The transformations of RP^2 that preserve a given conic -- such as the unit circle -- form a 3-parameter family (because there are 8 degrees of freedom in a projective transformation, and the space of conics is 5 parameters) with two connected components qua topological groups (the orientation-preserving component and the orientation-reversing component). This group of transformations is isomorphic to the group of isometries of the hyperbolic plane (proof: Beltrami-Klein model) which is has an index-2 subgroup (the orientation-preserving isometries) which is isomorphic to PSL(2, R) (proof: upper half-plane model). -- APG
Sent: Tuesday, May 21, 2019 at 7:36 PM From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Fwd: Re: "Solid" geometry
I’m pretty sure you can get self maps of the circle that aren’t rotations.
Imagine a circle with six equally-spaced points cyclically marked A,B,C,D,E,F. Hold it in front of you at a tilt, so that it looks like an ellipse with points A and D being the endpoints of the major axis. The visual angle between B and C should be different from the visual angle between E and F, since two of them are closer and two of them are farther away. So, if we put a flat plane where your retina is (ouch!), the projection gives an ellipse in which B is not opposite E and C is not opposite F. Now we can apply an orthogonal projection at an appropriate angle, turning this marked ellipse into a marked circle in which AD is a diameter but BE and CF are not.
(Actually, I’m not sure AD is still a diameter.)
Anyway: is this a Mobius transformation?
Jim Propp
On Tue, May 21, 2019 at 2:16 PM Tom Karzes <karzes@sonic.net> wrote:
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? > ----- >
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