[math-fun] Re: sections of quadratic surfaces
Bill Gosper writes:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval.
What is an oval? I thought it was just a word for a generic thing that looks kind of like an ellipse.
What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)?
Beats me! If you use the linear equation to eliminate one variable in a quadric, how can you get anything but a conic? But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section? Of course, what the line segment is trying to be is a line, which is a bona fide (albeit degenerate) conic section. (This might be a good example to use when introducing undergraduates to algebraic geometry: the intersection of a cone with a plane can't be a line segment because a line segment isn't Zariski-closed.) It'd be interesting to see an animation that shows, side by side, a plane cutting a cone and the associated quadratic plane curve (rotated in 3-space in some standard way) evolving as the latter becomes more and more eccentric, with fixed major axis AB; perhaps one could develop some intuition about the way in which the points on the line AB that don't lie between A and B "jump" into the locus when the cutting plane becomes tangent to the cone. Jim Propp
An oval is egg-shaped. The two ends have different curvatures. There's one axis of symmetry, while an ellipse has two. When a plane cuts a cone at an angle, the cut near the vertex cuts the cone more-nearly perpendicularly than the cut far from the vertex. Naively, the near-vertex cut should produce a curve that spreads out more than the far end of the cut, which should be somewhat pointy. So you expect an oval, not an ellipse. The trick is that the far cut is "bigger", which evens out the two ends of the ellipse. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of James Propp Sent: Wed 2/14/2007 3:40 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Re: sections of quadratic surfaces Bill Gosper writes:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval.
What is an oval? I thought it was just a word for a generic thing that looks kind of like an ellipse.
What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)?
Beats me! If you use the linear equation to eliminate one variable in a quadric, how can you get anything but a conic? But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section? Of course, what the line segment is trying to be is a line, which is a bona fide (albeit degenerate) conic section. (This might be a good example to use when introducing undergraduates to algebraic geometry: the intersection of a cone with a plane can't be a line segment because a line segment isn't Zariski-closed.) It'd be interesting to see an animation that shows, side by side, a plane cutting a cone and the associated quadratic plane curve (rotated in 3-space in some standard way) evolving as the latter becomes more and more eccentric, with fixed major axis AB; perhaps one could develop some intuition about the way in which the points on the line AB that don't lie between A and B "jump" into the locus when the cutting plane becomes tangent to the cone. Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Is there a technical definition of "egg-shaped"? A way to measure the ovality? Ellipses have eccentricity, do ovals then have eggcentricity? More seriously, what constraints make sense to give a well defined "egg shape"? You have one one axis of symmetry and two curves that must fit together smoothly, very smoothly. I don't see how you can glue halves of two different ellipses together to get the smoothness constraint. (Can you?) There should be lots of other "almost ellipse" definitions that will glue together. Any takers on a simple definition or process to make an egg? Err... oval? ________________________________ From: math-fun-bounces+mdtorge=sandia.gov@mailman.xmission.com on behalf of Schroeppel, Richard Sent: Wed 2/14/2007 5:02 PM To: math-fun; math-fun@mailman.xmission.com Cc: rcs@cs.arizona.edu Subject: RE: [math-fun] Re: sections of quadratic surfaces An oval is egg-shaped. The two ends have different curvatures. There's one axis of symmetry, while an ellipse has two. When a plane cuts a cone at an angle, the cut near the vertex cuts the cone more-nearly perpendicularly than the cut far from the vertex. Naively, the near-vertex cut should produce a curve that spreads out more than the far end of the cut, which should be somewhat pointy. So you expect an oval, not an ellipse. The trick is that the far cut is "bigger", which evens out the two ends of the ellipse. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of James Propp Sent: Wed 2/14/2007 3:40 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Re: sections of quadratic surfaces Bill Gosper writes:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval.
What is an oval? I thought it was just a word for a generic thing that looks kind of like an ellipse.
What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)?
Beats me! If you use the linear equation to eliminate one variable in a quadric, how can you get anything but a conic? But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section? Of course, what the line segment is trying to be is a line, which is a bona fide (albeit degenerate) conic section. (This might be a good example to use when introducing undergraduates to algebraic geometry: the intersection of a cone with a plane can't be a line segment because a line segment isn't Zariski-closed.) It'd be interesting to see an animation that shows, side by side, a plane cutting a cone and the associated quadratic plane curve (rotated in 3-space in some standard way) evolving as the latter becomes more and more eccentric, with fixed major axis AB; perhaps one could develop some intuition about the way in which the points on the line AB that don't lie between A and B "jump" into the locus when the cutting plane becomes tangent to the cone. Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Here's the simplest egg-shape known to me: http://img179.imageshack.us/img179/7915/eggxw7.gif. IIRC, it is the shape used by Arnault of Zwolle (ca. 1450) in describing his lute. Its construction is so simple that I think, before giving it, I should ask if others can figure it out first. David ----- Original Message ----- From: "Torgerson, Mark D" <mdtorge@sandia.gov> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, February 15, 2007 00:36 Subject: RE: [math-fun] Re: sections of quadratic surfaces Is there a technical definition of "egg-shaped"? A way to measure the ovality? Ellipses have eccentricity, do ovals then have eggcentricity? More seriously, what constraints make sense to give a well defined "egg shape"? You have one one axis of symmetry and two curves that must fit together smoothly, very smoothly. I don't see how you can glue halves of two different ellipses together to get the smoothness constraint. (Can you?) There should be lots of other "almost ellipse" definitions that will glue together. Any takers on a simple definition or process to make an egg? Err... oval? ________________________________ From: math-fun-bounces+mdtorge=sandia.gov@mailman.xmission.com on behalf of Schroeppel, Richard Sent: Wed 2/14/2007 5:02 PM To: math-fun; math-fun@mailman.xmission.com Cc: rcs@cs.arizona.edu Subject: RE: [math-fun] Re: sections of quadratic surfaces An oval is egg-shaped. The two ends have different curvatures. There's one axis of symmetry, while an ellipse has two. When a plane cuts a cone at an angle, the cut near the vertex cuts the cone more-nearly perpendicularly than the cut far from the vertex. Naively, the near-vertex cut should produce a curve that spreads out more than the far end of the cut, which should be somewhat pointy. So you expect an oval, not an ellipse. The trick is that the far cut is "bigger", which evens out the two ends of the ellipse. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of James Propp Sent: Wed 2/14/2007 3:40 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Re: sections of quadratic surfaces Bill Gosper writes:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval.
What is an oval? I thought it was just a word for a generic thing that looks kind of like an ellipse.
What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)?
Beats me! If you use the linear equation to eliminate one variable in a quadric, how can you get anything but a conic? But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section? Of course, what the line segment is trying to be is a line, which is a bona fide (albeit degenerate) conic section. (This might be a good example to use when introducing undergraduates to algebraic geometry: the intersection of a cone with a plane can't be a line segment because a line segment isn't Zariski-closed.) It'd be interesting to see an animation that shows, side by side, a plane cutting a cone and the associated quadratic plane curve (rotated in 3-space in some standard way) evolving as the latter becomes more and more eccentric, with fixed major axis AB; perhaps one could develop some intuition about the way in which the points on the line AB that don't lie between A and B "jump" into the locus when the cutting plane becomes tangent to the cone. Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun --------------------------------------------------------------------------------
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On 2/15/07, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
Here's the simplest egg-shape known to me: http://img179.imageshack.us/img179/7915/eggxw7.gif.
IIRC, it is the shape used by Arnault of Zwolle (ca. 1450) in describing his lute. Its construction is so simple that I think, before giving it, I should ask if others can figure it out first.
David
Huh --- I fiddled (or maybe luted) around here trying to join a circular arc to a parabolic arc, matching both tangent and curvature --- eventually realising to my disgust that it cannot be done! WFL
The construction is very simple, requiring only arcs of circles of three different radii. Surely someone can find it... David ----- Original Message ----- From: "Fred lunnon" <fred.lunnon@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, February 15, 2007 18:39 Subject: Re: [math-fun] Re: sections of quadratic surfaces
On 2/15/07, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
Here's the simplest egg-shape known to me: http://img179.imageshack.us/img179/7915/eggxw7.gif.
IIRC, it is the shape used by Arnault of Zwolle (ca. 1450) in describing his lute. Its construction is so simple that I think, before giving it, I should ask if others can figure it out first.
David
Huh --- I fiddled (or maybe luted) around here trying to join a circular arc to a parabolic arc, matching both tangent and curvature --- eventually realising to my disgust that it cannot be done! WFL
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Yesterday, I had said:
The construction is very simple, requiring only arcs of circles of three different radii. Surely someone can find it...
Hmm. Maybe not. :-( Anyway, the figure at <http://img255.imageshack.us/img255/498/eggconstructzn0.gif> should make the construction abundantly clear. I suspect that that beautiful egg shape was used a good bit during the Middle Ages (perhaps in architecture?) but have no supporting evidence. Having been a maker of historical instruments, I analyzed the outlines of many lutes (and some other instruments) from the Renaissance. The outlines were always done, it seemed, by ruler-and-compass constructions. But the outlines of lutes in the later Renaissance involved somewhat more complicated constructions than did Arnault's lute. David
----- Original Message ----- From: "Fred lunnon" <fred.lunnon@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, February 15, 2007 18:39 Subject: Re: [math-fun] Re: sections of quadratic surfaces
On 2/15/07, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
Here's the simplest egg-shape known to me: http://img179.imageshack.us/img179/7915/eggxw7.gif.
IIRC, it is the shape used by Arnault of Zwolle (ca. 1450) in describing his lute. Its construction is so simple that I think, before giving it, I should ask if others can figure it out first.
David
Huh --- I fiddled (or maybe luted) around here trying to join a circular arc to a parabolic arc, matching both tangent and curvature --- eventually realising to my disgust that it cannot be done! WFL
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On 2/16/07, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
Hmm. Maybe not. :-( Anyway, the figure at <http://img255.imageshack.us/img255/498/eggconstructzn0.gif> should make the construction abundantly clear. I suspect that that beautiful egg shape was used a good bit during the Middle Ages (perhaps in architecture?) but have no supporting evidence.
I looked at this diagram, afterwards attempted to reconstruct it mentally, and failed --- in spite of the fact that it has much in common with the construction of a Reuleaux (constant diameter) polygon.
Having been a maker of historical instruments, I analyzed the outlines of many lutes (and some other instruments) from the Renaissance. The outlines were always done, it seemed, by ruler-and-compass constructions. But the outlines of lutes in the later Renaissance involved somewhat more complicated constructions than did Arnault's lute.
A highly polished surface reflecting a bright light would show up the discontinuous curvature of such an elementary spline; as technology advanced, so perfectionist makers might well have become dissatisfied with this situation --- as have the designers of modern cars. Fred Lunnon
What about the locus of points such that d(A)+kd(B) is constant, where k is a real number and A,B are the focii. Wouldn't that make an oval? Steve Gray Fred lunnon wrote:
On 2/16/07, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
Hmm. Maybe not. :-( Anyway, the figure at <http://img255.imageshack.us/img255/498/eggconstructzn0.gif> should make the construction abundantly clear. I suspect that that beautiful egg shape was used a good bit during the Middle Ages (perhaps in architecture?) but have no supporting evidence.
On 2/17/07, Steve Gray <stevebg@adelphia.net> wrote:
What about the locus of points such that d(A)+kd(B) is constant, where k is a real number and A,B are the focii. Wouldn't that make an oval?
Steve Gray
This gives a rather nice pointy oval, if k is chosen just less than (string length)/(focal distance), being the critical value where the curve becomes limacon-like. The equation is ((x^2 + (y-a)^2) + c^2*(x^2 + (y+a)^2) - 4*b^2)^2 - 4*(x^2 + (y-a)^2)*c^2*(x^2 + (y+a)^2) = 0, where 2a = focal distance, 2b = string length, c = Gray weighting k; with say a = 1, b = 1.5, c = 1.35 inside a 3x3 box. WFL
MDT>Is there a technical definition of "egg-shaped"? A way to measure the ovality? Ellipses have eccentricity, do ovals then have eggcentricity? Ouch. Poach that with kosher salt. Based on my experience with (unsymmetric) ovoids (http://gosper.org/eleven.rtf), eggcentricity would probably be a two- or three-vector, at least.
More seriously, what constraints make sense to give a well defined "egg shape"? You have one one axis of symmetry and two curves that must fit together smoothly, very smoothly. I don't see how you can glue halves of two different ellipses together to get the smoothness constraint. (Can you?) There should be lots of other "almost ellipse" definitions that will glue together. Any takers on a simple definition or process to make an egg? Err... oval?
How about a "threelipse"? Use the loop and string with three tacks in an isosceles triangle instead of two (in a unit eccentricity ellipse!-). The curvatures are discontinuous, but people don't notice. It's like the "four-point ellipse" draftsmen use in isometric drawings to simulate a circle viewed at an angle, which is just four 90degree arcs taken alternately from two different circles. --rwg HEPTAGON PATHOGEN
On 2/14/07, James Propp <propp@math.wisc.edu> wrote:
It'd be interesting to see an animation that shows, side by side, a plane cutting a cone and the associated quadratic plane curve (rotated in 3-space in some standard way) evolving as the latter becomes more and more eccentric, with fixed major axis AB; perhaps one could develop some intuition about the way in which the points on the line AB that don't lie between A and B "jump" into the locus when the cutting plane becomes tangent to the cone.
Hi Jim and all, This kind of animation is easy to construct with, for example, Graphing Calculator. http://www.pacifict.com ... at least download the free viewer and see if you can run the demo from that, maybe. It's pretty fun. --Joshua Zucker
But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section?
Isn't this definition of a line segment just an ellipse whose minor axis is zero?
On 2/17/07, Emma Cohen <emma@don-eve.dyndns.org> wrote:
But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section?
Isn't this definition of a line segment just an ellipse whose minor axis is zero?
However, if you attempt to construct a finite line segment as the intersection of a plane with a cone --- the origin of the term "conic section" --- you will find it impossible, WFL
Just as if you attempt to construct two parallel lines by intersecting a plane with a cone. Or as if you attempt to construct a circle with the focus-directrix definition. The whole area of "limiting cases" is an embarrassment: We keep what's convenient, and tell students that the rest is nonsense. Nonsense. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Fred lunnon Sent: Sat 2/17/2007 7:04 AM To: math-fun Subject: Re: [math-fun] Re: sections of quadratic surfaces On 2/17/07, Emma Cohen <emma@don-eve.dyndns.org> wrote:
But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section?
Isn't this definition of a line segment just an ellipse whose minor axis is zero?
However, if you attempt to construct a finite line segment as the intersection of a plane with a cone --- the origin of the term "conic section" --- you will find it impossible, WFL _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (9)
-
David W. Cantrell -
Emma Cohen -
Fred lunnon -
James Propp -
Joshua Zucker -
R. William Gosper -
Schroeppel, Richard -
Steve Gray -
Torgerson, Mark D