[math-fun] Fwd: Re: "Solid" geometry
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? ----- _______________________________________________
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? -----
_______________________________________________ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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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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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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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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? -----
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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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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Right, I don't think AD remains a diameter, so I don't think this construction works. Tom James Propp writes:
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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participants (5)
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Adam P. Goucher -
Bill Gosper -
Fred Lunnon -
James Propp -
Tom Karzes