⚠️ Warning: This is a draft ⚠️

This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.

{{draft task}} When neglecting the influence of other objects, two celestial bodies orbit one another along a [[wp:conic section|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 [[wp:orbital elements|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 [[wp:perihelion and aphelion|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 [[wp:vis-viva equation|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 [[wp:cartesian coordinates|cartesian coordinates]], those two vectors constituting the so-called [[wp:orbital state vectors|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.)

## C

{{trans|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) {
}

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++

{{trans|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(
);
};

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;
}
```

{{out}}

```Position : (0.779423, 0.45, 0)
Speed    : (-0.552771, 0.957427, 0)
```

## C#

{{trans|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>(
);
}

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);
}
}
}
```

{{out}}

```Position : (0.77942284339868, 0.450000034653684, 0)
Speed    : (-0.552770840960444, 0.957427083179762, 0)
```

## D

{{trans|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) =>

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]);
}
```

{{out}}

```Position : (0.7794228433986798, 0.4500000346536842, 0.0000000000000000)
Speed    : (-0.5527708409604437, 0.9574270831797614, 0.0000000000000000)
```

## Go

{{trans|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 {
}

rotate := func(i, j vector, alpha float64) (vector, vector) {
}

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)
}
```

{{out}}

```
Position : (0.7794228433986797, 0.45000003465368416, 0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0)

```

## Java

{{trans|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[]{
};
}

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]);
}
}
```

{{out}}

```Position : (0.7794228433986797, 0.4500000346536842, 0.0000000000000000)
Speed : (-0.5527708409604438, 0.9574270831797618, 0.0000000000000000)
```

## Kotlin

{{trans|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) =

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")
}
```

{{out}}

```Position : (0.7794228433986797, 0.45000003465368416, 0.0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0.0)
```

## ooRexx

{{trans|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 'Perl6:'
pi=rxCalcpi(16)
ps=orbitalStateVectors(1,.1,pi/18,pi/6,pi/4,0) /*Perl6*/
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
speed=speed~'/'(speed~abs)
speed=speed~'*'(rxCalcsqrt(2.0/r-1.0/semimajorAxis,16))
Return .v2~new(position,speed)

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
res=.v2~new(xx,yy)
Return res

::requires 'rxmath' LIBRARY
```

{{out}}

```Position : (0.7794228433986798,0.4500000346536842,0)
Speed    : (-0.5527708409604436,0.9574270831797613,0)
Perl6:
Position : (0.2377712839822067,0.8609602616977158,0.1105090235720755)
Speed    : (-1.061933017480060,0.2758500205692495,0.1357470248655981)
```

## Perl

{{trans|Perl 6}}

```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
;
```

{{out}}

```\$VAR1 = {
'position' => bless( [
'0.77942284339868',
'0.450000034653684',
'0'
], 'Math::Vector::Real' ),
'speed' => bless( [
'-0.552770840960444',
'0.957427083179762',
'0'
], 'Math::Vector::Real' )
};
```

## Perl 6

We'll use the [https://github.com/grondilu/clifford 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;
```

{{out}}

```{position => 0.237771283982207*e0+0.860960261697716*e1+0.110509023572076*e2, speed => -1.06193301748006*e0+0.27585002056925*e1+0.135747024865598*e2}
```

## Phix

{{trans|Python}}

```function vabs(sequence v)
return sqrt(sum(sq_power(v,2)))
end function

function mulAdd(sequence v1, atom x1, sequence v2, atom x2)
end function

function rotate(sequence i, j, atom alpha)
atom ca = cos(alpha),
sa = sin(alpha)
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)
```

{{out}}

```
Position :{0.7872958014,0.4545454895,0}
Speed    :{-0.5477225997,0.9486832737,0}

```

## Python

```import math

class Vector:
def __init__(self, x, y, z):
self.x = x
self.y = y
self.z = z

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)

return v1 * x1 + v2 * x2

def rotate(i, j, 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]
```

{{out}}

```Position : (0.787295801413, 0.454545489549, 0.0)
Speed    : (-0.547722599684, 0.948683273698, 0.0)
```

## REXX

### version 1

{{trans|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
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)

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
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
```

{{out}}

```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
```

{{out|output|text= 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) =

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))
}

}
```

{{Out}}Best seen running in your browser either by [https://scalafiddle.io/sf/ac17jh2/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/2NQNgj4OQkazxZNvSzcexQ Scastie (remote JVM)].

## Sidef

{{trans|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) = (
)
}

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"
}
```

{{out}}

```
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

{{trans|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)")
```

{{out}}

```Position: 0.779423	0.450000	0.000000; Speed: -0.552771	0.957427	0.000000
```

## zkl

{{trans|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();
```

{{out}}

```L(L(0.779423,0.45,0),L(-0.552771,0.957427,0))
```