Talk:Antiparallel lines

Concerns

Please, can somebody create an image for this - I cannot understand this stub so far. 217.43.136.28 16:09, 24 June 2007 (UTC)

Drawing the diagram suggested just seems to make a regular parallel line to me. I have just come across the term antiparallel (in a physics text) used to describe a vector which is parallel to an axis but in the opposite sense. So perhaps antiparallel means something like this: if two vectors (or arrows or lines with direction) are parallel but in opposing senses then they are antiparallel. For example a left-to-right arrow is antiparallel to a right-to-left arrow. AP.

I rewrote the article.--Cronholm¹⁴⁴ 09:14, 13 July 2007 (UTC)

Error

The second sentence under Definitions doesn't make sense: there are two hanging ′and′ 's. Their purpose is unclear. Tweet7 (talk) 12:30, 18 December 2012 (UTC)

Fixed.--JohnBlackburne^words_deeds 14:04, 18 December 2012 (UTC)

What are antiparallel vectors?

The article contains "In a vector space over R (or some other ordered field), two nonzero vectors are called antiparallel if they are parallel but have opposite directions." That defines nothing, as, in a vector space there is no notion of parallel vectors. The only notion is "colinear". When vectors are represented with arrows, they all start from the origin (the vector 0). The notion makes sense only for vectors with a starting point in an Euclidean space, but this kind of vector is more correctly called a "bi-point", and does not belong to any vector space. D.Lazard (talk) 23:52, 19 January 2014 (UTC)

Antiparallel planes

Are two planes "antiparallel" if they do not intersect but have opposite normals. If so, should the article cover this? Acasson (talk) 12:37, 27 April 2021 (UTC)

If n is a normal to a plane, then –n is also a normal. So, "having opposite normals" is a nonsensical concept. D.Lazard (talk) 13:44, 27 April 2021 (UTC)

Two planes which are "antiparallel" in the sense of this article make congruent but opposite angles with a given cone. This comes up in Apollonius's conics, and is the key to proofs of theorems about the properties stereographic projection. –jacobolus (t) 19:35, 27 March 2023 (UTC)

"subcontrary"

It seems like the term "subcontrary" is also sometimes used for this idea. I need to figure out more precisely what the relations are between these terms. Cf. Malcolm Brown (1975) "Pappus, Plato and the Harmonic Mean", Phronesis 20(2): 173-184. –jacobolus (t) 19:27, 27 March 2023 (UTC)

parallel and antiparallel

Shouldn't the definition require that $l_{1},l_{2}$ are non-parallel? Unfortunately none of the sources and literature I've reviewed state that explicitly, but if you don't have that various properties and conclusions that are usually drawn seem to break. Moreover you can have a line being parallel and antiparallel at the same time (consider $l_{1},l_{2}$ being parallel and perpendicular to the 3rd line or angle bisector) --Kmhkmh (talk) 10:27, 20 February 2024 (UTC)

As far as I can tell, parallel and antiparallel at the same time is fine. –jacobolus (t) 14:50, 20 February 2024 (UTC)

That seems to be the case based on the given definition. However in literature (for instance in Johnson), they usually have a statement saying if $l_{1},l_{2}$ are antiparallel with respect to $m_{1},m_{2}$, then $m_{1},m_{2}$ are antiparallel with respect to $l_{1},l_{2}$ as well. But that claim seems to break if the case $l_{1},l_{2}$ being parallel and antiparallel is allowed, since parallel $l_{1},l_{2}$ don't intersect, but that was a requirement for for any two line being antiparallel with respect to $l_{1},l_{2}$.

As far as I can see to have the symmetry in the statement above, you either have to require $l_{1},l_{2}$ and $m_{1},m_{2}$ to intersect or allow both to have a parallel case (in which the angle bisector would be replaced by a middle parallel).--Kmhkmh (talk) 20:19, 20 February 2024 (UTC)

By the definition used here (which I didn't come up with) I think you could have two lines antiparallel to a set of parallel lines, but I am not an expert about what previous sources use as a definition. It could certainly be rewritten for clarity though. And it would be great to do a survey of existing sources' definitions.

In my opinion, the most sensible criterion for deciding whether a definition includes the right configurations is that two lines should be antiparallel with respect to two other lines if and only if the four intersections make a cyclic quadrilateral (assuming they are distinct; if there's a double intersection, this degenerates to a line tangent to the circumcircle); the sides of an isosceles trapezoid or rectangle naturally qualify. –jacobolus (t) 21:11, 20 February 2024 (UTC)

Davies & Peck's Mathematical Dictionary and Cyclopedia of Mathematical Science (1855) has: "If two straight lines, AB and CD, are intersected by two other straight lines, BD and CA, making the angle ACD equal to the angle ABD, the last two lines are sub-contrary with respect to the first two.The first two lines are also sub-contrary with respect to the last two. If the point O is at an infinite distance, the lines AB and CD will be parallel to each other, and AC, BD are anti-parallel". [O being the intersection of AB and CD] –jacobolus (t) 04:02, 22 February 2024 (UTC)

Actually I took a look at the literature again. Weißenstein (Mathworld), Johnson (Modern Geometry, p.172) and Lueger (German Encyclopedia) require one pair of lines to intersect (or an angle) while making no explicit requirement for the other pair of lines. That imho breaks the symmetry (of "with respect to") (you can't switch the line pairs anymore) and causes problem for some of the statements for antiparallel lines derived later on. This presumably can be by requiring the second pair of lines to intersects as well, so excluding any parallels.

But Ivanov (encyclopedia of mathematics) and Biaga both don't require any of two pairs of lines to intersect themselves and their definition is equivalent to requiring the 4 intersection points to form a concyclic quadrilateral. They are both listed under references already, accessible online and source the article's definition. So based on them everything seems fine, symmetry is maintained and a pair of lines can be parallel and antiparallel at the same time.--Kmhkmh (talk) 05:45, 22 February 2024 (UTC)

Do you have any idea when antiparallel/subcontrary was first introduced for pairs of lines? The concept for two planes relative to a cone (a slice of which gives antiparallel lines) comes from Apollonius. –jacobolus (t) 20:10, 22 February 2024 (UTC)

Here's Edmund Stone (1726) A New Mathematical Dictionary:

Anti-Parallels are those Lines that make the same Angles with two Lines cutting them; but contrary Ways: As Parallel-Lines make the same Angles on the same Side with the Lines that cut them, viz. AFE to ABC: So if AFE = ACB: The Lines FE, BC are called Anti-Parallels. When the Sides of a Triangle as AB, AC, be cut by a line FE, Anti-Parallel to the Base BC, the said Sides are cut reciprocally by the said Line FE.

–jacobolus (t) 20:33, 22 February 2024 (UTC)

(Aside: Eric W. Weisstein (not Weissenstein) spells his name with "ss" rather than "ß"; it's a German name but Weisstein is American.) I don't really consider Weisstein a very reliable source, and would recommend finding a different source for claims made in Wikipedia whenever possible. His website has been influential because it was reasonably comprehensive, easily available, had enough inbound links to be promoted by search engines, etc., but he's generally somewhat sloppy and happily reprints stuff he finds without double-checking it, as a result spreading a lot of mistakes. –jacobolus (t) 23:47, 22 February 2024 (UTC)