February 2007 - mathfun@mailman.xmission.com

Re: [math-fun] Re: sections of quadratic surfaces
by Daniel Asimov 19 Feb '07

19 Feb '07

Fred Lunnon wrote: [attributions have been edited for accuracy]: <<<< On 2/17/07, Emma Cohen <emma(a)don-eve.dyndns.org> wrote: << Jim Propp 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 >>>> This is an interesting point. It seems that by rights, a line segment (= an ellipse with |minor axis| = 0) *should* be within the family of conic sections, as a degenerate case. So if a is fixed, the locus of (*) (x/a)^2 + (y/b)^2 = 1 seems as if it ought to approach the locus of some limiting version of the equation (*) as b decreases to 0. This seems possible, in a sense, if conic sections are thought of as occurring in projective geometry. If we replace R^3 by its "projective compactification" by adding the P^2 of the "lines at infinity", the result is P^3, projective 3-space. A cone like, say, x^2 + y^2 = z^2 is replaced by the addition of a circle at infinity to it, resulting in a cool conoid shape in P^3 whose name I forget, but which (projectively) resembles a torus of revolution with one meridional circle pinched to a point, and which gets fatter the farther away one is from that point. Then the conic sections become the intersection of one of these conoids with an "affine projective plane" in P^3. All the nondegenerate intersections are equivalent to ellipses, but the segmentlike degenerate ellipse at issue becomes a circle. (At least it's compact.) --Dan

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[math-fun] How NOT to draw an oval ...
by Fred lunnon 18 Feb '07

18 Feb '07

Over the years I have spent many happy but largely fruitless hours attempting to construct a genuinely independent interpolation algorithm. Here, for the delectation of math funners, is my innovatory technique for building Bezier-like closed-curve splines, recently disinterred for service in another ovipository thread. Suppose we are given n control points P_i for i = 0,...,n-1, [as Cartesian 2-vectors in the plane, homogeneous 4-vectors in space, or whatever]. We wish to construct a closed curve Q(u), passing near(?) to each control as 0 <= u <= 1, with Q(0) = Q(1) and similarly for all derivatives. Consider the weight functions f_i(u) = (2/n)*cos(Pi*(u - i/n))^2 : these are analytic, periodic over 0 <= u <= 1, and for all u sum to 1 over 0 <= i <= n-1. Furthermore f_i is nonnegative, and has its maximum at u = i/n. So a reasonable looking spline function might be Q(u) = \sum_i f_i(u) P_i. Exercise: prove that, whatever the controls, Q(u) is always an ellipse! Fred Lunnon

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[math-fun] Re: sections of quadratic surfaces
by James Propp 17 Feb '07

17 Feb '07

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

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[math-fun] RE: amazing photo
by Guy Haworth 17 Feb '07

17 Feb '07

Wonderful: I wonder what such a mapping of the view from inside a regular or semi-regular Platonic solid would look like - the viewpoint being at the centre, or off-centre down some specific or random axis. The faces of the polyhedron could be 'windows' onto 'other worlds', though the view might be a little confusing with the non-convex, stellated ones. g

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[math-fun] amazing photo
by Thane Plambeck 17 Feb '07

17 Feb '07

>From a flickr user who's been taking photos of European churches and running them through the MathMap GIMP plugin, getting MC Escher-like results ** snip *** Saint-Etienne-du-Mont church in Paris, one of the most beautiful churches to visit. It is based on an equirectangular (see the immersive version!) panorama and it is transformed into the conformal Guyou Doubly Periodic mapping. *** snip ** click here http://www.flickr.com/photos/sbprzd/391549837/ -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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[math-fun] Quantum computer solves Sudoku
by Henry Baker 16 Feb '07

16 Feb '07

FYI -- I'm glad to see that quantum computers are working on useful things: http://www.nature.com/news/2007/070212/full/070212-8.html

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[math-fun] one-line/proof without words/elegant proof that coeffs of (1+x)^n approx a Gaussian?
by Henry Baker 15 Feb '07

15 Feb '07

Is there a simple/elegant/one-line proof that the coefficients of (1+x)^n approximate a Gaussian curve? I know that it is the convolution of n boxes, and that a large convolution of non-impulses approaches a Gaussian, but is there a quick/obvious proof of this that also gives you the appropriate mean & std deviation parameters?

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Re: [math-fun] one-line/proof without words/elegant proof thatcoeffs of (1+x)^n approx a Gaussian?
by Daniel Asimov 15 Feb '07

15 Feb '07

<< Why should a discrete distribution (the binomial coefficients) approach the continuous distribution ~ exp(-(x/sigma)^2/2), . . . >> Because the limit is taken of cumulative probability functions (cdf) F_n(x) = Prob{X_n <= x) (where X_n's are random variables with the distributions in question). --Dan

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[math-fun] sections of quadratic surfaces
by R. William Gosper 15 Feb '07

15 Feb '07

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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Re: [math-fun] sections of quadratic surfaces
by Daniel Asimov 15 Feb '07

15 Feb '07

Re "ovals" Duh -- it never occurred to me before now that the word "oval" derives from the word "ovum".. --Dan

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