Talk:Ruled surface

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Significance of ruled surface[edit]

Moved from main article:

question: what is the significance of ruled surfaces in the history of geometry? What is its place in today geometry? is it just of historical/curiosity interest? Foobar 09:34, 17 Jun 2004 (UTC)

Correctness of image caption[edit]

Is Image:Ruled hyperboloid.jpg indeed a hyperboloid of one sheet? The description of the image in Wikipedia and Wiki Commons were taken from the caption of item 51 in the original page at U. Arizona, but that caption seems incorrect. Indeed, this model looks suspiciously similar to the "quartic surface with double line" model (number 52 in that page). Check carefully the wires and how they cross, and compare with those of a true doubly ruled hyperboloid (number 68). The quartic surface, I gather, is obtained when two points of the generating line move along two parallel circles in opposite senses. All the best, --Jorge Stolfi (talk) 15:33, 26 June 2009 (UTC)[reply]

I repeat the question. Wikipedia does not need that embarassment...

Well, I fixed the caption to say what the picture actually seems to show. All the best, --Jorge Stolfi (talk) 21:52, 8 February 2010 (UTC)[reply]

Nope. It's an Elliptic hyperboloid of one sheet- and it's actually a doubly ruled surface.- Wolfkeeper 22:03, 8 February 2010 (UTC)[reply]

Look again. It is not. For one thing, it is self-intersecting and has two pinch points. All the best, --Jorge Stolfi (talk) 21:29, 21 February 2010 (UTC)[reply]

I claim that this image has parametric equation

\left\{{\begin{array}{l}X(u,v)=Au\sin(v)\\Y(u,v)=\cos(v)\\Z(u,v)=u\end{array}}\right.

The sweep lines are obtained by fixing v to different values. The two generating circles are obtained by setting u=+1/A and u=-1/A, respectively. The surface self-interesects along the segment X=Z=0,Y spanning [-1,+1].

The fact that the surface is (at least) 4th degree is proved by considering a line parallel to the YZ plane tilted 45 degrees relative to both axes and passing through the point (ε,0,0) where ε is a sufficiently small positive number. This line crosses the surface 4 times.

This surface is similar to Whitney's umbrella (a 3rd degree surface) except that the generating paabolas are replaced by circles.

All the best, --Jorge Stolfi (talk) 22:04, 21 February 2010 (UTC)[reply]

If you believe the caption of item 51 at U. Arizona is incorrect, I suggest you contact the authors of that page. Geometry guy 01:33, 22 February 2010 (UTC)[reply]

Good idea, it is done. --Jorge Stolfi (talk) 04:11, 22 February 2010 (UTC)[reply]

Let me know if/when you get a response. Geometry guy 22:46, 22 February 2010 (UTC)[reply]

I had to track down "item 51" from Geometry guy's link to find the caption: the frame is here. --Thnidu (talk) 16:15, 12 December 2010 (UTC)[reply]

It seems that the U Arizona page has been silently corrected. I have updated the image at commons, but it may take some time for the changes to become visible here. Sławomir Biały (talk) 14:24, 22 August 2010 (UTC)[reply]

To understand what Stolfi is saying, I made myself a POV-Ray model of the surface described, which is singly ruled, has a line of self-intersection, and if seen from some angles can be casually mistaken for a hyperboloid. I cannot find a viewpoint at which the two "limbs" of the figure are as symmetrical as in the photograph in question. Note also that in the photograph the threads are anchored in pairs, at opposite angles, along each rim. Perhaps Stolfi believes the photograph shows two copies of his object, with X and Y exchanged? —Tamfang (talk) 23:38, 29 June 2011 (UTC)[reply]

Never mind. Sławomir Biały changed the image file from the wrong one (described by Stolfi) to a correct hyperboloid. —Tamfang (talk) 07:19, 30 June 2011 (UTC)[reply]

equatorial "straight line segment" on hyperboloid is a circle[edit]

The caption of the main image, borrowed from Hyperboloid, said

A hyperboloid is a doubly-ruled surface generated by a set of straight wires, whose ends span two parallel circles in opposite senses. Note the isolated straight line segment that lies on the surface (at its narrowest part) and intercepts all the wires.

The second sentence does not appear in the caption at Hyperboloid. If I understand it at all correctly, it refers to the "equator" or "waistline" of the hyperboloid, which is not a straight line segment at all but a circle which contains the centers of the wires and is parallel to the circles containing their ends. Deleting that sentence. --Thnidu (talk) 16:10, 12 December 2010 (UTC)[reply]

That description fits the quartic surface that Jorge Stolfi talks about, above. —Tamfang (talk) 02:43, 30 June 2011 (UTC)[reply]

"The plane is the only surface which contains at least three distinct lines through each of its points."[edit]

I'd like to see a citation of this, I did not find this proposition elsewhere.212.179.21.194 (talk) 15:44, 12 September 2016 (UTC)[reply]