Talk:Parallel curve

Autoparallel

The notion of parallel from "autoparallel" doesn't seem to be quite the same as in the rest of the article. In fact, I can find no sources discussing offset curves to even mention these "autoparallel" curves. What I could find is a book on general relativity [1], which says: "in a flat space an auto-parallel curve is a straight line", with geodesics given as examples on the sphere. The notion of parallel there is that of parallel transport, of course. JMP EAX (talk) 16:27, 15 August 2014 (UTC)

Assuming the material about "autoparallel" curves in this article can be referenced, then autoparallel curve (currently a red link), should be a disambiguation. If no refs can be found for the use in this article, the material should be deleted and "autoparallel curve" redirected to "parallel transport". JMP EAX (talk) 16:49, 15 August 2014 (UTC)

I suppose there's fat chance of the IP who added the "autoparallel" material in 2004 to provide references now... [2]. JMP EAX (talk) 16:56, 15 August 2014 (UTC)

A quick search turns up [3] which defines auto-parallel curves as ones where the vector field of tangent vectors to the curve is parallel (in the sense of parallel transport). This seems completely different to saying it a curve is parallel to itself. I would say we remove this paragraph.--Salix alba (talk): 20:19, 15 August 2014 (UTC)

Distance field

It's effin' obvious these are related notions, see e.g. [4], but I can't find any source to cite for the relationship right now... JMP EAX (talk) 19:02, 15 August 2014 (UTC)

Parallel surface

Meanwhile the article contains more material about parallel surfaces. I suggest to move them to an article on "parallel surface".--Ag2gaeh (talk) 18:08, 1 May 2019 (UTC)

New geometric properties

I wrote this blog post after doing some recreational research into planar Bertrand curves. Planar Bertrand curves are parallel curves with nonvanishing curvature. I discussed some interesting results which are very relevant to the Geometric properties section of this article.

• In my blog post I describe parallel curves which touch the evolute and have no cusps. This is in contradiction to what is already written: "When parallel curves are constructed they will have cusps when the distance from the curve matches the radius of curvature. These are the points where the curve touches the evolute." For example, the parallel curve at distance 1 above the parabola $y={\frac {1}{2}}x^{2}$ has this property. It possesses a tangent at every point, even when it touches the evolute. This happens in general when $R_{d}(t)$ touches 0 at a point, but does not change sign on either side of that point.
• I also mention that, when they are defined, the osculating circles to planar Bertrand curves are concentric. This fact extends to parallel curves since existence of osculating circles implies nonvanishing curvature. Since this doesn't conflict with anything I have already included it.

FionaLovesCats (talk) 20:21, 8 May 2022 (UTC)

I've had a look at your blog post. In the case you mention, the parallel curve at a vertex of the curve where distance matches the radius of curvature. At the point on the parallel, it actually has a higher singularity, called an A2 or swallowtail singularity, while it may look smooth if you actually try and calculate its derivatives here you find they all vanish.

Your second point seems correct.--Salix alba (talk): 06:39, 9 May 2022 (UTC)