I think any rotation, with or without reflection, is possible. I.e., any 2D orthogonal matrix within that plane centered at that circle. Tom James Propp writes:
I wrote "... is a circle again" but should've written "... is the circle we started with".
Jim Propp
On Tue, May 21, 2019 at 1:36 PM James Propp <jamespropp@gmail.com> wrote:
I did not intend the two planes to be fixed; I was imagining a much more promiscuous groupoid!
Imagine a circle on a plane. Project it (using parallel rays of light, say) onto some other plane in 3-space; you've got an ellipse. Now take that ellipse and project it (this time using rays of light emanating from a point, say) onto some third plane. Keep doing this, over and over, until the millionth projection happens to return the locus back to the original plane, and lo and behold, the million-times-projected locus is a circle again. What sorts of self-maps of the circle can be generated in this way?
Jim Propp
On Tue, May 21, 2019 at 11:56 AM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the question intends the two planes to be given fixed planes, identified via some fixed linear correspondence between them? And the Möbius group M to be in 2-space, with dimension 6 ?
Does "projection" mean simply "perspectivity", excluding centres on either plane and including centres at infinity? Then G is not a groupoid, since the product of perspectivities is not in general a perspectivity. G has dimension 3 < 6, so --- no!
If on the other hand G is _generated_ by perspectivities, then it is some 15-dimensional subset of the 3-space projective group; though its action on the plane(s) is some 8-dimensional subset of the 2-space projective group.
Then the intersection of G with M equals the planar similarities, with dimension 3+1 = 4 < 6 , so once again --- no!
WFL
On 5/21/19, James Propp <jamespropp@gmail.com> wrote:
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? -----