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The Wiktionary lists under the meanings of paraxial:

  1. (mathematics) Parallel to one or more axes of the coordinate system
    paraxial rectangle

The respective information was added without any reference and given the following reasons I am skeptical about it.

  • Para means close to and similar, not parallel, which is for example reflected in another meaning:

    1. (physics) Near an optical axis
      paraxial radiation

    On the other hand, I am aware that the actual use of a word may depart from its etymology.

  • Other dictionaries do not report paraxial meaning axis-parallel, but this may be due to the term being rather rare and technical (some dictionaries don’t contain paraxial at all).

My question thus is: Is paraxial actually used in the meaning of axis-parallel by some communities? Be aware that in many contexts, close to the axis may imply axis-parallel or almost axis-parallel. Relatedly, I am interested in uses of paraxial for something that is parallel but not close to an axis.

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  • Googling "paraxial rectangle" returns a few, but not many, sites which use it for "axis-parallel rectangle", including some technical papers by German speakers, and some technical papers in computer science. I would guess there must be a similar word in German that is being translated this way (and computer scientists tend to form portmanteau words). – Peter Shor Feb 24 '15 at 12:30
  • The paraxial mesoderm is worth considering, as its etymology seems self-evident (parallel to the axis of the embryo), but since I have no authority to back me up on that, I'll have to leave this as a comment rather than answer. Also PM may be used as a fixed phrase, as opposed to "paraxial" serving as a generic adjective in embryology. – Dan Bron Feb 24 '15 at 12:31
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    @PeterShor: The results for "paraxial rectangle" seem to be dominated by Wiktionary clones and German resources. As for the latter: A popular German–English online dictionary lists paraxial as a translation of achsenparallel, with the latter literally meaning axis-parallel, but this may have very-well be originated from said Wiktionary entry and would not be the first mistake they made. – Wrzlprmft Feb 24 '15 at 12:38
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Yes. I easily find several citations, all from German software developers:

  • …let the coordinates of two paraxial rectangles in a cartesian coordinate system be given…

    The Computer—My Life

  • …which form the endpoints of the diagonal of a paraxial rectangle…

    Programming Smalltalk

  • …an oriented rectangle with anchor point (0, 0) width 100, height 10, and up vector (0, -1) is a paraxial rectangle with upper left corner (0, -10) and lower right corner (100, 0).

    YWorks: OrientedRectangle

It seems they have ported this word to English.

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M-W Unabridged defines paraxial as

relating to or being the space in the immediate neighborhood of the optical axis of a lens or mirror

One can approximate light rays in such a region as being parallel; the paraxial approximation is used to simplify calculations by modeling most of the propagation to be in a particular direction, to within a particular tolerance.

So paraxial would not be objectionable in the usage you're looking for, meaning "parallel to within a useful approximation;" however, you probably shouldn't use it to mean "axis-parallel."

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  • While it is not explicitly mentioned in the file you linked, the paraxial approximation does require rays to be close to the axis as well (though through the approach taken here, it’s rather that the axis is defined such that it is close to the rays, see page 5). – Wrzlprmft Feb 24 '15 at 17:59
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para- (prefix)

Wiktionary:

From Ancient Greek παρά (pará, “beside; next to, near, from; against, contrary to”).
1. above, beyond
2. beside, near, alongside
3. abnormal, incorrect
4. resembling
5. (chemistry) In isomeric benzene derivatives, having the two substituents in opposite positions (compare ortho- and meta-.)

ODO on para-:

1 Beside; adjacent to:
parataxis
parathyroid

See also: Why is “para” not used consistently? [closed]

However, in the OP's examples, parallel to and near are not contradictory or even much different. Nearly all words in all domains with the prefix para- have a broad sense of peripherality.

ODO on paraxial:

Anatomy & Zoology Situated alongside, or on each side of, an axis, especially the central axis of the body.

Note the "alongside, or on each side of, an axis" above.

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  • I fail to see how this addresses my question. I know that parallel and near are not contradictary and, moreover, near often implies parallel, but this does not mean that parallel implies near or that near means parallel. – Wrzlprmft Feb 24 '15 at 11:50
  • It doesn't have to. The instant question is whether anyone uses the para prefix to mean parallel to, right? – Kris Feb 24 '15 at 11:55
  • It’s rather specific to the combination of para and axial, but I do not even see the generalised variant of the question answered due to the above reasons. – Wrzlprmft Feb 24 '15 at 12:00
  • If you are looking to find a relation between para- and "parallel," as it apparently is the case -- the answer is not far to seek either: parallel: "1540s, from Middle French parallèle (16c.) and directly from Latin parallelus, from Greek parallelos "parallel," from para allelois "beside one another," from para- "beside" (see para- (1)) + allelois "each other," from allos "other" (see alias (adv.))." (etymonline) HTH :) – Kris Feb 24 '15 at 12:06
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    Good job, @Kris. Both of the questions the OP asked he also answered within his own question. 1) Paraxial is defined as parallel to an axis. Therefore it means (at least in some contexts) parallel to an axis. 2) Paraxial is listed in the dictionary (a descriptive tool) as parallel to the axis, demonstrating that it is used in some communities as axis-parallel. For a line to remain close to the axis, it must be parallel to the axis. – ScotM Feb 24 '15 at 15:34

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