Talk:Angle of parallelism

Bonola and Halsted
In 1912 Roberto Bonola published a textbook with Open Court in Chicago on the topic of non-Euclidean geometry. When Dover re-issued the book half a century later, it included work by G.B. Halsted that significantly improved coverage of the subject, particularly a translation of Lobachevski. It is that appendix that is cited for an English reference for the topic "angle of parallelism". Recently the Bonola text has been scanned into Archive.org. One can now find the link at the end of Non-Euclidean geometry. Since the text scanned was the original 1912 edition it does not include Halsted's translation of Lobachevski. Hence I have removed the link as inappropriate for this article.Rgdboer (talk) 22:45, 19 February 2008 (UTC)
 * The Halsted manuscript has become available Google Books so the old print source in the back of the Dover edition of Bonola is unnecessary.
 * The text is available by template:.
 * Bonola's original edition did not include Halsted and the paging is unclear in editions that include the translation.Rgdboer (talk) 01:24, 10 February 2015 (UTC)

Napier ?
The second of the equivalent descriptions of angle of parallelism is
 * $$ \tan(\tfrac{1}{2}\Pi(a)) = e^{-a} .$$

Here the e is the base of natural logarithms. The historic English reference cited is Halsted 1891, now on GB:

Unfortunately Halsted considers unit of length in connection with selection of a base for logarithm, and in doing so refers to Napierian logarithm, a topic frequently confused with natural logarithm, but is in fact a rather different function. All reference to Napierian logarithm should be avoided; better references might be found to complete the thought Halsted meant to express before this misdirection.Rgdboer (talk) 02:04, 11 February 2015 (UTC) Correct date 1891.Rgdboer (talk) 02:07, 11 February 2015 (UTC)

Consequently, the following quotation from Halsted was removed:
 * ...$$ \tan(\tfrac{1}{2} \Pi(x)) = \theta ^{-x} $$ where $$ \theta $$ may be any arbitrary number, which is geater than unity, since $$ \Pi(x) = 0 $$ for $$ x = \infty $$.
 * Since the unit by which lines are measured are arbitrary, so we may also understand by $$ \theta $$ the base of Napierian logarithms.

The quotation is found on page 41 of Halsted's translation of Lobachevsky.Rgdboer (talk) 02:21, 20 February 2015 (UTC)

Negative angle
The following confusing contribution was removed:
 * By definition for a negative angle p:
 * $$ \Pi(p) + \Pi(-p) = \pi $$

Before the text was made consistent with a as variable segment length, some uses were p. It does not make sense to then call p an angle. Negative segment length doesn't make sense either. Perhaps a discussion here can clarify the idea, or a reference can be produced to justify this addition.Rgdboer (talk) 20:08, 7 March 2015 (UTC)


 * Reference: Halsteds translation of Lobachevsky's "Geometrische Untersuchungen zur Theory der Parallellinien" page 20-21, end of paragraph 23 (taken from Bonola):
 * Since we are wholy at liberty we will understand by the symbol $$ \Pi(p) $$ when the line p is expressed by a negative number we will assume
 * $$ \Pi(p) + \Pi(-p) = \pi $$,
 * as equation which shall hold for all values of p, positive as well as negative and for p = 0.
 * Maybe my shortening of this was not really clear, or maybe it is later given another meaning
 * WillemienH (talk) 20:51, 9 March 2015 (UTC)

Compass and straightedge construction
The following section was moved here for discussion. The third step assumes b and l intersect (with no justification). The section has no attribution, and the appeal to Lambert quadrilateral is strange because there is no link to this article there. Recommend finding feature of this angle of parallelism in that quadrilateral and documenting it before another effort like this:
 * The angle of parallelism between an point P and a line l can be constructed by:


 * Construct line b through point P perpendicular to line l
 * Construct line d through point P perpendicular to line b
 * Point B is the intersection of line b and line l
 * Construct circle a with center P going trough B
 * Construct circle c with center B going trough P
 * Point C is one of the intersections of line l and circle c
 * construct line e through point C perpendicular to line d
 * Points D and E are where line e intersects with circle a (D near C)
 * The lines PD and PE are the limiting parallels trough P to l
 * The angle ∠BPD is the angle of parallelism for segment PB.


 * The lines l, b, d and e are the sides of a Lambert quadrilateral with the non right angle at C.

Hyperbolic geometry is now a classical subject with numerous sources, some of which may support the idea sparking this contribution. — Rgdboer (talk) 03:16, 9 November 2015 (UTC)


 * The construction is from Bonola's "Non euclidean Geometry" on page 217, and is also in Coxeter's "Non euclidean Geometry" on page 204. I did simplify the construction a bit by fixing the distance BC to be equal PB. (which is not really needed, BC must be equal to PD and PE)
 * I don't undertstand your remark "The third step assumes b and l intersect (with no justification)", b is the line from point P perpendicular to line l. How can b and l be perpendicular but not intersecting? WillemienH (talk) 22:40, 11 November 2015 (UTC)

Good to see some references (will check them out soon). Also, now the construction of the Lambert quadrilateral makes sense to me (sorry about slowness). But still there is the question of relation to this article, Angle of parallelism. The construction belongs on the quadrilateral page. Now the important thing here is the question: Is ∠BPD the angle of parallelism for segment PB ? — Rgdboer (talk) 00:52, 12 November 2015 (UTC)


 * Yes, ∠BPD is the angle of parallelism for segment PB. (if we assume CD < CE )  It is about the construction of an angle of parallelism, so why should it on the page for the lambert quadrilateral? i just added that remark to make the construction more understandable. Long time I thought that ∠C was right while in fact it is acute, the ((unnamed) angle opposite ∠B is right. I updated the construction above with these remarks. Ps I cannot follow the proof of this construction, maybe you can add a bit about that as well. WillemienH (talk) 01:08, 12 November 2015 (UTC)

Have put the construction in the Quadrilateral article. Must rush now to check Coxeter before Library shuts. — Rgdboer (talk) 01:16, 12 November 2015 (UTC)

Coxeter is making use of Cayley-Klein metric theory in his "proof" of the construction of the parallels through a given point not on a reference line. Straightedge and compass constructions are part of Euclidean geometry and can be used for the model Coxeter gives as figure 10.4A on page 204. On the other hand, the hyperbolic plane is a curved surface where geodesics replace straight lines and the "geodesic-straightedge" is science fiction. Coxeter gives as references Baldus (1927) and Mohrmann (1930) for his construction of the parallels. They may be helpful. As for this article on Angle of parallelism, it is the function &Pi;(x) that Lobachevski introduced that forms the subject. Nevertheless, the configuration in Coxeter's diagram is very instructive and may well have a place somewhere in the encyclopedia, perhaps in Beltrami-Klein model. But that model does not represent angles that agree with their Euclidean magnitudes (it is not conformal), whereas the Poincare half-plane model used in this article does have the conformal property. — Rgdboer (talk) 22:45, 12 November 2015 (UTC)


 * I would like to add the construction back to the article. the article is on the angle of parallelism in the hyperbolic plane, (not just on how to measure/ calculate it) and this is just a good way to construct it as it is independent of the model used.
 * Bonola gives some other proofs of the construction, but they all seem rather complicated. I need to study them in more detail myself. WillemienH (talk) 10:00, 13 November 2015 (UTC)

Check out Compass-and-straightedge construction and see that it is presuming Euclidean geometry. There is no straightedge to be used for constructions in Lobachevsky's plane. — Rgdboer (talk) 02:37, 14 November 2015 (UTC)

Unclear description and unclear illustration that do not correspond to each other
The introduction begins with these two paragraphs:

"In hyperbolic geometry, the angle of parallelism  $$ \Pi(a) $$, is the angle at one vertex of a right hyperbolic triangle that has two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.

Given a point not on a line, if we drop a perpendicular to the line from the point, then a is the distance along this perpendicular segment, and &phi; or $$ \Pi(a) $$ is the least angle such that the line drawn through the point at that angle does not intersect the given line."

Normally "the distance along a segment" is much more clearly described as the length of the segment.

But the accompanying illustration emphatically does not show the lowercase letter a as denoting the length of the same segment. (Instead, it shows a along the asymptotic hypotenuse of the triangle, nowhere near the perpendicular segment.)

I hope that someone knowledgeable in this area will fix these problems.50.205.142.50 (talk) 19:38, 6 June 2020 (UTC)


 * You are correct about the figure being mislabeled. I have removed it and also fixed some language issues. I can replace the figure with a correct one in a day or so. --Bill Cherowitzo (talk) 21:32, 6 June 2020 (UTC)
 * ✅--Bill Cherowitzo (talk) 18:22, 15 June 2020 (UTC)

Should the $Π$ symbol be italicized?
That is, should we have upright $Π(a)$ or italic $Π(a)$? Lobachevsky (Halsted translation) and some other old sources (e.g. Becker & Van Orstrand) use the italic version, but a Roman (upright) font is easier in LaTeX documents where upright, $$\Pi(a),$$ is default and italic  , $$\mathit{\Pi}(a),$$ takes extra typing or an explicit definition. I haven't done any serious literature survey. –jacobolus (t) 21:23, 13 March 2023 (UTC)
 * Rule of thumb: if its a variable use italic. Here the function is a constant object, don't use italic. For instance, imaginary unit i should not be italic. — Rgdboer (talk) 01:26, 14 March 2023 (UTC)
 * This "rule of thumb" does not match prevailing practice in pretty much any area of mathematics. So I’m not sure how useful it is. –jacobolus (t) 03:56, 14 March 2023 (UTC)