.779423, 0.45, 0)
Speed    : (-0.552771, 0.957427, 0)```

## D

Translation of: Kotlin
`import std.math;import std.stdio;import std.typecons; struct Vector {    double x, y, z;     auto opBinary(string op : "+")(Vector rhs) {        return Vector(x+rhs.x, y+rhs.y, z+rhs.z);    }     auto opBinary(string op : "*")(double m) {        return Vector(x*m, y*m, z*m);    }    auto opOpAssign(string op : "*")(double m) {        this.x *= m;        this.y *= m;        this.z *= m;        return this;    }     auto opBinary(string op : "/")(double d) {        return Vector(x/d, y/d, z/d);    }    auto opOpAssign(string op : "/")(double m) {        this.x /= m;        this.y /= m;        this.z /= m;        return this;    }     auto abs() {        return sqrt(x * x + y * y + z * z);    }     void toString(scope void delegate(const(char)[]) sink) const {        import std.format;        sink("(");        formattedWrite(sink, "%.16f", x);        sink(", ");        formattedWrite(sink, "%.16f", y);        sink(", ");        formattedWrite(sink, "%.16f", z);        sink(")");    }} auto orbitalStateVectors(    double semiMajorAxis,    double eccentricity,    double inclination,    double longitudeOfAscendingNode,    double argumentOfPeriapsis,    double trueAnomaly) {    auto i = Vector(1.0, 0.0, 0.0);    auto j = Vector(0.0, 1.0, 0.0);    auto k = Vector(0.0, 0.0, 1.0);     auto mulAdd = (Vector v1, double x1, Vector v2, double x2) => v1 * x1 + v2 * x2;     auto rotate = (Vector i, Vector j, double alpha) =>        tuple(mulAdd(i, +cos(alpha), j, sin(alpha)),              mulAdd(i, -sin(alpha), j, cos(alpha)));     auto p = rotate(i, j, longitudeOfAscendingNode);    i = p[0]; j = p[1];    p = rotate(j, k, inclination);    j = p[0];    p = rotate(i, j, argumentOfPeriapsis);    i = p[0]; j = p[1];     auto l = semiMajorAxis * ((eccentricity == 1.0) ? 2.0 : (1.0 - eccentricity * eccentricity));    auto c = cos(trueAnomaly);    auto s = sin(trueAnomaly);    auto r = l / (1.0 + eccentricity * c);    auto rprime = s * r * r / l;    auto position = mulAdd(i, c, j, s) * r;    auto speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c);    speed /= speed.abs();    speed *= sqrt(2.0 / r - 1.0 / semiMajorAxis);    return tuple(position, speed);} void main() {    auto res = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0);    writeln("Position : ", res[0]);    writeln("Speed    : ", res[1]);}`
Output:
```Position : (0.7794228433986798, 0.4500000346536842, 0.0000000000000000)
Speed    : (-0.5527708409604437, 0.9574270831797614, 0.0000000000000000)```

## Go

Translation of: Kotlin
`package main import (    "fmt"    "math") type vector struct{ x, y, z float64 } func (v vector) add(w vector) vector {    return vector{v.x + w.x, v.y + w.y, v.z + w.z}} func (v vector) mul(m float64) vector {    return vector{v.x * m, v.y * m, v.z * m}} func (v vector) div(d float64) vector {    return v.mul(1.0 / d)} func (v vector) abs() float64 {    return math.Sqrt(v.x*v.x + v.y*v.y + v.z*v.z)} func (v vector) String() string {    return fmt.Sprintf("(%g, %g, %g)", v.x, v.y, v.z)} func orbitalStateVectors(    semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode,    argumentOfPeriapsis, trueAnomaly float64) (position vector, speed vector) {     i := vector{1, 0, 0}    j := vector{0, 1, 0}    k := vector{0, 0, 1}     mulAdd := func(v1, v2 vector, x1, x2 float64) vector {        return v1.mul(x1).add(v2.mul(x2))    }     rotate := func(i, j vector, alpha float64) (vector, vector) {        return mulAdd(i, j, math.Cos(alpha), math.Sin(alpha)),            mulAdd(i, j, -math.Sin(alpha), math.Cos(alpha))    }     i, j = rotate(i, j, longitudeOfAscendingNode)    j, _ = rotate(j, k, inclination)    i, j = rotate(i, j, argumentOfPeriapsis)     l := 2.0    if eccentricity != 1.0 {        l = 1.0 - eccentricity*eccentricity    }    l *= semimajorAxis    c := math.Cos(trueAnomaly)    s := math.Sin(trueAnomaly)    r := l / (1.0 + eccentricity*c)    rprime := s * r * r / l    position = mulAdd(i, j, c, s).mul(r)    speed = mulAdd(i, j, rprime*c-r*s, rprime*s+r*c)    speed = speed.div(speed.abs())    speed = speed.mul(math.Sqrt(2.0/r - 1.0/semimajorAxis))    return} func main() {    long := 355.0 / (113.0 * 6.0)    position, speed := orbitalStateVectors(1.0, 0.1, 0.0, long, 0.0, 0.0)    fmt.Println("Position :", position)    fmt.Println("Speed    :", speed)}`
Output:
```Position : (0.7794228433986797, 0.45000003465368416, 0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0)
```

## Java

Translation of: Kotlin
`public class OrbitalElements {    private static class Vector {        private double x, y, z;         public Vector(double x, double y, double z) {            this.x = x;            this.y = y;            this.z = z;        }         public Vector plus(Vector rhs) {            return new Vector(x + rhs.x, y + rhs.y, z + rhs.z);        }         public Vector times(double s) {            return new Vector(s * x, s * y, s * z);        }         public Vector div(double d) {            return new Vector(x / d, y / d, z / d);        }         public double abs() {            return Math.sqrt(x * x + y * y + z * z);        }         @Override        public String toString() {            return String.format("(%.16f, %.16f, %.16f)", x, y, z);        }    }     private static Vector mulAdd(Vector v1, Double x1, Vector v2, Double x2) {        return v1.times(x1).plus(v2.times(x2));    }     private static Vector[] rotate(Vector i, Vector j, double alpha) {        return new Vector[]{            mulAdd(i, Math.cos(alpha), j, Math.sin(alpha)),            mulAdd(i, -Math.sin(alpha), j, Math.cos(alpha))        };    }     private static Vector[] orbitalStateVectors(        double semimajorAxis, double eccentricity,         double inclination, double longitudeOfAscendingNode,         double argumentOfPeriapsis, double trueAnomaly    ) {        Vector i = new Vector(1, 0, 0);        Vector j = new Vector(0, 1, 0);        Vector k = new Vector(0, 0, 1);         Vector[] p = rotate(i, j, longitudeOfAscendingNode);        i = p[0];        j = p[1];        p = rotate(j, k, inclination);        j = p[0];        p = rotate(i, j, argumentOfPeriapsis);        i = p[0];        j = p[1];         double l = (eccentricity == 1.0) ? 2.0 : 1.0 - eccentricity * eccentricity;        l *= semimajorAxis;        double c = Math.cos(trueAnomaly);        double s = Math.sin(trueAnomaly);        double r = l / (1.0 + eccentricity * c);        double rprime = s * r * r / l;        Vector position = mulAdd(i, c, j, s).times(r);        Vector speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c);        speed = speed.div(speed.abs());        speed = speed.times(Math.sqrt(2.0 / r - 1.0 / semimajorAxis));         return new Vector[]{position, speed};    }     public static void main(String[] args) {        Vector[] ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0);        System.out.printf("Position : %s\n", ps[0]);        System.out.printf("Speed : %s\n", ps[1]);    }}`
Output:
```Position : (0.7794228433986797, 0.4500000346536842, 0.0000000000000000)
Speed : (-0.5527708409604438, 0.9574270831797618, 0.0000000000000000)```

## jq

Translation of: Wren
Works with: jq

Works with gojq, the Go implementation of jq

`# Array/vector operationsdef addVectors: transpose | map(add); def multiply(\$m): map(. * \$m); def divide(\$d): map(1/\$d); def abs: map(.*.) | add | sqrt;  def orbitalStateVectors(semimajorAxis; eccentricity; inclination;      longitudeOfAscendingNode; argumentOfPeriapsis; trueAnomaly):  def mulAdd(\$v1; \$x1; \$v2; \$x2):    [(\$v1|multiply(\$x1)), (\$v2|multiply(\$x2))] | addVectors;   def rotate(\$i; \$j; \$alpha):    [mulAdd(\$i;  \$alpha|cos; \$j; \$alpha|sin),     mulAdd(\$i; -\$alpha|sin; \$j; \$alpha|cos)];   [1, 0, 0] as \$i  | [0, 1, 0] as \$j  | [0, 0, 1] as \$k  | rotate(\$i; \$j; longitudeOfAscendingNode) as [\$i, \$j]  | rotate(\$j; \$k; inclination) as [\$j, \$_]  | rotate(\$i; \$j; argumentOfPeriapsis) as [\$i, \$j]  | (semimajorAxis * (if (eccentricity == 1) then 2 else (1 - eccentricity * eccentricity) end)) as \$l  | (trueAnomaly|cos) as \$c  | (trueAnomaly|sin) as \$s  | (\$l / (1 + eccentricity * \$c)) as \$r  | (\$s * \$r * \$r / \$l) as \$rprime   | mulAdd(\$i; \$c; \$j; \$s) | multiply(\$r) as \$position   | mulAdd(\$i; \$rprime * \$c - \$r * \$s; \$j; \$rprime * \$s + \$r * \$c)  | divide(abs)  | multiply( ((2 / \$r) - (1 / semimajorAxis))|sqrt) as \$speed  | [\$position, \$speed] ;`

The Task

`orbitalStateVectors(1; 0.1; 0; 355 / (113 * 6); 0; 0)| "Position : \(.[0])",  "Speed    : \(.[1])"`
Output:
```Position : [0.7794228433986797,0.45000003465368416,0]
Speed    : [1.228379551983482,1.228379551983482,1.228379551983482]
```

## Julia

Translation of: Kotlin
`using GeometryTypesimport Base.abs, Base.print Vect = Point3Base.abs(p::Vect) = sqrt(sum(x -> x*x, p))Base.print(io::IO, p::Vect) = print(io, "(", p[1], ", ", p[2], ", ", p[3], ")")muladd(v1, x1, v2, x2) = v1 * x1 + v2 * x2rotate(i, j, a) = Pair(muladd(i, cos(a), j, sin(a)), muladd(i, -sin(a), j, cos(a))) function orbitalStateVectors(semimajorAxis, eccentricity, inclination,    longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly)     i, j, k = Vect(1.0, 0.0, 0.0), Vect(0.0, 1.0, 0.0), Vect(0.0, 0.0, 1.0)    p = rotate(i, j, longitudeOfAscendingNode)    i, j = p    p = rotate(j, k, inclination)    p = rotate(i, p[1], argumentOfPeriapsis)    i, j = p     l = semimajorAxis * (eccentricity == 1.0 ? 2.0 : (1.0 - eccentricity * eccentricity))    c, s = cos(trueAnomaly), sin(trueAnomaly)    r = l / (1.0 + eccentricity * c)    rprime, position = s * r * r / l, muladd(i, c, j, s) * r    speed = muladd(i, rprime * c - r * s, j, rprime * s + r * c)    speed /= abs(speed)    speed *= sqrt(2.0 / r - 1.0 / semimajorAxis)    return Pair(position, speed)end function testorbitalmath()    (position, speed) = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)    println("Position : \$position\nSpeed    : \$speed")end testorbitalmath() `
Output:
```Position : (0.7794228433986797, 0.45000003465368416, 0.0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0.0)
```

## Kotlin

Translation of: Sidef
`// version 1.1.4-3 class Vector(val x: Double, val y: Double, val z: Double) {     operator fun plus(other: Vector) = Vector(x + other.x, y + other.y, z + other.z)     operator fun times(m: Double) = Vector(x * m, y * m, z * m)     operator fun div(d: Double) = this * (1.0 / d)     fun abs() = Math.sqrt(x * x + y * y + z * z)     override fun toString() = "(\$x, \$y, \$z)"} fun orbitalStateVectors(    semimajorAxis: Double,    eccentricity: Double,    inclination: Double,    longitudeOfAscendingNode: Double,    argumentOfPeriapsis: Double,    trueAnomaly: Double): Pair<Vector, Vector> {    var i = Vector(1.0, 0.0, 0.0)    var j = Vector(0.0, 1.0, 0.0)    var k = Vector(0.0, 0.0, 1.0)     fun mulAdd(v1: Vector, x1: Double, v2: Vector, x2: Double) = v1 * x1 + v2 * x2     fun rotate(i: Vector, j: Vector, alpha: Double) =         Pair(mulAdd(i, +Math.cos(alpha), j, Math.sin(alpha)),             mulAdd(i, -Math.sin(alpha), j, Math.cos(alpha)))     var p = rotate(i, j, longitudeOfAscendingNode)    i = p.first; j = p.second    p = rotate(j, k, inclination)    j = p.first    p = rotate(i, j, argumentOfPeriapsis)    i = p.first; j = p.second     val l = semimajorAxis * (if (eccentricity == 1.0) 2.0 else (1.0 - eccentricity * eccentricity))    val c = Math.cos(trueAnomaly)    val s = Math.sin(trueAnomaly)    val r = l / (1.0 + eccentricity * c)    val rprime = s * r * r / l    val position = mulAdd(i, c, j, s) * r    var speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c)    speed /= speed.abs()    speed *= Math.sqrt(2.0 / r - 1.0 / semimajorAxis)    return Pair(position, speed)} fun main(args: Array<String>) {    val (position, speed) = orbitalStateVectors(        semimajorAxis = 1.0,        eccentricity = 0.1,        inclination = 0.0,        longitudeOfAscendingNode = 355.0 / (113.0 * 6.0),        argumentOfPeriapsis = 0.0,        trueAnomaly = 0.0    )     println("Position : \$position")    println("Speed    : \$speed")}`
Output:
```Position : (0.7794228433986797, 0.45000003465368416, 0.0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0.0)```

## Nim

Translation of: Kotlin
`import math, strformat type Vector = tuple[x, y, z: float] func `+`(v1, v2: Vector): Vector = (v1.x + v2.x, v1.y + v2.y, v1.z + v2.z)func `*`(v: Vector; m: float): Vector = (v.x * m, v.y * m, v.z * m)func `*=`(v: var Vector; m: float) = v.x *= m; v.y *= m; v.z *= mfunc `/=`(v: var Vector; d: float) = v.x /= d; v.y /= d; v.z /= dfunc abs(v: Vector): float = sqrt(v.x * v.x + v.y * v.y + v.z * v.z)func `\$`(v: Vector): string = &"({v.x}, {v.y}, {v.z})" func orbitalStateVectors(semimajorAxis: float,                         eccentricity: float,                         inclination: float,                         longitudeOfAscendingNode: float,                         argumentOfPeriapsis: float,                         trueAnomaly: float): tuple[position, speed: Vector] =   var    i: Vector = (1.0, 0.0, 0.0)    j: Vector = (0.0, 1.0, 0.0)    k: Vector = (0.0, 0.0, 1.0)    func mulAdd(v1: Vector; x1: float; v2: Vector; x2: float): Vector = v1 * x1 + v2 * x2   func rotate(a, b: Vector; alpha: float): (Vector, Vector) =    (mulAdd(a, cos(alpha), b, sin(alpha)), mulAdd(a, -sin(alpha), b, cos(alpha)))   var p = rotate(i, j, longitudeOfAscendingNode)  (i, j) = p  p = rotate(j, k, inclination)  j = p[0]  p = rotate(i, j, argumentOfPeriapsis)  (i, j) = p   let    l = semimajorAxis * (if eccentricity == 1: 2.0 else: 1.0 - eccentricity * eccentricity)    c = cos(trueAnomaly)    s = sin(trueAnomaly)    r = l / (1.0 + eccentricity * c)    rprime = s * r * r / l   result.position = mulAdd(i, c, j, s) * r  result.speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c)  result.speed /= abs(result.speed)  result.speed *= sqrt(2 / r - 1 / semimajorAxis)  let (position, speed) = orbitalStateVectors(semimajorAxis = 1.0,                                            eccentricity = 0.1,                                            inclination = 0.0,                                            longitudeOfAscendingNode = 355.0 / (113.0 * 6.0),                                            argumentOfPeriapsis = 0.0,                                            trueAnomaly = 0.0)echo "Position: ", positionecho "Speed:    ", speed`
Output:
```Position: (0.7794228433986797, 0.4500000346536842, 0.0)
Speed:    (-0.5527708409604438, 0.9574270831797618, 0.0)```

## ooRexx

Translation of: Java
`/* REXX */Numeric Digits 16ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)Say "Position :" ps~x~tostringSay "Speed    :" ps~y~tostringSay 'Raku:'pi=rxCalcpi(16)ps=orbitalStateVectors(1,.1,pi/18,pi/6,pi/4,0) /*Raku*/Say "Position :" ps~x~tostringSay "Speed    :" ps~y~tostring ::class v2::method init  expose x y  Use Arg x,y::attribute x::attribute y ::class vector::method init  expose x y z  use strict arg x = 0, y = 0, z = 0  -- defaults to 0 for any non-specified coordinates ::attribute x::attribute y::attribute z ::method print  expose x y z  Numeric Digits 16  Say 'Vector:'||x'/'y'/'z ::method tostring  expose x y z  Return '('||x','y','z')' ::method abs  expose x y z  Numeric Digits 16  Return rxCalcsqrt(x**2+y**2+z**2,16) ::method '*'  expose x y z  Parse Arg f  Numeric Digits 16  Return .vector~new(x*f,y*f,z*f) ::method '/'  expose x y z  Parse Arg f  Numeric Digits 16  Return .vector~new(x/f,y/f,z/f) ::method '+'  expose x y z  Use Arg v2  Numeric Digits 16  Return .vector~new(x+v2~x,y+v2~y,z+v2~z) ::routine orbitalStateVectorsUse Arg  semimajorAxis,,         eccentricity,,         inclination,,         longitudeOfAscendingNode,,         argumentOfPeriapsis,,         trueAnomalyNumeric Digits 16i = .vector~new(1, 0, 0)j = .vector~new(0, 1, 0)k = .vector~new(0, 0, 1)p = rotate(i, j, longitudeOfAscendingNode)i = p~xj = p~yp = rotate(j, k, inclination)j = p~xp = rotate(i, j, argumentOfPeriapsis)i = p~xj = p~yIf eccentricity=1 Then l=2Else l=1-eccentricity*eccentricityl*=semimajorAxisc=rxCalccos(trueAnomaly,16,'R')s=rxCalcsin(trueAnomaly,16,'R')r=l/(1+eccentricity*c)rprime=s*r*r/lposition=mulAdd(i,c,j,s)~'*'(r)speed=mulAdd(i,rprime*c-r*s,j,rprime*s+r*c)speed=speed~'/'(speed~abs)speed=speed~'*'(rxCalcsqrt(2.0/r-1.0/semimajorAxis,16))Return .v2~new(position,speed) ::routine muladd  Use Arg v1,x1,v2,x2  Numeric Digits 16  w1=v1~'*'(x1)  w2=v2~'*'(x2)  Return w1~'+'(w2) ::routine rotate  Use Arg i,j,alpha  Numeric Digits 16  xx=mulAdd(i,rxCalccos(alpha,16,'R'),j,rxCalcsin(alpha,16,'R'))  yy=mulAdd(i,-rxCalcsin(alpha,16,'R'),j,rxCalccos(alpha,16,'R'))  res=.v2~new(xx,yy)  Return res ::requires 'rxmath' LIBRARY`
Output:
```Position : (0.7794228433986798,0.4500000346536842,0)
Speed    : (-0.5527708409604436,0.9574270831797613,0)
Raku:
Position : (0.2377712839822067,0.8609602616977158,0.1105090235720755)
Speed    : (-1.061933017480060,0.2758500205692495,0.1357470248655981)```

## Perl

Translation of: Raku
`use strict;use warnings;use Math::Vector::Real; sub orbital_state_vectors {    my (        \$semimajor_axis,        \$eccentricity,        \$inclination,        \$longitude_of_ascending_node,        \$argument_of_periapsis,        \$true_anomaly    ) = @_[0..5];     my (\$i, \$j, \$k) = (V(1,0,0), V(0,1,0), V(0,0,1));     sub rotate {        my \$alpha = shift;        @_[0,1] = (            +cos(\$alpha)*\$_[0] + sin(\$alpha)*\$_[1],            -sin(\$alpha)*\$_[0] + cos(\$alpha)*\$_[1]        );    }     rotate \$longitude_of_ascending_node, \$i, \$j;    rotate \$inclination,                 \$j, \$k;    rotate \$argument_of_periapsis,       \$i, \$j;     my \$l = \$eccentricity == 1 ? # PARABOLIC CASE        2*\$semimajor_axis :        \$semimajor_axis*(1 - \$eccentricity**2);     my (\$c, \$s) = (cos(\$true_anomaly), sin(\$true_anomaly));     my \$r = \$l/(1 + \$eccentricity*\$c);    my \$rprime = \$s*\$r**2/\$l;     my \$position = \$r*(\$c*\$i + \$s*\$j);     my \$speed =     (\$rprime*\$c - \$r*\$s)*\$i + (\$rprime*\$s + \$r*\$c)*\$j;    \$speed /= abs(\$speed);    \$speed *= sqrt(2/\$r - 1/\$semimajor_axis);     {        position => \$position,        speed    => \$speed    }} use Data::Dumper; print Dumper orbital_state_vectors    1,                             # semimajor axis    0.1,                           # eccentricity    0,                             # inclination    355/113/6,                     # longitude of ascending node    0,                             # argument of periapsis    0                              # true-anomaly    ;`
Output:
```\$VAR1 = {
'position' => bless( [
'0.77942284339868',
'0.450000034653684',
'0'
], 'Math::Vector::Real' ),
'speed' => bless( [
'-0.552770840960444',
'0.957427083179762',
'0'
], 'Math::Vector::Real' )
};```

## Phix

Translation of: Python
`function vabs(sequence v)    return sqrt(sum(sq_power(v,2)))end function  function mulAdd(sequence v1, atom x1, sequence v2, atom x2)    return sq_add(sq_mul(v1,x1),sq_mul(v2,x2))end function function rotate(sequence i, j, atom alpha)    atom ca = cos(alpha),         sa = sin(alpha)    return {mulAdd(i,ca,j,sa),mulAdd(i,-sa,j,ca)}end function procedure orbitalStateVectors(atom semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly)    sequence i = {1, 0, 0},             j = {0, 1, 0},             k = {0, 0, 1}     {i,j} = rotate(i, j, longitudeOfAscendingNode)    {j} = rotate(j, k, inclination)    {i,j} = rotate(i, j, argumentOfPeriapsis)     atom l = iff(eccentricity=1?2:1-eccentricity*eccentricity)*semimajorAxis,         c = cos(trueAnomaly),         s = sin(trueAnomaly),         r = 1 / (1+eccentricity*c),         rprime = s * r * r / l    sequence posn = sq_mul(mulAdd(i, c, j, s),r),             speed = mulAdd(i, rprime*c-r*s, j, rprime*s+r*c)    speed = sq_div(speed,vabs(speed))    speed = sq_mul(speed,sqrt(2/r - 1/semimajorAxis))     puts(1,"Position :") ?posn    puts(1,"Speed    :") ?speedend procedure orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)`
Output:
```Position :{0.7872958014,0.4545454895,0}
Speed    :{-0.5477225997,0.9486832737,0}
```

## Prolog

Translation of: C#

This implementation uses the CLP/R library of swi-prolog, but doesn't have to. This removes the need for a vector divide and has limited capability to reverse the functionality (eg: given the position/speed find some orbital elements).

`:- use_module(library(clpr)). v3_add(v(X1,Y1,Z1),v(X2,Y2,Z2),v(X,Y,Z)) :-	{ X = X1 + X2 },	{ Y = Y1 + Y2 },	{ Z = Z1 + Z2 }. v3_mul(v(X1,Y1,Z1),M,v(X,Y,Z)) :-	{ X = X1 * M },	{ Y = Y1 * M },	{ Z = Z1 * M }. v3_muladd(V1,X1,V2,X2,R) :-	v3_mul(V1,X1,V1X1),	v3_mul(V2,X2,V2X2),	v3_add(V1X1,V2X2,R). v3_rotate(IV, JV, Alpha, R1, R2) :-	{ SinA = sin(Alpha) },	{ CosA = cos(Alpha) },	{ NegSinA = 0 - SinA },	v3_muladd(IV, CosA, JV, SinA, R1),	v3_muladd(IV, NegSinA, JV, CosA, R2). v3_abs(v(X,Y,Z), Abs) :- { Abs = (X * X + Y * Y + Z * Z) ^ 0.5 }. orbital_state_vectors(	o(SemiMajor,Ecc,Inc,LongAscNode,ArgPer,TrueAnon), 	Position,	Speed) :- 	v3_rotate(v(1,0,0),v(0,1,0),LongAscNode,I1,J1),		v3_rotate(J1,v(0,0,1),Inc,J2,_),	v3_rotate(I1,J2,ArgPer,I,J), 	find_l(Ecc, SemiMajor, L), 	{ C = cos(TrueAnon) },	{ S = sin(TrueAnon) },	{ R = L / (1.0 + Ecc * C ) },	{ RPrime = S * R * R / L }, 	v3_muladd(I, C, J, S, P1),	v3_mul(P1, R, Position), 	{ SpeedIr = RPrime * C - R * S },	{ SpeedJr = RPrime * S + R * C },	v3_muladd(I, SpeedIr, J, SpeedJr, SpeedA),	v3_abs(SpeedA, SpeedAbs),	v3_mul(SpeedDiv, SpeedAbs, SpeedA),	{ Sf = (2.0 / R - 1.0 / SemiMajor ) ^ 0.5 },	v3_mul(SpeedDiv, Sf, Speed).  find_l(1.0, SemiMajor, L) :- 	{ L = SemiMajor * 2.0 }.find_l(Ecc, SemiMajor, L) :- 	dif(Ecc,1.0), 	{ L = SemiMajor * (1.0 - Ecc * Ecc) }.`
Output:
```?- { T = 355 / (113 * 6) }, orbital_state_vectors(o(1.0,0.1,0.0,T,0,0), P, S).
T = 0.523598820058997,
P = v(0.7794228433986797, 0.45000003465368416, 0.0),
S = v(-0.5527708409604438, 0.9574270831797618, 0.0) .

?-```

## Python

`import math class Vector:    def __init__(self, x, y, z):        self.x = x        self.y = y        self.z = z     def __add__(self, other):        return Vector(self.x + other.x, self.y + other.y, self.z + other.z)     def __mul__(self, other):        return Vector(self.x * other, self.y * other, self.z * other)     def __div__(self, other):        return Vector(self.x / other, self.y / other, self.z / other)     def __str__(self):        return '({x}, {y}, {z})'.format(x=self.x, y=self.y, z=self.z)     def abs(self):        return math.sqrt(self.x*self.x + self.y*self.y + self.z*self.z) def mulAdd(v1, x1, v2, x2):    return v1 * x1 + v2 * x2 def rotate(i, j, alpha):    return [mulAdd(i,math.cos(alpha),j,math.sin(alpha)), mulAdd(i,-math.sin(alpha),j,math.cos(alpha))] def orbitalStateVectors(semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly):    i = Vector(1, 0, 0)    j = Vector(0, 1, 0)    k = Vector(0, 0, 1)     p = rotate(i, j, longitudeOfAscendingNode)    i = p[0]    j = p[1]    p = rotate(j, k, inclination)    j = p[0]    p  =rotate(i, j, argumentOfPeriapsis)    i = p[0]    j = p[1]     l = 2.0 if (eccentricity == 1.0) else 1.0 - eccentricity * eccentricity    l *= semimajorAxis    c = math.cos(trueAnomaly)    s = math.sin(trueAnomaly)    r = 1 / (1.0 + eccentricity * c)    rprime = s * r * r / l    position = mulAdd(i, c, j, s) * r    speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c)    speed = speed / speed.abs()    speed = speed * math.sqrt(2.0 / r - 1.0 / semimajorAxis)     return [position, speed] ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)print "Position :", ps[0]print "Speed    :", ps[1]`
Output:
```Position : (0.787295801413, 0.454545489549, 0.0)
Speed    : (-0.547722599684, 0.948683273698, 0.0)```

## Raku

(formerly Perl 6) We'll use the Clifford geometric algebra library but only for the vector operations.

`sub orbital-state-vectors(    Real :\$semimajor-axis where * >= 0,    Real :\$eccentricity   where * >= 0,    Real :\$inclination,    Real :\$longitude-of-ascending-node,    Real :\$argument-of-periapsis,    Real :\$true-anomaly) {    use Clifford;    my (\$i, \$j, \$k) = @e[^3];     sub rotate(\$a is rw, \$b is rw, Real \α) {        (\$a, \$b) = cos(α)*\$a + sin(α)*\$b, -sin(α)*\$a + cos(α)*\$b;    }    rotate(\$i, \$j, \$longitude-of-ascending-node);    rotate(\$j, \$k, \$inclination);    rotate(\$i, \$j, \$argument-of-periapsis);     my \l = \$eccentricity == 1 ?? # PARABOLIC CASE        2*\$semimajor-axis !!        \$semimajor-axis*(1 - \$eccentricity**2);     my (\$c, \$s) = .cos, .sin given \$true-anomaly;     my \r = l/(1 + \$eccentricity*\$c);    my \rprime = \$s*r**2/l;     my \$position = r*(\$c*\$i + \$s*\$j);     my \$speed =     (rprime*\$c - r*\$s)*\$i + (rprime*\$s + r*\$c)*\$j;    \$speed /= sqrt(\$speed**2);    \$speed *= sqrt(2/r - 1/\$semimajor-axis);     { :\$position, :\$speed }} say orbital-state-vectors    semimajor-axis => 1,    eccentricity => 0.1,    inclination => pi/18,    longitude-of-ascending-node => pi/6,    argument-of-periapsis => pi/4,    true-anomaly => 0;`
Output:
`{position => 0.237771283982207*e0+0.860960261697716*e1+0.110509023572076*e2, speed => -1.06193301748006*e0+0.27585002056925*e1+0.135747024865598*e2}`

## REXX

### version 1

Translation of: Java

Vectors are represented by strings: 'x/y/z'

`/* REXX */Numeric Digits 16  Parse Value orbitalStateVectors(1.0,0.1,0.0,355.0/(113.0*6.0),0.0,0.0),      With position speed  Say "Position :" tostring(position)  Say "Speed    :" tostring(speed)  Exit orbitalStateVectors: Procedure  Parse Arg semimajorAxis,,            eccentricity,,            inclination,,            longitudeOfAscendingNode,,            argumentOfPeriapsis,,            trueAnomaly  i='1/0/0'  j='0/1/0'  k='0/0/1'  Parse Value rotate(i, j, longitudeOfAscendingNode) With i j  Parse Value rotate(j, k, inclination) With j p  Parse Value rotate(i, j, argumentOfPeriapsis) With i j  If eccentricity=1 Then l=2  Else l=1-eccentricity*eccentricity  l=l*semimajorAxis  c=my_cos(trueAnomaly,16)  s=my_sin(trueAnomaly,16)  r=l/(1+eccentricity*c)  rprime=s*r*r/l  position=vmultiply(mulAdd(i,c,j,s),r)  speed=mulAdd(i,rprime*c-r*s,j,rprime*s+r*c)  speed=vdivide(speed,abs(speed))  speed=vmultiply(speed,my_sqrt(2.0/r-1.0/semimajorAxis,16))  Return position speed abs: Procedure  Parse Arg v.x '/' v.y '/' v.z  Return my_sqrt(v.x**2+v.y**2+v.z**2,16) muladd: Procedure  Parse Arg v1,x1,v2,x2  Parse Var v1 v1.x '/' v1.y '/' v1.z  Parse Var v2 v2.x '/' v2.y '/' v2.z  z=(v1.x*x1+v2.x*x2)||'/'||(v1.y*x1+v2.y*x2)||'/'||(v1.z*x1+v2.z*x2)  Return z rotate: ProcedureParse Arg i,j,alpha  xx=mulAdd(i,my_cos(alpha,16,'R'),j,my_sin(alpha,16))  yy=mulAdd(i,-my_sin(alpha,16,'R'),j,my_cos(alpha,16))  Return xx yy vmultiply: Procedure  Parse Arg v,d  Parse Var v v.x '/' v.y '/' v.z  Return (v.x*d)||'/'||(v.y*d)||'/'||(v.z*d) vdivide: Procedure  Parse Arg v,d  Parse Var v v.x '/' v.y '/' v.z  Return (v.x/d)||'/'||(v.y/d)||'/'||(v.z/d) tostring:  Parse Arg v.x '/' v.y '/' v.z  Return '('v.x','v.y','v.z')' my_sqrt: Procedure/* REXX **************************************************************** EXEC to calculate the square root of a = 2 with high precision**********************************************************************/  Parse Arg x,prec  If prec<9 Then prec=9  prec1=2*prec  eps=10**(-prec1)  k = 1  Numeric Digits 3  r0= x  r = 1  Do i=1 By 1 Until r=r0 | ('ABS'(r*r-x)<eps)    r0 = r    r  = (r + x/r) / 2    k  = min(prec1,2*k)    Numeric Digits (k + 5)    End  Numeric Digits prec  Return r+0 my_sin: Procedure/* REXX ***************************************************************** Return my_sin(x<,p>) -- with the specified precision* my_sin(x) = x-(x**3/3!)+(x**5/5!)-(x**7/7!)+-...***********************************************************************/  Parse Arg x,prec  If prec='' Then prec=9  Numeric Digits (2*prec)  Numeric Fuzz   3  pi=left('3.1415926535897932384626433832795028841971693993751058209749445923',2*prec+1)  Do While x>pi    x=x-pi    End  Do While x<-pi    x=x+pi    End  o=x  u=1  r=x  Do i=3 By 2    ra=r    o=-o*x*x    u=u*i*(i-1)    r=r+(o/u)    If r=ra Then Leave    End  Numeric Digits prec  Return r+0 my_cos: Procedure/* REXX ***************************************************************** Return my_cos(x) -- with specified precision* my_cos(x) = 1-(x**2/2!)+(x**4/4!)-(x**6/6!)+-...***********************************************************************/  Parse Arg x,prec  If prec='' Then prec=9  Numeric Digits (2*prec)  Numeric Fuzz 3  o=1  u=1  r=1  Do i=1 By 2    ra=r    o=-o*x*x    u=u*i*(i+1)    r=r+(o/u)    If r=ra Then Leave    End  Numeric Digits prec  Return r+0`
Output:
```Position : (0.7794228433986798,0.4500000346536842,0)
Speed    : (-0.5527708409604436,0.9574270831797613,0)```

### version 2

Re-coding of REXX version 1,   but with greater decimal digits precision.

`/*REXX pgm converts orbital elements ──► orbital state vectors  (angles are in radians).*/numeric digits length( pi() )  -  length(.)      /*limited to pi len, but show 1/3 digs.*/call orbV 1,   .1,   0,    355/113/6,    0,    0 /*orbital elements taken from:  Java   */call orbV 1,   .1,  pi/18,      pi/6,  pi/4,   0 /*   "        "      "     "    Perl 6 */exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/orbV: procedure;  parse arg  semiMaj, eccentricity, inclination, node, periapsis, anomaly      say;     say center(' orbital elements ', 99, "═")      say '            semi-major axis:'  fmt(semiMaj)      say '               eccentricity:'  fmt(eccentricity)      say '                inclination:'  fmt(inclination)      say '   ascending node longitude:'  fmt(node)      say '      argument of periapsis:'  fmt(periapsis)      say '               true anomaly:'  fmt(anomaly)      i= 1 0 0;          j= 0 1 0;        k= 0 0 1    /*define the  I,  J,  K   vectors.*/      parse value rot(i, j, node)        with i '~' j /*rotate ascending node longitude.*/      parse value rot(j, k, inclination) with j '~'   /*rotate the inclination.         */      parse value rot(i, j, periapsis)   with i '~' j /*rotate the argument of periapsis*/      if eccentricity=1  then L= 2                         else L= 1 - eccentricity**2      L= L * semiMaj                                  /*calculate the semi─latus rectum.*/      c= cos(anomaly);               s= sin(anomaly)  /*calculate COS and SIN of anomaly*/      r= L / (1 + eccentricity * c)      @= s*r**2 / L;        speed= MA(i,  @*c - r*s,  j,   @*s + r*c)      speed=    mulV( divV( speed, absV(speed) ), sqrt(2 / r  - 1 / semiMaj) )      say '                   position:'  show( mulV( MA(i, c, j, s),  r) )      say '                      speed:'  show( speed);            return/*──────────────────────────────────────────────────────────────────────────────────────*/absV: procedure; parse arg x y z;              return sqrt(x**2  +  y**2  +  z**2)divV: procedure; parse arg x y z, div;         return  (x / div)    (y / div)    (z / div)mulV: procedure; parse arg x y z, mul;         return  (x * mul)    (y * mul)    (z * mul)show: procedure; parse arg a b c;              return '('fmt(a)","   fmt(b)','   fmt(c)")"fmt:  procedure; parse arg #;  return strip( left( left('', #>=0)# / 1, digits() %3), 'T')MA:   procedure; parse arg x y z,@,a b c,\$;    return  (x*@ + a*\$) (y*@ + b*\$) (z*@ + c*\$)pi:   pi= 3.1415926535897932384626433832795028841971693993751058209749445923;    return pirot:  procedure; parse arg i,j,\$; return MA(i,cos(\$),j,sin(\$))'~'MA(i, -sin(\$), j, cos(\$))r2r:  return arg(1)  //  (pi() * 2)                /*normalize radians ──► a unit circle*/.sinCos: arg z 1 _,i; do k=2 by 2 until p=z; p=z; _= -_*\$ /(k*(k+i)); z=z+_; end; return z/*──────────────────────────────────────────────────────────────────────────────────────*/cos:  procedure; arg x;  x= r2r(x);   if x=0  then return 1;    a= abs(x);    Hpi= pi * .5      numeric fuzz min(6, digits() - 3);        if a=pi       then return -1      if a=Hpi | a=Hpi*3  then return   0;      if a=pi / 3   then return .5      if a=pi * 2 / 3     then return '-.5';    \$= x * x;          return .sinCos(1, -1)/*──────────────────────────────────────────────────────────────────────────────────────*/sin:  procedure; arg x;  x= r2r(x);   numeric fuzz min(5, max(1, digits() - 3) )      if x=0  then return 0;   if x=pi*.5  then return 1;   if x==pi*1.5  then return -1      if abs(x)=pi  then return 0;              \$= x * x;          return .sinCos(x, 1)/*──────────────────────────────────────────────────────────────────────────────────────*/sqrt: procedure; arg x;  if x=0  then return 0;  d= digits();  numeric form; m.= 9; h= d+6      numeric digits;  parse value format(x,2,1,,0) 'E0' with g 'E' _ .;  g= g *.5'e'_ % 2        do j=0  while h>9;        m.j= h;              h= h % 2  +  1;    end        do k=j+5  to 0  by '-1';  numeric digits m.k;  g= (g+x/g) * .5;   end;    return g`
output   when using the default internal inputs:
```════════════════════════════════════════ orbital elements ═════════════════════════════════════════
semi-major axis:  1
eccentricity:  0.1
inclination:  0
ascending node longitude:  0.523598820058997050
argument of periapsis:  0
true anomaly:  0
position: ( 0.779422843398679832,  0.450000034653684237,  0)
speed: (-0.552770840960443759,  0.957427083179761535,  0)

════════════════════════════════════════ orbital elements ═════════════════════════════════════════
semi-major axis:  1
eccentricity:  0.1
inclination:  0.174532925199432957
ascending node longitude:  0.523598775598298873
argument of periapsis:  0.785398163397448309
true anomaly:  0
position: ( 0.237771283982206547,  0.860960261697715834,  0.110509023572075562)
speed: (-1.061933017480060047,  0.275850020569249507,  0.135747024865598167)
```

## Scala

`import scala.language.existentials object OrbitalElements extends App {  private val ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)  println(f"Position : \${ps(0)}%s%nSpeed    : \${ps(1)}%s")   private def orbitalStateVectors(semimajorAxis: Double,                                  eccentricity: Double,                                  inclination: Double,                                  longitudeOfAscendingNode: Double,                                  argumentOfPeriapsis: Double,                                  trueAnomaly: Double) = {     def mulAdd(v1: Vector, x1: Double, v2: Vector, x2: Double) = v1 * x1 + v2 * x2     case class Vector(x: Double, y: Double, z: Double) {      def +(term: Vector) =        Vector(x + term.x, y + term.y, z + term.z)      def *(factor: Double) = Vector(factor * x, factor * y, factor * z)      def /(divisor: Double) = Vector(x / divisor, y / divisor, z / divisor)      def abs: Double = math.sqrt(x * x + y * y + z * z)      override def toString: String = f"(\$x%.16f, \$y%.16f, \$z%.16f)"    }     def rotate(i: Vector, j: Vector, alpha: Double) =      Array[Vector](mulAdd(i, math.cos(alpha), j, math.sin(alpha)),        mulAdd(i, -math.sin(alpha), j, math.cos(alpha)))     val p = rotate(Vector(1, 0, 0), Vector(0, 1, 0), longitudeOfAscendingNode)    val p2 = rotate(p(0),      rotate(p(1), Vector(0, 0, 1), inclination)(0),      argumentOfPeriapsis)    val l = semimajorAxis *      (if (eccentricity == 1.0) 2.0 else 1.0 - eccentricity * eccentricity)    val (c, s) = (math.cos(trueAnomaly), math.sin(trueAnomaly))    val r = l / (1.0 + eccentricity * c)    val rprime = s * r * r / l    val speed = mulAdd(p2(0), rprime * c - r * s, p2(1), rprime * s + r * c)    Array[Vector](mulAdd(p(0), c, p2(1), s) * r,      speed / speed.abs * math.sqrt(2.0 / r - 1.0 / semimajorAxis))  } }`
Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

## Sidef

Translation of: Perl
`func orbital_state_vectors(    semimajor_axis,    eccentricity,    inclination,    longitude_of_ascending_node,    argument_of_periapsis,    true_anomaly) {     var (i, j, k) = (        Vector(1, 0, 0),        Vector(0, 1, 0),        Vector(0, 0, 1),    )     func muladd(v1, x1, v2, x2) {        (v1 * x1) + (v2 * x2)    }     func rotate(Ref i, Ref j, α) {        (*i, *j) = (            muladd(*i, +cos(α), *j, sin(α)),            muladd(*i, -sin(α), *j, cos(α)),        )    }     rotate(\i, \j, longitude_of_ascending_node)    rotate(\j, \k, inclination)    rotate(\i, \j, argument_of_periapsis)     var l = (eccentricity == 1 ? 2*semimajor_axis                               : semimajor_axis*(1 - eccentricity**2))     var (c, s) = with(true_anomaly) { (.cos, .sin) }     var r = l/(1 + eccentricity*c)    var rprime = (s * r**2 / l)    var position = muladd(i, c, j, s)*r     var speed = muladd(i, rprime*c - r*s, j, rprime*s + r*c)    speed /= speed.abs    speed *= sqrt(2/r - 1/semimajor_axis)     struct Result { position, speed }    Result(position, speed)} for args in ([    [1, 0.1, 0, 355/(113*6), 0, 0],    [1, 0.1, Num.pi/18, Num.pi/6, Num.pi/4, 0]]) {    var r = orbital_state_vectors(args...)     say "Arguments: #{args}:"    say "Position : #{r.position}"    say "Speed    : #{r.speed}\n"}`
Output:
```Arguments: [1, 1/10, 0, 355/678, 0, 0]:
Position : Vector(0.779422843398679832042176328223663037464703527986, 0.450000034653684237432302249506712706822033851071, 0)
Speed    : Vector(-0.552770840960443759673279062314259546277084494097, 0.957427083179761535246200368614952095349966503287, 0)

Arguments: [1, 1/10, 0.174532925199432957692369076848861271344287188854, 0.523598775598298873077107230546583814032861566563, 0.785398163397448309615660845819875721049292349844, 0]:
Position : Vector(0.23777128398220654779107184959165027147748809404, 0.860960261697715834668966272382699039216399966872, 0.110509023572075562109405412890808505271310143909)
Speed    : Vector(-1.06193301748006004757467368094494935655538772696, 0.275850020569249507846452830330085489348356659642, 0.135747024865598167166145512759280712986072818844)
```

## Swift

Translation of: Kotlin
`import Foundation public struct Vector {  public var x = 0.0  public var y = 0.0  public var z = 0.0   public init(x: Double, y: Double, z: Double) {    (self.x, self.y, self.z) = (x, y, z)  }   public func mod() -> Double {    (x * x + y * y + z * z).squareRoot()  }   public static func + (lhs: Vector, rhs: Vector) -> Vector {    return Vector(      x: lhs.x + rhs.x,      y: lhs.y + rhs.y,      z: lhs.z + rhs.z    )  }   public static func * (lhs: Vector, rhs: Double) -> Vector {    return Vector(      x: lhs.x * rhs,      y: lhs.y * rhs,      z: lhs.z * rhs    )  }   public static func *= (lhs: inout Vector, rhs: Double) {    lhs.x *= rhs    lhs.y *= rhs    lhs.z *= rhs  }   public static func / (lhs: Vector, rhs: Double) -> Vector {    return lhs * (1 / rhs)  }   public static func /= (lhs: inout Vector, rhs: Double) {    lhs = lhs * (1 / rhs)  }} extension Vector: CustomStringConvertible {  public var description: String {    return String(format: "%.6f\t%.6f\t%.6f", x, y, z)  }} private func mulAdd(v1: Vector, x1: Double, v2: Vector, x2: Double) -> Vector {  return v1 * x1 + v2 * x2} private func rotate(_ i: Vector, _ j: Vector, alpha: Double) -> (Vector, Vector) {  return (    mulAdd(v1: i, x1: +cos(alpha), v2: j, x2: sin(alpha)),    mulAdd(v1: i, x1: -sin(alpha), v2: j, x2: cos(alpha))  )} public func orbitalStateVectors(  semimajorAxis: Double,  eccentricity: Double,  inclination: Double,  longitudeOfAscendingNode: Double,  argumentOfPeriapsis: Double,  trueAnomaly: Double) -> (Vector, Vector) {  var i = Vector(x: 1.0, y: 0.0, z: 0.0)  var j = Vector(x: 0.0, y: 1.0, z: 0.0)  let k = Vector(x: 0.0, y: 0.0, z: 1.0)   (i, j) = rotate(i, j, alpha: longitudeOfAscendingNode)  (j, _) = rotate(j, k, alpha: inclination)  (i, j) = rotate(i, j, alpha: argumentOfPeriapsis)   let l = eccentricity == 1.0 ? 2.0 : 1.0 - eccentricity * eccentricity  let c = cos(trueAnomaly)  let s = sin(trueAnomaly)  let r = l / (1.0 + eccentricity * c)  let rPrime = s * r * r / l  let position = mulAdd(v1: i, x1: c, v2: j, x2: s) * r  var speed = mulAdd(v1: i, x1: rPrime * c - r * s, v2: j, x2: rPrime * s + r * c)   speed /= speed.mod()  speed *= (2.0 / r - 1.0 / semimajorAxis).squareRoot()   return (position, speed)} let (position, speed) = orbitalStateVectors(  semimajorAxis: 1.0,  eccentricity: 0.1,  inclination: 0.0,  longitudeOfAscendingNode: 355.0 / (113.0 * 6.0),  argumentOfPeriapsis: 0.0,  trueAnomaly: 0.0) print("Position: \(position); Speed: \(speed)")`
Output:
`Position: 0.779423	0.450000	0.000000; Speed: -0.552771	0.957427	0.000000`

## Wren

Translation of: Kotlin
`class Vector {    construct new(x, y, z) {        _x = x        _y = y        _z = z    }     x { _x }    y { _y }    z { _z }     +(other) { Vector.new(_x + other.x, _y + other.y, _z + other.z) }     *(m) { Vector.new(_x * m, _y * m, _z * m) }     /(d) { this * (1/d) }     abs { (_x *_x + _y *_y + _z * _z).sqrt }     toString { "(%(_x), %(_y), %(_z))" }} var orbitalStateVectors = Fn.new { |semimajorAxis, eccentricity, inclination,                                    longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly|    var i = Vector.new(1, 0, 0)    var j = Vector.new(0, 1, 0)    var k = Vector.new(0, 0, 1)     var mulAdd = Fn.new { |v1, x1, v2, x2| v1 * x1 + v2 * x2 }     var rotate = Fn.new { |i, j, alpha|        return [mulAdd.call(i,  alpha.cos, j, alpha.sin),                 mulAdd.call(i, -alpha.sin, j, alpha.cos)]    }     var p = rotate.call(i, j, longitudeOfAscendingNode)    i = p[0]    j = p[1]    p = rotate.call(j, k, inclination)    j = p[0]    p = rotate.call(i, j, argumentOfPeriapsis)    i = p[0]    j = p[1]     var l = semimajorAxis * ((eccentricity == 1) ? 2 : (1 - eccentricity * eccentricity))    var c = trueAnomaly.cos    var s = trueAnomaly.sin    var r = l / (1 + eccentricity * c)    var rprime = s * r * r / l    var position = mulAdd.call(i, c, j, s) * r    var speed = mulAdd.call(i, rprime * c - r * s, j, rprime * s + r * c)    speed = speed / speed.abs    speed = speed * (2 / r - 1 / semimajorAxis).sqrt    return [position, speed]} var ps = orbitalStateVectors.call(1, 0.1, 0, 355 / (113 * 6), 0, 0)System.print("Position : %(ps[0])")System.print("Speed    : %(ps[1])")`
Output:
```Position : (0.77942284339868, 0.45000003465368, 0)
Speed    : (-0.55277084096044, 0.95742708317976, 0)
```

## zkl

Translation of: Perl
`fcn orbital_state_vectors(semimajor_axis, eccentricity, inclination,         longitude_of_ascending_node, argument_of_periapsis, true_anomaly){   i,j,k:=T(1.0, 0.0, 0.0), T(0.0, 1.0, 0.0), T(0.0, 0.0, 1.0);    vdot:=fcn(c,vector){ vector.apply('*,c) };   vsum:=fcn(v1,v2)   { v1.zipWith('+,v2)  };   rotate:='wrap(alpha, a,b){  // a&b are vectors: (x,y,z)      return(vsum(vdot( alpha.cos(),a), vdot(alpha.sin(),b)), #cos(alpha)*a + sin(alpha)*b             vsum(vdot(-alpha.sin(),a), vdot(alpha.cos(),b)));   };   i,j=rotate(longitude_of_ascending_node,i,j);   j,k=rotate(inclination,		  j,k);   i,j=rotate(argument_of_periapsis,      i,j);    l:=if(eccentricity==1)   # PARABOLIC CASE        semimajor_axis*2  else        semimajor_axis*(1.0 - eccentricity.pow(2));;   c,s,r:=true_anomaly.cos(), true_anomaly.sin(), l/(eccentricity*c + 1);   rprime:=s*r.pow(2)/l;    position:=vdot(r,vsum(vdot(c,i), vdot(s,j)));  #r*(c*i + s*j)    speed:=vsum(vdot(rprime*c - r*s,i), vdot(rprime*s + r*c,j)); #(rprime*c - r*s)*i + (rprime*s + r*c)*j   z:=speed.zipWith('*,speed).sum(0.0).sqrt();  #sqrt(speed**2)   speed=vdot(1.0/z,speed);			#speed/z    speed=vdot((2.0/r - 1.0/semimajor_axis).sqrt(),speed); #speed*sqrt(2/r - 1/semimajor_axis)    return(position,speed);}`
`orbital_state_vectors(    1.0,                           # semimajor axis    0.1,                           # eccentricity    0.0,                           # inclination    (0.0).pi/6,                    # longitude of ascending node    0.0,                           # argument of periapsis    0.0                            # true-anomaly).println();`
Output:
`L(L(0.779423,0.45,0),L(-0.552771,0.957427,0))`