How to calculate position and velocity from given orbital elements?

Question

I have some orbit, with a given semi-major axis, inclination, eccentricity, longitude of ascending node, argument of periapsis, and true anomaly. How can I, from this, calculate the position and velocity as Cartesian vectors?

For the sake of this, assume that the parent body is stationary at the origin.

Does this answer your question? Converting Orbital Elements to Cartesian State Vectors — Organic Marble, Apr 18 '20 at 22:49
How can we best :“round up” the most sought-after Keplerian-to-state-vector answers so they are easier to find? Might starting a community Wiki on this question be the best place, since the question is so short and simple? — uhoh, Apr 19 '20 at 00:21
@OrganicMarble Thanks for hunting all of those down! Rather than close this question now, how about preserving that work as the Community Wiki answer below? This way we can all add to it and build a "road map" to the other answers. — uhoh, Apr 19 '20 at 00:27
About the above links: The first one contains links to pdfs which explain the algorithm, but they're difficult to parse (and I think SE wants the answer to not depend on external links). The top answer depends on dead links. The second one talks about how to compute osculating vs. mean elements (more general than my question, at the cost of being much more complicated by forgoing orbital dynamics). The only answer depends entirely on links to things behind a paywall (not a good answer for this site). — Jarred Allen, Apr 19 '20 at 04:05
The third one is about going the other direction, is just someone who wants their code debugged, and has no answers posted. The fourth link has a lot of equations, and the answer only explains some of them. The fifth link talks about non-Keplerian mechanics, which is not what I was talking about and, therefore, the answers don't answer my question. If a wiki of some kind to link all questions on this topic is being made, they may be useful things to link to, but none of them answer the question. — Jarred Allen, Apr 19 '20 at 04:05
@JarredAllen I think the conversion algorithm may be too long to not rely on external links. I have a comment on one of those QAs that points to a PDF I found very useful, but I have no desire to restate the contents of the PDF in an answer here, since I will add little and may possibly introduce errors in so doing. — Russell Borogove, Apr 19 '20 at 04:17
Voting to leave open. I agree that this could be made a comprehensive resource, unlike the linked answers. — William R. Ebenezer, Apr 20 '20 at 12:56

score 2 · Answer 1 · edited Apr 20 '20 at 18:04

I've started a Community Wiki answer so that we can consolidate all the best answers as links here, accompanied by short explanations.

I've chosen this question to do so because the question is simple and short and so it requires the most general answer.

From @OrganicMarble's comments:

Does this answer your question? Converting Orbital Elements to Cartesian State Vectors This answer seems to be the best. It also houses an implementation of the conversion methodology in code, which may or may not answer this question. However the link in the question itself Memorandum #1 Keplerian Orbit Elements → Cartesian State Vectors seems to be what we're looking for, though not yet sure.
Or this: Keplerian Orbital Elements ↔ Cartesian ECEI with WGS84 Corrections Answers here provide some potentially helpful references but don't answer this question
Or this: Need help with calculating state vectors from Orbital Elements - C# This is unanswered
Related: What is this algorithm to calculate orbital state vectors? @DavidHammen's answer has several helpful equations but isn't a complete answer to this question, since it's purely 2D.
Related: Why are Keplerian elements used in TLEs instead of Cartesian state vectors? Of no real help here because it's about TLEs.
- Here is the CSPICE function that does this; CSPICE is open source. The CSPICE function here does the opposite.

I believe there are other answers that contain actual step-by-step instructions how to go from a full set of Keplerian elements to a state vector, but I haven't found them yet.

One way to find more answers will be to search for "eccentricity vector" since I think that that's central to the conversion in either direction.

You may remove the line which says "rotted PDFs", I got that fixed. — William R. Ebenezer, Apr 20 '20 at 13:11
@WilliamR.Ebenezer That's great! This is a community wiki answer, anybody can and is encouraged to edit this to improve it. Go for it! — uhoh, Apr 20 '20 at 16:57

How to calculate position and velocity from given orbital elements?

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