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# File:Angular Parameters of Elliptical Orbit.png

 Description Raytraced image showing the concepts of inclination, longitude of the ascending node, and argument of the periapsis for a "minor" object in an elliptic orbit around a larger object. Date 23 November 2005 (original upload date) Source No machine-readable source provided. Own work assumed (based on copyright claims). Author No machine-readable author provided. Peo~commonswiki assumed (based on copyright claims).

## Summary

### Description (English)

Raytraced image showing the concepts of inclination, longitude of the ascending node, and argument of the periapsis for a "minor" object in an elliptic orbit around a larger object.

#### Legend

Letters in the image denote:

• A – Minor, orbiting body
• B – Major body being orbited by A
• C – Reference plane, e.g. the ecliptic
• D – Orbital plane of A
• E – Descending node
• F – Periapsis
• G – Ascending node
• H – Apoapsis
• i – Inclination
• J – Reference direction; for orbits in or near the ecliptic, usually the vernal point
• Ω – Longitude of the ascending node
• ω – Argument of the periapsis

The red line is the line of apsides; going through the periapsis (F) and apoapsis (H); this line coincides wíth the major axis in the elliptical shape of the orbit

The green line is the node line; going through the ascending (G) and descending node (E); this is where the reference plane (C) intersects the orbital plane (D).

#### Raytracing

This image was created using the Persistence of Vision Raytracer and the scene description code below: You can use this free raytracing package and the scene description below to re-render the image in new resolutions, or modify the description and thus the image being rendered.

A few notes of caution for those who want to do their own renditions of this image:

• The "camera" (viewpoint) assumes that the image format is square (i.e. has the same number of pixels in width and height) - to achieve this, use the +w and +h command line options to set the same number of pixels in width and height, respectively.
• This image comes complete with the letter annotations, and for this, the POV-Ray installation needs access to the TrueType fonts timesbi.ttf (Times new roman, bold and italic) and symbol.ttf (for greek letters). These come as standard on a Microsoft Windows installation, so this image should at least be able to render in POV-Ray for Windows.
• A little "dirty trick" is used to put those annotations there; they are text objects placed right in front of the "camera" that "sees" the scenario. Because of this, if you modify the camera location and/or look_at-point in the code, you need to either delete the annotations or make sure they "move with" the camera.

#### Four images in one the same description

Rendering the scene description as shown below renders this image, showing all three angular out of the six orbital elements. Some found this image to be a little too crammed and overwhelming, I modified the original description to render not only this combined image, but also three other, similar images, each showing only one of the three angles.

Line 10 in the description reads:

```#declare View=0;
```

As described in the comments starting from line 11, the 0 in the above line results in the combined image showing all three angles. Replacing the 0 with a 1, 2 or a 3 gives images that demonstrates one of the angles;

1. Argument of the periapsis (see Image:Argument of Periapsis in Elliptical Orbit.png)
2. Longitude of the ascending (see Image:Longitude of Ascending Node in Elliptical Orbit.png)
3. Inclination (see Image:Inclination in Elliptical Orbit.png)

### Beskrivelse (Dansk)

Raytracet billede der demonstrerer inklination, den opstigende knudes længde og periapsisargumentet for et mindre himmellegeme i elliptisk kredsløb om et større.

#### Nøgle

Bogstaverne i billedet angiver:

• A – Det mindre himmellegeme
• B – Det større himmellegeme
• C – Referenceplan, f.eks. da:ekliptika
• D – Baneplan for A's omløb
• F – Periapsis
• G – Opstigende knude
• H – Apoapsis
• i – Inklination
• J – Referenceretning; for baner i eller nær ekliptikas plan typisk forårspunktet i Vædderen
• Ω – Opstigende knudes længde
• ω – Periapsisargument

#### Raytracing

Billedet er lavet med raytracin-programmet Persistence of Vision Raytracer, samt den scenarie-beskrivelse der er vist nedenfor. Du kan bruge dette gratis raytracing-program og beskrivelsen nedenfor til at renderer billedet i nye opløsninger, eller lave ændringer i beskrivelsen og dermed også i det endelige billede.

Et par detaljer man skal være opmærksom på hvis man vil rendere billedet:

• "Kameraet" (betragtningspunktet) i billedet går ud fra at det færdige billede får et kvadratisk format, dvs. har lige mange pixels i bredden og højden. Man bør derfor bruge kommandolinje-ordrerne +w og +h til at specificere det samme antal pixels i respektive bredden og højden.
• Billedet leveres "komplet", inklusiv bogstav-annotationerne. For at lave disse, Persistence of Vision-programmet have adgang til Truetype-skrifttyperne timesbi.ttf (Times New Roman i fed og kursiv) og symbol.ttf (for græske bogstaver). Disse er standard i en Microsoft Windows-installation, så denne scenarie-beskrivelse skulle kunne køre fejlfrit med Persistence of Vision Raytracer for Windows.
• Der er brugt et lille "sidegade-kneb" til at lave bogstav-annotationerne; de er text-objekter anbragt lige foran camera'et, så hvis man flytter på synsretningen mod motivet, skal man enten sørge for at annotationerne "flytter med" synsretningen, eller helt fjerne dem.

#### Fire billeder ud af én beskrivelse

Hvis man renderer scenariebeskrivelsen som den er vist nedenfor, får man dette billede der viser alle de tre parametre for en omløbsbane der er vinkler. Nogen synes at der er lidt for meget overvældende detaljemylder i billedet, så jeg ændrede beskrivelsen så den kan bruges til ikke blot hosstående billede, men også tre andre tilsvarende billeder, der blot kun beskriver én af vinklerne "ad gangen".

I linje 10 i beskrivelsen står der:

```#declare View=0;
```

Som beskrevet i de kommentarer der starter fra linje 11, giver 0'et i ovenstående linje det kombinerede billeder der viser alle tre vinkler. Erstatter man 0'et med enten 1, 2 eller 3, får man billeder der viser én vinkel:

1. for periapsisargumentet (se Image:Argument of Periapsis in Elliptical Orbit.png)
2. for den opstigende knudes længde (se Image:Longitude of Ascending Node in Elliptical Orbit.png)
3. for banehældning (se Image:Inclination in Elliptical Orbit.png)

## Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.

## POV-Ray scene description

POV-ray image description:

```/*
================================================
Three Angular Parameters of the Elliptical Orbit
------------------------------------------------
Created by Søren Peo Pedersen - see my user page
at http://da.wikipedia.org/wiki/Bruger:Peo
================================================
*/

#declare View=0;
// 0 for all three angles
// 1 for argument of the periapsis only
// 2 for longitude of the ascending node only
// 3 for inclination only

#declare txtLatLonGrid=texture {  // Texture for latitude
pigment {color rgb <.4,.7,1>}   // and longnitude lines
finish {ambient .6}             // on planet
}

#declare txtPlanet=texture {      // Texture for planet
pigment {color rgb <0,.5,1>}
finish {ambient .6}
}

#local txtLatitudes=texture {     // Texture with latitudes only
texture_map {
[0 txtPlanet]
#local Cnt=-9;
#while (Cnt<9)
[.5+sin(Cnt*.174533-.02)/2 txtPlanet]
[.5+sin(Cnt*.174533-.02)/2 txtLatLonGrid]
[.5+sin(Cnt*.174533+.02)/2 txtLatLonGrid]
[.5+sin(Cnt*.174533+.02)/2 txtPlanet]
#local Cnt=Cnt+3;
#end
[1 txtPlanet]
}
translate <0,-.5,0>
scale 10
}

box {<-5,-.002,0>,<0,.002,5> rotate <0,45,0> scale <1,1,3>}
plane {<0,0,-1>,-1.5}
}

merge {
difference {
}

pigment {color rgb <1,1,1>}
finish {ambient 1}
}

#end

sphere {0,5   // "Main" planet
texture {
object {
union {
#local Cnt=0;
#while (Cnt<18)
box {<-.1,-8,-8>,<.1,8,8> rotate <0,10*Cnt+11,0>}
#local Cnt=Cnt+3;
#end
}
texture {txtLatitudes}
texture {txtLatLonGrid}
}
}
}

#if (View=0 | View=2)
merge {                                   // Arrow
box {<-.1,-.001,0>,<.1,.001,-23>}       // pointing to
#object {Arrowhead translate <0,0,-24>} // reference
pigment {color rgb<.8,.4,1>}            // point
finish {ambient 1 diffuse 0}            // (typically
rotate <0,0,0>                          // vernal point)
}
#end

#local Sma=20;  // Semimajor axis
#local Smi=16;  // Semiminor axis

#local Incl=60; // Inclination

#if (View=0|View=2)
#object { // Measures longitude of ascending node
#if (View=2)
AngleArc(60,20)
#else
AngleArc(60,7.5)
#end
rotate <0,210,0>}
#end

#local txtOrbitPlane=texture {    // Orbit
pigment {color rgbt<1,.9,0,.5>} // plane
finish {ambient .4}             // texture
}

#local txtOrbitMarking=texture {  // Texture for
pigment {color rgb<1,.9,0>}     // markings on
finish {ambient 1 diffuse 0}    // orbit plane
}

union {
disc {0,<0,1,0>,1,0   // Elliptic "disc" indicating
scale <Sma,1,Smi>   // the area inside the orbit
translate <sqrt(Sma*Sma-Smi*Smi),0,0>
texture {
#if (View=0|View=3)
object {
difference {
box {<-1,-1,#if (View=0) -9 #else -18 #end>,<1,1,0>}
box {<-.8,-2,-1.8>,<.6,2,1>}
box {<-2,-2,-99>,<.6,2,-2>}
#if (View=0)
translate <16,0,0>
#else
translate <7,0,0>
#end
rotate <0,-40,0>
}
texture {txtOrbitPlane}
texture {txtOrbitMarking}
}
#else
txtOrbitPlane
#end
}
}

difference {  // Orbit edge outline
cylinder {<0,-.001,0>,<0,.001,0>,1
scale <Sma+.15,1,Smi+.15>
}
cylinder {<0,-1,0>,<0,1,0>,1
scale <Sma-.15,1,Smi-.15>
}
translate <sqrt(Sma*Sma-Smi*Smi),0,0>
pigment {
color_map {
[0 color rgbt <1,1,1,0>]
[0.1 color rgbt <1,.95,.5,0>]
[0.3 color rgbt <1,.9,0,0>]
[0.7 color rgbt <1,.9,0,0>]
[.9 color rgbt <1,.9,0,1>]
[1 color rgbt <1,.9,0,1>]
}
rotate <0,-90,0>
}
finish {ambient 1 diffuse 0}
}

#if (View=0)
#object {AngleArc(140,6.5)} // Measures argument of the periapsis
#end
#if (View=1)
#object {AngleArc(140,9)}   // Larger arc for argument of periapsis only
#end

#if (View<2)
cylinder {  // Line of apsides
<sqrt(Sma*Sma-Smi*Smi)-Sma-5,0,0>,
<sqrt(Sma*Sma-Smi*Smi)+Sma+5,0,0>,.1
pigment {color rgb<1,0,0>}
finish {ambient 1 diffuse 0}
}
#end

#if (View<2)
sphere {<sqrt(Sma*Sma-Smi*Smi)-Sma,0,0>,.5  // Periapsis
pigment {color rgb 1} finish {ambient 1 diffuse 0}
}
#end

#if (View=0)
sphere {<sqrt(Sma*Sma-Smi*Smi)+Sma,0,0>,.5  // Apoapsis
pigment {color rgb 1} finish {ambient 1 diffuse 0}
}
#end

sphere {  // Yellow, orbiting "moon"
<0,0,Smi*Smi/Sma>,1
pigment {color rgb <1,.8,0>}
finish {ambient .6}
}

rotate <0,130,Incl>
rotate <0,-60,0>
}

union { // line of nodes
cylinder {<-30,0,0>,<30,0,0>,.1 pigment {color rgb<.3,1,.1>} finish {ambient 1 diffuse 0}}

#if (View!=3)
sphere {<23.6,0,0>,.5    // Ascending node
pigment {color rgb 1} finish {ambient 1 diffuse 0}
}
#end
#if (View=0)
sphere {<-8.8,0,0>,.5    // Descending node
pigment {color rgb 1} finish {ambient 1 diffuse 0}
}
#end
#if (View=0)
#object {AngleArc(60,8)  // Measures inclination
rotate <90,-90,0> translate <16.8,0,0>}
#end
#if (View=3)
#object {AngleArc(60,17) // Measures inclination
rotate <90,-90,0> translate <7.8,0,0>}
#end

rotate <0,30,0>
}

#local RefPlaneChecker=texture {  // Texture for
pigment {checker                // reference
color rgbt<.6,.7,1,.5>        // plane
color rgbt<.48,.56,.8,.5>
scale 3
}
finish {ambient .4}
}

#local RefPlaneMark=texture {     // Texture for
pigment {checker                // markings on
color rgbt<.6,.7,1,0>         // reference
color rgbt<.48,.56,.8,0>      // plane
scale 3
}
finish {ambient 1 diffuse 0}
}

merge { // The reference plane
triangle {<-9,0,-21>,<21,0,-21>,<-9,0,9>}
triangle {<21,0,9>,<21,0,-21>,<-9,0,9>}
texture {
#if (View=0|View=3)
object {
difference {
box {<-1,-1,#if (View=0) -9 #else -18 #end>,<1,1,0>}
box {<-.8,-2,-1.8>,<.6,2,1>}
box {<-2,-2,-99>,<.6,2,-2>}
#if (View=0)
translate <16,0,0>
#else
translate <7,0,0>
#end
rotate <0,30,0>
}
texture {RefPlaneChecker}
texture {RefPlaneMark}
}
#else
RefPlaneChecker
#end
}
}

union {
// A, B, C, and D are common for all four images...:
text {ttf "timesbi.ttf","A",.001,0         // A: Orbiting body
scale .0035 translate <.0045,.0132,0>}
text {ttf "timesbi.ttf","B",.001,0         // B: Body being orbited
scale .0035 translate <-.0045,.0092,0>}
text {ttf "timesbi.ttf","C",.001,0         // C: Reference plane
scale .0035 translate <-.016,-.002,0>}
text {ttf "timesbi.ttf","D",-.001,0        // D: Orbital plane of A
scale .0035 translate <-.002,-.014,0>}

#switch (View) // Deal with "special cases" in each of the four images:
#case (0)  // Letter markings for viewing all three angles
text {ttf "symbol.ttf","W",.001,0          // "Upper-case" Omega at the
scale .0035 translate <-.002,.003,0>}    // longitude of ascending node
text {ttf "symbol.ttf","w",.001,0          // "Lower-case" omega at the
scale .0035 translate <.0009,.0158,0>}   // argument of the periapsis
text {ttf "timesbi.ttf","i",.001,0         // "Lower-case" i at
scale .0035 translate <.0045,-.0083,0>}  // the inclination
text {ttf "timesbi.ttf","E",.001,0         // E: Descending node
scale .0035 translate <-.011,.013,0>}
text {ttf "timesbi.ttf","F",.001,0         // F: Periapsis
scale .0035 translate <-.008,.0175,0>}
text {ttf "timesbi.ttf","G",.001,0         // G: Ascending node
scale .0035 translate <.0149,-.003,0>}
text {ttf "timesbi.ttf","H",.001,0         // H: Apoapsis
scale .0035 translate <.006,-.0182,0>}
text {ttf "timesbi.ttf","J",.001,0         // J: Reference direction,
scale .0035 translate <-.0145,-.014,0>}  //    e.g. vernal point
#break

#case (1)  // Letter markings for viewing only argument of periapsis:
text {ttf "symbol.ttf","w",.001,0          // "lower-case" omega at
scale .007 translate <.0053,.01,0>}      // argument of periapsis
text {ttf "timesbi.ttf","E",.001,0         // E: Ascending node
scale .0035 translate <.0149,-.003,0>}
text {ttf "timesbi.ttf","F",-.001,0        // F: Periapsis
scale .0035 translate <-.0085,.0167,0>}
#break

#case (2)  // Letter markings for viewing only longitude of ascending node:
text {ttf "symbol.ttf","W",.001,0          // "Upper-case" Omega at
scale .007 translate <.004,-.01,0>}      // longitude of ascending node
text {ttf "timesbi.ttf","E",.001,0         // E: Ascending node
scale .0035 translate <.0149,-.003,0>}
text {ttf "timesbi.ttf","F",.001,0         // F: Reference direction,
scale .0035 translate <-.0145,-.014,0>}  //    e.g. vernal point
pigment {color rgb<1,1,1>}
#break

#case (3)  // Letter markings for viewing only the inclination:
text {ttf "timesbi.ttf","i",.001,0         // "Lower-case" i at
scale .007 translate <-.011,-.012,0>}    // the inclination
#break
#end

pigment {color rgb<1,1,1>}   // Common settings
finish {ambient 1 diffuse 0} // for the letters
translate <0,0,.04>
rotate <51.3765,-13.62699,0>
translate <11,26,-33>
}

camera {                   // Viewpoint - DO NOT CHANGE without recalculating the
right <1,0,0> up <0,1,0> // translate and rotate above - they align the letter
location <11,26,-33>     // markings in the image with the camera's viewing
look_at <3,-16.5,0>      // angle!!
angle 55
}

light_source {<10000,5000,-5000> color rgb 1}
```

### Captions

Raytraced image showing the concepts of inclination, longitude of the ascending node, and argument of the periapsis for a "minor" object in an elliptic orbit around a larger object.