[math-fun] sections of quadratic surfaces
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. 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)? --rwg o o o o o o o o o o o o o o o o o o o o o o o o o kosher kosher salt
Dandelin's proof (which I call the "ice cream cone proof") of ellipses from sections of cones is one of the coolest proofs from "The Book". Too bad my high school teacher didn't use it instead of "some math, some garbage"... http://en.wikipedia.org/wiki/Dandelin_spheres BTW, "oval" at MIT used to mean "Springfield oval" (Google it!) At 12:05 PM 2/14/2007, R. William Gosper wrote:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. 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)? --rwg
Regarding the term "oval" in geometry: it's not well-defined. See http://en.wikipedia.org/wiki/Oval_(geometry) -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
Imagine a (half) cone with its bottom pointed vertically down. Now put a ball bearing inside the cone & send it around the inside of the cone. If the ball bearing is sufficiently small, and the friction is sufficiently small, the ball should revolve around the center of the cone in a cyclic manner. I would guess that this cycle would remain in a plane, but this would have to be proven. Q. What shape does this cycle have when the plane of the cycle isn't horizontal? (I don't know the answer.) At 12:05 PM 2/14/2007, R. William Gosper wrote:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. 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)? --rwg
o o o o o o o o o o o o o o o o o o o o o o o o o
kosher kosher salt
If the cycle were in a plane, it would be an ellipse. Since the potential is something like c*x*sin(theta), for the energy to be conserved would require c*x'*sin(theta) + x'*x'' + y'*y'' = 0, which indicates that y'' = 0, or that there is no net force in the y direction (and the y-component of the velocity is constant). Possibly, this is true, but it seems unlikely at first glance. Bill C. ----------------------------------- Henry Baker: Imagine a (half) cone with its bottom pointed vertically down. Now put a ball bearing inside the cone & send it around the inside of the cone. If the ball bearing is sufficiently small, and the friction is sufficiently small, the ball should revolve around the center of the cone in a cyclic manner. I would guess that this cycle would remain in a plane, but this would have to be proven. Q. What shape does this cycle have when the plane of the cycle isn't horizontal? (I don't know the answer.) At 12:05 PM 2/14/2007, R. William Gosper wrote:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. 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)? --rwg
o o o o o o o o o o o o o o o o o o o o o o o o o
kosher kosher salt
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You can see this for real... http://youtube.com/watch?v=MqDAf_lg9Xs --Michael Kleber On 2/15/07, Henry Baker <hbaker1@pipeline.com> wrote:
Imagine a (half) cone with its bottom pointed vertically down.
Now put a ball bearing inside the cone & send it around the inside of the cone. If the ball bearing is sufficiently small, and the friction is sufficiently small, the ball should revolve around the center of the cone in a cyclic manner. I would guess that this cycle would remain in a plane, but this would have to be proven.
Q. What shape does this cycle have when the plane of the cycle isn't horizontal? (I don't know the answer.)
At 12:05 PM 2/14/2007, R. William Gosper wrote:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval. 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)? --rwg
o o o o o o o o o o o o o o o o o o o o o o o o o
kosher kosher salt
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participants (5)
-
Cordwell, William R -
Henry Baker -
Michael Kleber -
Mike Stay -
R. William Gosper