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Orbital elements

From Rosetta Code
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 polar 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)

Ada

Translation of: Kotlin
with Ada.Text_IO;   use Ada.Text_IO;
with Ada.Numerics.Generic_Real_Arrays;
with Ada.Numerics.Generic_Elementary_Functions;

procedure Orbit is

   type Real is new Long_Float;

   package Real_Arrays is
      new Ada.Numerics.Generic_Real_Arrays (Real => Real);
   use Real_Arrays;

   package Math is
      new Ada.Numerics.Generic_Elementary_Functions (Float_Type => Real);

   subtype Vector_3D is Real_Vector (1 .. 3);

   procedure Put (V : Vector_3D) is
      package Real_IO is
        new Ada.Text_Io.Float_IO (Num => Real);
   begin
      Put ("(");
      Real_IO.Put (V (1), Exp => 0, Aft => 6);  Put (",");
      Real_IO.Put (V (2), Exp => 0, Aft => 6);  Put (",");
      Real_IO.Put (V (3), Exp => 0, Aft => 6);  Put (")");
   end Put;

   function Mul_Add (V1 : Vector_3D;
                     X1 : Real;
                     V2 : Vector_3D;
                     X2 : Real)
                    return Vector_3D is
   begin
      return V1 * X1 + V2 * X2;
   end Mul_Add;

   procedure Rotate (R1    : out Vector_3D;
                     R2    : out Vector_3D;
                     I     :     Vector_3D;
                     J     :     Vector_3D;
                     Alpha :     Real) is
   begin
      R1 := Mul_Add (I, +Math.Cos (Alpha), J, Math.Sin (Alpha));
      R2 := Mul_Add (I, -Math.Sin (Alpha), J, Math.Cos (Alpha));
   end Rotate;

   type Orbital_State_Vectors is record
      Position : Vector_3D;
      Speed    : Vector_3D;
   end record;

   function Calculate_Orbital_State
      (Semimajor_Axis             : Real;
      Eccentricity                : Real;
      Inclination                 : Real;
      Longitude_Of_Ascending_Node : Real;
      Argument_Of_Periapsis       : Real;
      True_Anomaly                : Real)
     return Orbital_State_Vectors
   is
      I : Vector_3D := (1.0, 0.0, 0.0);
      J : Vector_3D := (0.0, 1.0, 0.0);
      K : constant Vector_3D := (0.0, 0.0, 1.0);
      P_R1, P_R2 : Vector_3D;
      State    : Orbital_State_Vectors;
      Position : Vector_3D renames State.Position;
      Speed    : Vector_3D renames State.Speed;
   begin
      Rotate (P_R1, P_R2, I, J, Longitude_Of_Ascending_Node);
      I := P_R1; J := P_R2;
      Rotate (P_R1, P_R2, J, K, Inclination);
      J := P_R1;
      Rotate (P_R1, P_R2, I, J, Argument_Of_Periapsis);
      I := P_R1; J := P_R2;
      declare
         L : constant Real :=
           Semimajor_Axis * (if (Eccentricity = 1.0) then 2.0
           else (1.0 - Eccentricity * Eccentricity));
         C : constant Real := Math.Cos (True_Anomaly);
         S : constant Real := Math.Sin (True_Anomaly);
         R : constant Real := L / (1.0 + Eccentricity * C);
         Rprime   : constant Real      := S * R * R / L;
      begin
         Position := Mul_Add (I, C, J, S) * R;
         Speed    := Mul_Add (I, Rprime * C - R * S,
                              J, Rprime * S + R * C);
         Speed := Speed / abs (Speed);
         Speed := Speed * Math.Sqrt (2.0 / R - 1.0 / Semimajor_Axis);
      end;
      return State;
   end Calculate_Orbital_State;

   Longitude : constant Real := 355.000 / (113.000 * 6.000);
   State     : constant Orbital_State_Vectors :=

     Calculate_Orbital_State
       (Semimajor_Axis              => 1.000,
        Eccentricity                => 0.100,
        Inclination                 => 0.000,
        Longitude_Of_Ascending_Node => Longitude,
        Argument_Of_Periapsis       => 0.000,
        True_Anomaly                => 0.000);
begin
   Put ("Position : "); Put (State.Position); New_Line;
   Put ("Speed    : "); Put (State.Speed);    New_Line;
end Orbit;
Output:
Position : ( 0.779423, 0.450000, 0.000000)
Speed    : (-0.552771, 0.957427, 0.000000)

ALGOL 68

Translation of: Lua

(which is a translation of C which is...) 

BEGIN # orbital elements #

    MODE VECTOR = STRUCT( REAL x, y, z );

    OP +   = ( VECTOR v, w )VECTOR: ( x OF v + x OF w, y OF v + y OF w, z OF v + z OF w );
    OP *   = ( VECTOR v, REAL m )VECTOR: ( x OF v * m, y OF v * m, z OF v * m );
    OP /   = ( VECTOR v, REAL d )VECTOR: v * ( 1 / d );
    OP ABS = ( VECTOR v )REAL: sqrt( x OF v * x OF v + y OF v * y OF v + z OF v * z OF v );

    PROC muladd = ( VECTOR  v1, v2, REAL x1, x2 )VECTOR: ( v1 * x1 ) + ( v2 * x2 );
    PROC set v  = ( REF VECTOR v, w, []VECTOR ps )VOID:
         BEGIN v := ps[ LWB ps ]; w := ps[ UPB ps ] END;
    PROC rotate = ( VECTOR i, j, REAL alpha )[]VECTOR:
         ( muladd( i, j,  cos( alpha ), sin( alpha ) ), muladd( i, j, -sin( alpha ), cos( alpha ) ) );
    PROC orbital state vectors = ( REAL semimajor axis, eccentricity, inclination
                                      , longitude of ascending node, argument of periapsis
                                      , true anomaly
                                 , REF VECTOR position, speed 
                                 ) VOID:
         BEGIN
            VECTOR i := ( 1.0, 0.0, 0.0 ), j := ( 0.0, 1.0, 0.0 ), k := ( 0.0, 0.0, 1.0 );
            set v( i, j,          rotate( i, j, longitude of ascending node ) );
            set v( j, LOC VECTOR, rotate( j, k, inclination                 ) );
            set v( i, j,          rotate( i, j, argument of periapsis       ) );
            REAL l = IF eccentricity /= 1 THEN 1 - eccentricity * eccentricity ELSE 2 FI
                   * semimajor axis;
            REAL c       = cos( true anomaly ), s = sin( true anomaly );
            REAL r       = l / ( 1.0 + eccentricity * c );
            REAL rprime  = s * r * r / l;
            position    := muladd( i, j, c, s ) * r;
            speed       := muladd( i, j, rprime * c - r * s, rprime * s + r * c );
            speed       := speed / ABS speed;
            speed       := speed * sqrt( 2 / r - 1 / semimajor axis )
         END # orbital state vectors # ;

    OP   FMT = ( REAL v )STRING:
         BEGIN
            STRING result := fixed( ABS v, 0, 15 );
            IF result[ LWB result ] = "." THEN "0" +=: result FI;
            WHILE result[ UPB result ] = "0" DO result := result[ : UPB result - 1 ] OD;
            IF result[ UPB result ] = "." THEN result := result[ : UPB result - 1 ] FI;
            IF v < 0 THEN "-" + result ELSE result FI
         END # FMT # ;
    OP TOSTRING = ( VECTOR v )STRING: "(" + FMT x OF v + ", " + FMT y OF v + ", " + FMT z OF v + ")";

    BEGIN
        REAL longitude = 355 / ( 113 * 6 );
        VECTOR position, speed;
        orbital state vectors( 1.0, 0.1, 0.0, longitude, 0.0, 0.0, position, speed );
        print( ( "Position : ", TOSTRING position, newline ) );
        print( ( "Speed    : ", TOSTRING speed,    newline ) )
    END
END
Output:
Position : (0.77942284339868, 0.450000034653684, 0)
Speed    : (-0.552770840960444, 0.957427083179762, 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    )
    end
end.
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)

EasyLang

Translation of: 11l
sysconf radians
func[] vmul v[] a .
   return [ v[1] * a v[2] * a v[3] * a ]
.
func[] vadd a[] b[] .
   return [ a[1] + b[1] a[2] + b[2] a[3] + b[3] ]
.
func[] vnorm v[] .
   lng = sqrt (v[1] * v[1] + v[2] * v[2] + v[3] * v[3])
   return [ v[1] / lng v[2] / lng v[3] / lng ]
.
func[] mulAdd v1[] x1 v2[] x2 .
   return vadd vmul v1[] x1 vmul v2[] x2
.
func[][] rotate i[] j[] alpha .
   r[][] &= mulAdd i[] cos alpha j[] sin alpha
   r[][] &= mulAdd i[] -sin alpha j[] cos alpha
   return r[][]
.
func[][] orbStateVectors semimajorAxis eccentricity inclination longOfAscNode argOfPeriapsis trueAnomaly .
   i[] = [ 1 0 0 ]
   j[] = [ 0 1 0 ]
   k[] = [ 0 0 1 ]
   p[][] = rotate i[] j[] longOfAscNode
   i[] = p[1][]
   j[] = p[2][]
   p[][] = rotate j[] k[] inclination
   j[] = p[1][]
   p[][] = rotate i[] j[] argOfPeriapsis
   i[] = p[1][]
   j[] = p[2][]
   l = 2
   if eccentricity <> 1
      l = 1 - eccentricity * eccentricity
   .
   l *= semimajorAxis
   c = cos trueAnomaly
   s = sin trueAnomaly
   r = 1 / (1 + eccentricity * c)
   rprime = s * r * r / l
   position[] = vmul mulAdd i[] c j[] s r
   speed[] = mulAdd i[] (rprime * c - r * s) j[] (rprime * s + r * c)
   speed[] = vmul vnorm speed[] sqrt (2 / r - 1 / semimajorAxis)
   return [ position[] speed[] ]
.
ps[][] = orbStateVectors 1 0.1 0 (355 / (113 * 6)) 0 0
print "Position: " & ps[1][]
print "Speed:    " & ps[2][]
Output:
Position: [ 0.79 0.45 0 ]
Speed:    [ -0.55 0.95 0 ]

FreeBASIC

Translation of: Phix
Sub vabs(v() As Double, Byref result As Double)
    result = 0
    For i As Integer = Lbound(v) To Ubound(v)
        result += v(i) * v(i)
    Next i
    result = Sqr(result)
End Sub

Sub mulAdd(v1() As Double, x1 As Double, v2() As Double, x2 As Double, result() As Double)
    For i As Integer = Lbound(v1) To Ubound(v1)
        result(i) = v1(i) * x1 + v2(i) * x2
    Next i
End Sub

Sub rotate(i() As Double, j() As Double, alfa As Double, result1() As Double, result2() As Double)
    Dim As Double ca = Cos(alfa), sa = Sin(alfa)
    mulAdd(i(), ca, j(), sa, result1())
    mulAdd(i(), -sa, j(), ca, result2())
End Sub

Sub orbitalStateVectors(semimajorAxis As Double, eccentricity As Double, inclination As Double, longitudeOfAscendingNode As Double, argumentOfPeriapsis As Double, trueAnomaly As Double)
    Dim As Double i(1 To 3) = {1, 0, 0}
    Dim As Double j(1 To 3) = {0, 1, 0}
    Dim As Double k(1 To 3) = {0, 0, 1}
    Dim As Double temp1(1 To 3), temp2(1 To 3)
    Dim As Integer index, t
    
    rotate(i(), j(), longitudeOfAscendingNode, temp1(), temp2())
    For index = 1 To 3
        i(index) = temp1(index)
        j(index) = temp2(index)
    Next index
    rotate(j(), k(), inclination, temp1(), temp2())
    For index = 1 To 3
        j(index) = temp1(index)
    Next index
    rotate(i(), j(), argumentOfPeriapsis, temp1(), temp2())
    For index = 1 To 3
        i(index) = temp1(index)
        j(index) = temp2(index)
    Next index
    
    Dim As Double l = Iif(eccentricity = 1, 2, 1 - eccentricity * eccentricity) * semimajorAxis
    Dim As Double c = Cos(trueAnomaly), s = Sin(trueAnomaly)
    Dim As Double r = 1 / (1 + eccentricity * c)
    Dim As Double rprime = s * r * r / l
    Dim As Double posn(1 To 3), speed(1 To 3), vabsResult
    
    mulAdd(i(), c, j(), s, posn())
    For t = Lbound(posn) To Ubound(posn)
        posn(t) *= r
    Next t
    mulAdd(i(), rprime * c - r * s, j(), rprime * s + r * c, speed())
    vabs(speed(), vabsResult)
    mulAdd(speed(), 1 / vabsResult, speed(), 0, speed())
    For t = Lbound(speed) To Ubound(speed)
        speed(t) *= Sqr(2 / r - 1 / semimajorAxis)
    Next t
    Print Using "Position : {&, &, &}"; posn(1); posn(2); posn(3)
    Print Using "Speed    : {&, &, &}"; speed(1); speed(2); speed(3)
End Sub

orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)

Sleep
Output:
Position : {0.7872958014128079, 0.454545489549176, 0}
Speed    : {-0.5477225996842874, 0.9486832736983857, 0}

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)

J

Translation of: Raku
NB. euler rotation matrix, left hand rule
NB. x: axis (0, 1 or 2), y: angle in radians
R=: {{ ((2 1,:1 2) o.(,-)y*_1^2|x)(,&.>/~0 1 2-.x)} =i.3 }}
X=: +/ .* NB. inner product
norm=: % %:@X~

orbitalStateVectors=: {{ 'a e i Om w f'=. y
  NB.  a: semi-major axis
  NB.  e: eccentricity
  NB.  i: inclination
  NB. Om: Longitude of the ascending node
  NB.  w: argument of Periapsis (the other "omega")
  NB.  f: true anomaly (varies with time)
  L=. a*2:`]@.*1-*:e
  'c s'=. 2{.,F=. 2 R f
  ra=. L % 1+ e*c
  rp=. s*ra*ra%L
  ijk=. F X (2 R w)X(0 R i)X(2 R Om)
  position=. ra*{.ijk
  speed=. (%:(2%ra)-%a)*norm(rp,ra,0) X ijk
  position,:speed
}}

The true anomaly, argument of Periapsis, Longitude of the ascending node and inclination are all angles. And we use the dot product of their rotation matrices (in that order) to find the orientation of the orbit and the object's position in that orbit. Here, R finds the rotation matrix for a given angle around a given axis. Here's an example of what R gives us for a sixty degree angle: 

   0 1 2 R&.> 60r180p1 NB. rotate around first, second or third axis
┌────────────────────┬────────────────────┬────────────────────┐
│1         0        0│     0.5 0 _0.866025│      0.5 0.866025 0│
│0       0.5 0.866025│       0 1         0│_0.866025      0.5 0│
│0 _0.866025      0.5│0.866025 0       0.5│        0        0 1│
└────────────────────┴────────────────────┴────────────────────┘

Task example:

   orbitalStateVectors 1 0.1 0 355r678 0 0
 0.779423     0.45 0
_0.552771 0.957427 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 operations
def 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 GeometryTypes
import Base.abs, Base.print

Vect = Point3
Base.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 * x2
rotate(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)

Lua

Translation of: C

...which is translation of Kotlin which is ... 

do -- orbital elements

    local function Vector( x, y, z )
        return { x = x, y= y, z = z }
    end

    local function add( v, w )
        return Vector( v.x + w.x, v.y + w.y, v.z + w.z )
    end

    local function mul( v, m )
        return Vector( v.x * m, v.y * m, v.z * m )
    end

    local function div( v, d )
        return mul( v, 1.0 / d )
    end

    local function vabs( v )
        return math.sqrt( v.x * v.x + v.y * v.y + v.z * v.z )
    end

    local function mulAdd( v1,  v2,  x1,  x2 )
        return add( mul( v1, x1 ), mul( v2, x2 ) ) 
    end

    local function vecAsStr( v )
        return string.format( "(%.17g", v.x )..string.format( ", %.17g", v.y )..string.format( ", %.17g)", v.z )
    end

    local function rotate( i,  j, alpha )
        return mulAdd( i, j,  math.cos( alpha ), math.sin( alpha ) )
             , mulAdd( i, j, -math.sin( alpha ), math.cos( alpha ) )
    end
 
    local function orbitalStateVectors( semimajorAxis, eccentricity, inclination
                                      , longitudeOfAscendingNode, argumentOfPeriapsis
                                      , trueAnomaly
                                      )

        local i, j, k = Vector( 1.0, 0.0, 0.0 ), Vector( 0.0, 1.0, 0.0 ), Vector( 0.0, 0.0, 1.0 )
        local L = 2.0
        i, j = rotate( i, j, longitudeOfAscendingNode )
        j, _ = rotate( j, k, inclination )
        i, j = rotate( i, j, argumentOfPeriapsis )
        if eccentricity ~= 1.0 then L = 1.0 - eccentricity * eccentricity end
        L = L * semimajorAxis
        local c, s = math.cos( trueAnomaly ), math.sin( trueAnomaly )
        local r = L / ( 1.0 + eccentricity * c )
        local rprime   = s * r * r / L;
        local position = mul( mulAdd( i, j, c, s ), r )
        local speed    = mulAdd( i, j, rprime * c - r * s, rprime * s + r * c )
        speed = div( speed, vabs( speed ) )
        speed = mul( speed, math.sqrt( 2.0 / r - 1.0 / semimajorAxis ) )
        return position, speed
    end

    local longitude = 355.0 / ( 113.0 * 6.0 )
    local position, speed = orbitalStateVectors( 1.0, 0.1, 0.0, longitude, 0.0, 0.0 )
    print( "Position : "..vecAsStr( position ) )
    print( "Speed    : "..vecAsStr( speed    ) )
end
Output:
Position : (0.77942284339867973, 0.45000003465368416, 0)
Speed    : (-0.55277084096044382, 0.95742708317976177, 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 *= m
func `/=`(v: var Vector; d: float) = v.x /= d; v.y /= d; v.z /= d
func 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: ", position
echo "Speed:    ", speed
Output:
Position: (0.7794228433986797, 0.4500000346536842, 0.0)
Speed:    (-0.5527708409604438, 0.9574270831797618, 0.0)

ooRexx

Translation of: Java
/* REXX */
Numeric Digits 16
ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)
Say "Position :" ps~x~tostring
Say "Speed    :" ps~y~tostring
Say 'Raku:'
pi=rxCalcpi(16)
ps=orbitalStateVectors(1,.1,pi/18,pi/6,pi/4,0) /*Raku*/
Say "Position :" ps~x~tostring
Say "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 orbitalStateVectors
Use Arg  semimajorAxis,,
         eccentricity,,
         inclination,,
         longitudeOfAscendingNode,,
         argumentOfPeriapsis,,
         trueAnomaly
Numeric Digits 16
i = .vector~new(1, 0, 0)
j = .vector~new(0, 1, 0)
k = .vector~new(0, 0, 1)
p = rotate(i, j, longitudeOfAscendingNode)
i = p~x
j = p~y
p = rotate(j, k, inclination)
j = p~x
p = rotate(i, j, argumentOfPeriapsis)
i = p~x
j = p~y
If eccentricity=1 Then l=2
Else l=1-eccentricity*eccentricity
l*=semimajorAxis
c=rxCalccos(trueAnomaly,16,'R')
s=rxCalcsin(trueAnomaly,16,'R')
r=l/(1+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~'*'(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
with javascript_semantics
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    :") ?speed
end 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: Procedure
Parse 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 pi
rot:  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
Library: Wren-vector
import "./vector" for Vector3

var orbitalStateVectors = Fn.new { |semimajorAxis, eccentricity, inclination,
                                    longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly|
    var i = Vector3.new(1, 0, 0)
    var j = Vector3.new(0, 1, 0)
    var k = Vector3.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.length
    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)

XPL0

Translation of: C
include xpllib; \for VCopy, VAdd, VMul, VDiv, VMag and Print

proc VShow(A);
real A;
Print("(%1.6f, %1.6f, %1.6f)\n", A(0), A(1), A(2));

func real VMulAdd(V0, V1, V2, X1, X2);
real V0, V1, V2, X1, X2, V3(3), V4(3);
return VAdd(V0, VMul(V3, V1, X1), VMul(V4, V2, X2)); 

proc Rotate(I, J, Alpha, PS);
real I, J, Alpha, PS;
[VMulAdd(PS(0), I, J,  Cos(Alpha), Sin(Alpha));
 VMulAdd(PS(1), I, J, -Sin(Alpha), Cos(Alpha));
];

proc OrbitalStateVectors; real SemiMajorAxis, Eccentricity, Inclination,
        LongitudeOfAscendingNode, ArgumentOfPeriapsis, TrueAnomaly, PS;
real I, J, K, L, QS(2,3), C, S, R, RPrime;
[I:= [1.0, 0.0, 0.0];
 J:= [0.0, 1.0, 0.0];
 K:= [0.0, 0.0, 1.0];
L:= 2.0;
Rotate(I, J, LongitudeOfAscendingNode, QS);
VCopy(I, QS(0));  VCopy(J, QS(1));
Rotate(J, K, Inclination, QS);
VCopy(J, QS(0));
Rotate(I, J, ArgumentOfPeriapsis, QS);
VCopy(I, QS(0));  VCopy(J, QS(1));
if Eccentricity # 1.0 then  L:= 1.0 - Eccentricity*Eccentricity;
L:= L * SemiMajorAxis;
C:= Cos(TrueAnomaly);
S:= Sin(TrueAnomaly);
R:= L / (1.0 + Eccentricity*C);
RPrime:= S * R * R / L;
VMulAdd(PS(0), I, J, C, S);
VMul(PS(0), PS(0), R);
VMulAdd(PS(1), I,  J,  RPrime*C - R*S,  RPrime*S + R*C);
VDiv(PS(1), PS(1), VMag(PS(1)));
VMul(PS(1), PS(1), sqrt(2.0/R - 1.0/SemiMajorAxis));
];

def  Longitude = 355.0 / (113.0 * 6.0);
real PS(2,3);
[OrbitalStateVectors(1.0, 0.1, 0.0, Longitude, 0.0, 0.0, PS);
Print("Position : ");  VShow(PS(0));
Print("Speed    : ");  VShow(PS(1));
]
Output:
Position : (0.779423, 0.450000, 0.000000)
Speed    : (-0.552771, 0.957427, 0.000000)

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))
Retrieved from "https://rosettacode.org/wiki/Orbital_elements?oldid=372575"