Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
6)are all German mathematicians like Peter Roquette, Gunther Schmidt KarlOtto Stöhr as dumb as Franz, teachi ng a Conic section is ellipse, when in truth it is an oval?
Replies:
11
Last Post:
Oct 4, 2017 7:05 PM




Re: Princeton math dept agreeing with Franz for none has spotted the error as dumb as Franz, teaching a Conic section is ellipse, when in truth it is an oval?
Posted:
Oct 2, 2017 12:11 PM


On Monday, October 2, 2017 at 8:00:29 AM UTC7, Archimedes Plutonium wrote: > On Monday, October 2, 2017 at 1:37:38 AM UTC5, Me wrote: > > Let's consider the Sectioning of a Cylinder and a Cone. > > > > ^ x > > E > > + > > .'  `. > > /  \ > > .  . > > G  +c  H > > .  . > > \  / > > `.  ´ > > y <+ ´ > > F > > > > > The above is a view of a ellipse with center c and is produced by the > > > Sectioning of a Cylinder as long as the cut is not perpendicular to the base, > > > and as long as the cut involves two points not larger than the height of the > > > cylinder walls. What we want to prove is that the cut is always a ellipse, > > > which is a [certain] plane figure of two axes of symmetry with a Major Axis > > > and Minor Axis and center at c. > > > > > > So, what is the proof that [cut] figure EGFH is always an ellipse in the > > > cylinder section [as well as in the cone section]? > > > > Here's is an easy proof for it: > > > > Cylinder (side view): > > > >    > > ++ <= x = h > >   ´ > >   ´  > >  ´  > >  ´   > >  ´   > > x = 0 => ´ > >  r   > > > > d(x) = r  (2r/h)x > > > > y^2 = r^2  d(x)^2 = r^2  r^2(2x/h  1)^2 = r^2(1  4(x  h)^2/h^2 > > > > => (1/r^2)y^2 + (4/h^2)(x  h)^2 = 1 ...equation of an ellipse > > > > Considerations: > > > > => y(h/2 + x')^2 = sqrt(r^2  r^2(2(h/2 + x')/h  1)^2) = r^2  r^2(2x'/h)^2 > > > > => y(h/2 + x') = r * (sqrt(1  (2x'/h)^2) ...symmetric relative to h/2 > > > > => y(h/2) = r (= Gc = cH) > > > > Cone (side view): > > > > . > > /\ > > /  \ > > /b  \ > > /+´ <= x = h > > / ´ \ > > / ´  \ > > / ´  \ > > x = 0 => ´+\ > > / a  \ > > > > r(x) = a  ((ab)/h)x > > d(x) = a  ((a+b)/h)x > > > > y(x)^2 = r(x)^2  d(x)^2 = ab  ab(2x/h  1)^2 = ab(1  4(x  h)^2/h^2 > > > > => (1/ab)y(x)^2 + (4/h^2)(x  h)^2 = 1 ...equation of an ellipse > > > > Considerations: > > > > => y(h/2 + x')^2 = sqrt(ab  ab(2(h/2 + x')/h  1)^2) = ab  ab(2x'/h)^2 > > > > => y(h/2 + x') = sqrt(ab) * (sqrt(1  (2x'/h)^2) ...symmetric relative to h/2 > > > > => y(h/2) = sqrt(ab) (= Gc = cH) > > > > ====================================================== > > > > It turns out that a cylinder can be considered as a special case of a cone here. Actually, the latter proof works for both cases, cone and cylinder. > > > > Cone/Cylinder (side view): > > > > /  \ > > /b  \ > > /+´ <= x = h > > / ´ \ > > / ´  \ > > / ´  \ > > x = 0 => ´+\ > > / a  \ > > > > (cone: b < a; cylinder: a = b = r) > > > > r(x) = a  ((ab)/h)x > > d(x) = a  ((a+b)/h)x > > > > y(x)^2 = r(x)^2  d(x)^2 = ab  ab(2x/h  1)^2 = ab(1  4(x  h)^2/h^2 > > > > => (1/ab)y(x)^2 + (4/h^2)(x  h)^2 = 1 ...equation of an ellipse > > > > Considerations: > > > > => y(h/2 + x')^2 = sqrt(ab  ab(2(h/2 + x')/h  1)^2) = ab  ab(2x'/h)^2 > > > > => y(h/2 + x') = sqrt(ab) * (sqrt(1  (2x'/h)^2) ...symmetric relative to h/2 > > > > => y(h/2) = sqrt(ab) (= Gc = cH) > > > > ====================================================== > > > > @Archie: Yes, this proves that (certain) cone sections "as depicted in my diagram" as well as (certain) cylinder sections (as described by you) are ellipses. qed > > > > Note, Archie, that there is no reference to Dandelin Spheres whatsoever. > > > > Still not convinced? Can you point out an error in my simple calculation (of the shape of the cone/cylinder section) above? > > > Princeton University Math dept > > Michael Aizenman Professor > > Zahra Aminzare Lecturer > > Manjul Bhargava Professor > > Nathaniel Bottman Postdoctoral Research Fellow > > Nicolas Boumal Instructor > > Jean Bourgain Visiting Lecturer with Rank of Professor > Mathematics > > William Browder Professor Emeritus > > Tristan Buckmaster Assistant Professor > > Francesc Castella Instructor > > SunYung Alice Chang Professor > > Otis Chodosh Veblen Research Instructor > zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz > Maria Chudnovsky Professor > > Peter Constantin Professor of Mathematics and Director of PACM > > John Conway Professor Emeritus > > Mihalis Dafermos Professor > > Gabriele Di Cerbo Assistant Professor > > Hansheng Diao Instructor > > Theodore Drivas Postdoctoral Research Fellow > > Zeev Dvir Associate Professor > > Weinan E Professor > > Tarek Elgindi Instructor > > Tolga Etgü Visiting Fellow > > Charles Fefferman Professor > > Jonathan Fickenscher Associate Research Scholar > > David Gabai Chair, Professor > > Ziyang Gao Instructor > > > Javier GómezSerrano Assistant Professor, Director of Graduate Studies > > Robert C. Gunning Professor > > Jonathan Hanselman Assistant Professor > > Helmut Hofer Visiting Lecturer with Rank of Professor > Mathematics > > Henry Horton Postdoctoral Research Associate > > Yong Hou Lecturer > Mathematics > Tatiana Howard Lecturer > > WuChung Hsiang Professor Emeritus > > June Huh Veblen Fellow > > Mihaela Ignatova Instructor > > Alexandru Ionescu Professor > > Jennifer M. Johnson Senior Lecturer, Associate Departmental Representative > > Nicholas Katz Professor > > Casey Kelleher Postdoctoral Research Fellow > > Daniel Ketover Instructor > > Ilya Khayutin Veblen Research Instructor > > Seongtag Kim Visiting Fellow > > Sergiu Klainerman Professor > > Simon Kochen Professor Emeritus > > Joseph Kohn Professor Emeritus > > János Kollár Professor, Department Representative > > Elliott Lieb Professor Emeritus > > Francesco Lin Veblen Research Instructor > > YuehJu Lin Instructor > > ChunHung Liu Instructor > > Robert MacPhersonVisiting Lecturer with Rank of Professor > Mathematics > > Adam Marcus Assistant Professor > > Fernando Codá Marques Professor > > Mark McConnell Senior Lecturer > > Stephen McKeown Postdoctoral Research Associate > > Ana Menezes Assistant Professor > > Sophie Morel Professor > > Assaf Naor Professor > > Evita Nestoridi Instructor > > Huy Quang Nguyen Postdoctoral Research Associate > > Oanh Nguyen Instructor > > Peter Ozsváth Professor, Director of Graduate Studies > > John Pardon Professor > > Fabio Pusateri Assistant Professor > > Igor Rodnianski Professor > > Vermont Rutherfoord Postdoctoral Research Associate > > Peter Sarnak Professor > > Paul D. Seymour Professor > > Tatyana Shcherbyna Assistant Professor > > Nicholas Sheridan Assistant Professor > > Goro Shimura Professor Emeritus > > Yakov ShlapentokhRothman Instructor > > Yakov Sinai Professor > > Amit Singer Professor > > Christopher Skinner Professor > > Allan Sly Professor > > Elias Stein Professor Emeritus > > Zoltán Szabó Professor > > Yunqing Tang Instructor > > Richard Taylor Visiting Lecturer with Rank of Professor > > Christine Taylor Senior Lecturer > > Gang Tian Professor > > Konstantin Tikhomirov Instructor > > Hale Trotter Professor Emeritus > > Karen Uhlenbeck Visiting Research Scholar > > Vlad Vicol Assistant Professor > > Ilya Vinogradov Lecturer > > Rafael von Känel Postdoctoral Research Fellow > > Joseph Waldron Instructor > > Guangbo Xu Associate Research Scholar > > Paul C. Yang Professor > > Ian Zemke Postdoctoral Research Fellow > > ShouWu Zhang Professor > > Yongbin Zhang Visiting Research Scholar
The lists are getting longer. Is this part of your "proof"? Or are you finally getting completely unglued?



