Funny that we're discussing this; over the past week I was marveling over the fact that a circle or ellipse remains elliptical (and hence has Klein 4-group symmetry) whether you project it to a plane by parallel projection OR project it to a plane by central projection. Intuitively it seems to me that these facts are in conflict with each other, but I guess they're not, since parallel projection is a limiting case of central projection. Here's a question: Let G be the groupoid of parallel and central projection maps taking one plane in R^3 to another. Let H be the subgroupoid that carries a fixed circle to itself. This subgroupoid is actually a group. Is it the group of Mobius transformations? Jim Propp On Tue, May 21, 2019 at 5:38 AM Bill Gosper <billgosper@gmail.com> wrote:
Thanks wfl, brad klee, etc for all the (resoundingly affirmative) responses. When you all work on such a pure math-y question, do you feel a twinge of irrelevance? I mean, who sections elliptical cones?
Dishware from Dong Lai Shun Restaurant in Mountain View: https://s3-media4.fl.yelpcdn.com/bphoto/We4TAw4FEsYATsaBogX9Xw/o.jpg
The symmetry of the elongated ellipse from slicing an already elliptical cone is perhaps even more surprising than in the circular conic construction, with no Dandelin spheres to save you. —rwg
-------- Original Message -------- Subject: Re: [math-fun] "Solid" geometry Date: 2019-05-20 17:54 From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Reply-To: Dan Asimov <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com>
If we apply a linear transformation to the elliptical cone to get a round one, the sectioning plane goes to another plane cutting a conic section, so the inverse transformation shows the answer is Yes, since linear images of conic sections are still conic.
—Dan
----- Is a section of an elliptical cone a Conic Section? -----
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