I’m going to talk to my daughter’s 8th grade class about conic sections. I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry. But I found it surprisingly tricky to work out examples. Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4. Using Pythagoras’ theorem produces an equation with a bunch of square roots. Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms. Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2. Am I doing something wrong? Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.? Of course, I then want to talk about light waves bouncing from one focus to another… I’m not sure how to justify this without a little calculus. - Cris
My answer isn't going to please you, but here it is: It's a war crime these days to torture kids by forcing them to do algebraic manipulations, when computers do this stuff so much better. Are we still forcing kids to do 5-digit multiplications & divisions by hand rather than using a pocket calculator [or their cellphone calculator] ? Get Maxima, Mathematica or Maple and teach her that. Re conic sections: There are excellent reasons why vector algebra was invented by Yale Prof. J. Willard Gibbs (Gibbs's vector algebra decomposed quaternion products into dot products and cross products); it takes a *lot* of algebraic garbage out of manipulations like this. I'm talking about dot products, cross products, etc. I recall the first time I had to do 2D graphics calculations, and I thought: I'm good at math, I aced all of my algebra classes, this should be a piece of cake; I don't have to read anything about computer graphics, because I know it all, already! Well, I struggled with the code and eventually got it to work, but it was a holy mess, with lots and lots of excess sqrt calculations, and I finally understood why vector algebra was such an advance. I threw my original code away, and replaced it with code that was one-tenth the size and about 1000x more perspicuous. BTW, the NASA Apollo program went through precisely the same sequence of events, before they developed a language incorporating vector algebra (e.g., "A double precision vector cross-product programmed in 'Interpretive' took about 5 milliseconds" on the AGC [Apollo Guidance Computer]"). Your daughter could have a lot of fun with the Postscript programming language -- freely available as "Ghostscript" -- for hacking 2D graphics. Basically, Postscript is Forth+graphics, and is a reverse polish interpreter remarkably similar to that used in the AGC. Unfortunately, I don't believe that either Forth or Postscript has built-in dot and cross products, although each of these routines is a one-liner. At 05:43 AM 11/8/2018, Cris Moore wrote:
I'm going to talk to my daughter's 8th grade class about conic sections.
I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples.
Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4.
Using Pythagoras' theorem produces an equation with a bunch of square roots.
Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2.
Am I doing something wrong?
Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to anotherÂ
I'm not sure how to justify this without a little calculus.
- Cris
PostScript has matrix operations, including concat (multiply) for 2D affine transformations. On 08-Nov-18 10:25, Henry Baker wrote:
My answer isn't going to please you, but here it is:
It's a war crime these days to torture kids by forcing them to do algebraic manipulations, when computers do this stuff so much better. Are we still forcing kids to do 5-digit multiplications & divisions by hand rather than using a pocket calculator [or their cellphone calculator] ?
Get Maxima, Mathematica or Maple and teach her that.
Re conic sections:
There are excellent reasons why vector algebra was invented by Yale Prof. J. Willard Gibbs (Gibbs's vector algebra decomposed quaternion products into dot products and cross products); it takes a *lot* of algebraic garbage out of manipulations like this. I'm talking about dot products, cross products, etc.
I recall the first time I had to do 2D graphics calculations, and I thought: I'm good at math, I aced all of my algebra classes, this should be a piece of cake; I don't have to read anything about computer graphics, because I know it all, already!
Well, I struggled with the code and eventually got it to work, but it was a holy mess, with lots and lots of excess sqrt calculations, and I finally understood why vector algebra was such an advance. I threw my original code away, and replaced it with code that was one-tenth the size and about 1000x more perspicuous.
BTW, the NASA Apollo program went through precisely the same sequence of events, before they developed a language incorporating vector algebra (e.g., "A double precision vector cross-product programmed in 'Interpretive' took about 5 milliseconds" on the AGC [Apollo Guidance Computer]").
Your daughter could have a lot of fun with the Postscript programming language -- freely available as "Ghostscript" -- for hacking 2D graphics. Basically, Postscript is Forth+graphics, and is a reverse polish interpreter remarkably similar to that used in the AGC. Unfortunately, I don't believe that either Forth or Postscript has built-in dot and cross products, although each of these routines is a one-liner.
At 05:43 AM 11/8/2018, Cris Moore wrote:
I'm going to talk to my daughter's 8th grade class about conic sections.
I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples.
Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4.
Using Pythagoras' theorem produces an equation with a bunch of square roots.
Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2.
Am I doing something wrong?
Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to another…
I'm not sure how to justify this without a little calculus.
- Cris
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes, it does have these operations, but they aren't "general", but have to do specifically with transforming the current coordinate system. (Yes, one could save the coordinate system, utilize these operations, and then restore the coordinate system -- which might be a good way to implement a more general transform, but that more general transform isn't "built in", although it is also a one-liner.) Notice also, that the 3x3 matrix ops in Postscript aren't general matrix ops, but are to implement *homogeneous* representations: the triple (x,y,z) represents the 2D point (x/z,y/z). So Cris also needs to teach his daughter homogeneous representations! BTW, lots of kids in the early 1980's were already hacking Apple ]['s and assembly language, so 8th grade is NOT too early to start learning computer graphics. At 08:01 AM 11/8/2018, Mike Speciner wrote:
PostScript has matrix operations, including concat (multiply) for 2D affine transformations.
These comments are all very helpful… sort of :-) My goal is to show students the unity between algebra and geometry. I don’t mind doing some algebra on the board, and it’s not _that_ hard. They know Pythagoras’ theorem, but they haven’t really used it to go back and forth between algebra and circles or ellipses. They know what a parabola is, and they’ve heard that parabolic dishes collect incoming rays, but they don’t know quite what this means, or why it might be true. And they haven’t seen the physical way of drawing an ellipse with a string connected to two thumbtacks... And I have one class period to do this. - Cris
On Nov 8, 2018, at 9:26 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Yes, it does have these operations, but they aren't "general", but have to do specifically with transforming the current coordinate system.
(Yes, one could save the coordinate system, utilize these operations, and then restore the coordinate system -- which might be a good way to implement a more general transform, but that more general transform isn't "built in", although it is also a one-liner.)
Notice also, that the 3x3 matrix ops in Postscript aren't general matrix ops, but are to implement *homogeneous* representations:
the triple (x,y,z) represents the 2D point (x/z,y/z).
So Cris also needs to teach his daughter homogeneous representations!
BTW, lots of kids in the early 1980's were already hacking Apple ]['s and assembly language, so 8th grade is NOT too early to start learning computer graphics.
At 08:01 AM 11/8/2018, Mike Speciner wrote:
PostScript has matrix operations, including concat (multiply) for 2D affine transformations.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Actually, using the focus-directrix definition of a parabola, and the fastest-path property of light, it's easy to see that parabolic dishes focus light at the focus. (Fastest path defaults to shortest path when there's only one medium; refraction happens with different media.) And the focus to focus reflection of ellipses is easy to see when using the constant sum of distances to foci property. And, by the way, the 3-D proof of the properties of conic sections (with tangent spheres to cone and plane cutting cone) is really quite beautiful if they don't know it. --ms On 08-Nov-18 11:39, Cris Moore wrote:
These comments are all very helpful… sort of :-)
My goal is to show students the unity between algebra and geometry. I don’t mind doing some algebra on the board, and it’s not _that_ hard. They know Pythagoras’ theorem, but they haven’t really used it to go back and forth between algebra and circles or ellipses. They know what a parabola is, and they’ve heard that parabolic dishes collect incoming rays, but they don’t know quite what this means, or why it might be true. And they haven’t seen the physical way of drawing an ellipse with a string connected to two thumbtacks...
And I have one class period to do this.
- Cris
On Nov 8, 2018, at 9:26 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Yes, it does have these operations, but they aren't "general", but have to do specifically with transforming the current coordinate system.
(Yes, one could save the coordinate system, utilize these operations, and then restore the coordinate system -- which might be a good way to implement a more general transform, but that more general transform isn't "built in", although it is also a one-liner.)
Notice also, that the 3x3 matrix ops in Postscript aren't general matrix ops, but are to implement *homogeneous* representations:
the triple (x,y,z) represents the 2D point (x/z,y/z).
So Cris also needs to teach his daughter homogeneous representations!
BTW, lots of kids in the early 1980's were already hacking Apple ]['s and assembly language, so 8th grade is NOT too early to start learning computer graphics.
At 08:01 AM 11/8/2018, Mike Speciner wrote:
PostScript has matrix operations, including concat (multiply) for 2D affine transformations.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes - with a focus at (0,1) and a directrix on the x axis, the parabola is y = (x^2+1)/2, nice and simple. And I can certainly sketch the reflection property. And I agree that it’s cool to take the cone z^2=x^2+y^2 and substitute various linear functions of x and y into z to find the curves in various slices! Cris
On Nov 8, 2018, at 10:06 AM, Mike Speciner <ms@alum.mit.edu> wrote:
Actually, using the focus-directrix definition of a parabola, and the fastest-path property of light, it's easy to see that parabolic dishes focus light at the focus. (Fastest path defaults to shortest path when there's only one medium; refraction happens with different media.)
And the focus to focus reflection of ellipses is easy to see when using the constant sum of distances to foci property.
And, by the way, the 3-D proof of the properties of conic sections (with tangent spheres to cone and plane cutting cone) is really quite beautiful if they don't know it.
--ms
On 08-Nov-18 11:39, Cris Moore wrote:
These comments are all very helpful… sort of :-)
My goal is to show students the unity between algebra and geometry. I don’t mind doing some algebra on the board, and it’s not _that_ hard. They know Pythagoras’ theorem, but they haven’t really used it to go back and forth between algebra and circles or ellipses. They know what a parabola is, and they’ve heard that parabolic dishes collect incoming rays, but they don’t know quite what this means, or why it might be true. And they haven’t seen the physical way of drawing an ellipse with a string connected to two thumbtacks...
And I have one class period to do this.
- Cris
On Nov 8, 2018, at 9:26 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Yes, it does have these operations, but they aren't "general", but have to do specifically with transforming the current coordinate system.
(Yes, one could save the coordinate system, utilize these operations, and then restore the coordinate system -- which might be a good way to implement a more general transform, but that more general transform isn't "built in", although it is also a one-liner.)
Notice also, that the 3x3 matrix ops in Postscript aren't general matrix ops, but are to implement *homogeneous* representations:
the triple (x,y,z) represents the 2D point (x/z,y/z).
So Cris also needs to teach his daughter homogeneous representations!
BTW, lots of kids in the early 1980's were already hacking Apple ]['s and assembly language, so 8th grade is NOT too early to start learning computer graphics.
At 08:01 AM 11/8/2018, Mike Speciner wrote:
PostScript has matrix operations, including concat (multiply) for 2D affine transformations.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute I can think of nothing more dangerous, more divisive, or more self-destructive than the effort to prey on what is called 'white backlash'... it is dangerous because it threatens to vest power in the hands of second-rate men whose only qualification is their ability to pander to other men's fears. I think it divides this nation at a very critical time—and therefore it weakens us as a united country. — Lyndon Johnson, 1966
On Nov 8, 2018, at 10:06 AM, Mike Speciner <ms@alum.mit.edu> wrote:
And the focus to focus reflection of ellipses is easy to see when using the constant sum of distances to foci property.
I guess I need to sketch a proof that at each point x on the ellipse, the curve is perpendicular to the bisector of the lines connecting x to the two foci. C
One of my favourite Useful Coincidences in Mathematics is that a paraboloid: (a) has the reflector property; (b) is the shape assumed by the surface of rotating liquid; which together enable the construction of 'liquid mirror telescopes': https://cp4space.wordpress.com/2014/05/28/liquid-mirror-telescopes/ ^^ this also has a calculus-free proof of the reflector property. Best wishes, Adam P. Goucher
Sent: Thursday, November 08, 2018 at 5:06 PM From: "Mike Speciner" <ms@alum.mit.edu> To: "Cris Moore" <moore@santafe.edu>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] conic sections
Actually, using the focus-directrix definition of a parabola, and the fastest-path property of light, it's easy to see that parabolic dishes focus light at the focus. (Fastest path defaults to shortest path when there's only one medium; refraction happens with different media.)
And the focus to focus reflection of ellipses is easy to see when using the constant sum of distances to foci property.
And, by the way, the 3-D proof of the properties of conic sections (with tangent spheres to cone and plane cutting cone) is really quite beautiful if they don't know it.
--ms
On 08-Nov-18 11:39, Cris Moore wrote:
These comments are all very helpful… sort of :-)
My goal is to show students the unity between algebra and geometry. I don’t mind doing some algebra on the board, and it’s not _that_ hard. They know Pythagoras’ theorem, but they haven’t really used it to go back and forth between algebra and circles or ellipses. They know what a parabola is, and they’ve heard that parabolic dishes collect incoming rays, but they don’t know quite what this means, or why it might be true. And they haven’t seen the physical way of drawing an ellipse with a string connected to two thumbtacks...
And I have one class period to do this.
- Cris
On Nov 8, 2018, at 9:26 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Yes, it does have these operations, but they aren't "general", but have to do specifically with transforming the current coordinate system.
(Yes, one could save the coordinate system, utilize these operations, and then restore the coordinate system -- which might be a good way to implement a more general transform, but that more general transform isn't "built in", although it is also a one-liner.)
Notice also, that the 3x3 matrix ops in Postscript aren't general matrix ops, but are to implement *homogeneous* representations:
the triple (x,y,z) represents the 2D point (x/z,y/z).
So Cris also needs to teach his daughter homogeneous representations!
BTW, lots of kids in the early 1980's were already hacking Apple ]['s and assembly language, so 8th grade is NOT too early to start learning computer graphics.
At 08:01 AM 11/8/2018, Mike Speciner wrote:
PostScript has matrix operations, including concat (multiply) for 2D affine transformations.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes they are general. You can use them for either the CTM (current transformation matrix) or with an arbitrary matrix: concatmatrix does a matrix multiply with two given matrices, pretty much all the other matrix operators take an optional matrix as final argument, using CTM otherwise. And yes, the matrices are actually 6 element arrays, [a b c d e f], interpreted as 3x3 matrices representing 2D affine transformations in homogeneous coordinates [a b 0] [c d 0] [e f 1] with points x y being row vectors [x y 1] so x y [a b c d e f] transform -> ax+cy+e bx+dy+f --ms On 08-Nov-18 11:26, Henry Baker wrote:
Yes, it does have these operations, but they aren't "general", but have to do specifically with transforming the current coordinate system.
(Yes, one could save the coordinate system, utilize these operations, and then restore the coordinate system -- which might be a good way to implement a more general transform, but that more general transform isn't "built in", although it is also a one-liner.)
Notice also, that the 3x3 matrix ops in Postscript aren't general matrix ops, but are to implement *homogeneous* representations:
the triple (x,y,z) represents the 2D point (x/z,y/z).
So Cris also needs to teach his daughter homogeneous representations!
BTW, lots of kids in the early 1980's were already hacking Apple ]['s and assembly language, so 8th grade is NOT too early to start learning computer graphics.
At 08:01 AM 11/8/2018, Mike Speciner wrote:
PostScript has matrix operations, including concat (multiply) for 2D affine transformations.
Some long time ago, I worked out for fun the reverse problem: given a general quadratic Axx + Bxy + Cyy + Dx + Ey + F, find foci, directrices, and eccentricity. It turned out to be somewhat interesting... 30 Aug 1982 focus u,v directrix px + qy + r = 0 eccentricity g (x-u)^2 + (y-v)^2 = g^2 (px+qy+r)^2 / (p^2+q^2) = (Px+Qy+R)^2 where P := gp(p^2+q^2)^-.5 Q := gq(p^2+q^2)^-.5 R := gr(p^2+q^2)^-.5 x^2-2ux+u^2+y^2-2vy+v^2 = P^2 x^2 + 2PQxy + Q^2 y^2 + 2PRx + 2QRy + R^2 x^2(1-P^2) + xy(-2PQ) + y^2(1-Q^2) + (-2u-2PR)x + (-2v-2QR)y + (u^2+v^2-R^2) = 0 a b c d e f A/k B/k C/k D/k E/k F/k a+c = 2-(P^2+Q^2) = 2-g^2 Note: g = 0 <=> P=Q=0 b^2-4ac = 4 P^2 Q^2 - 4 - 4 P^2 Q^2 + 4 P^2 + 4 Q^2 = 4g^2 - 4 Case 0: B^2-4AC = 0 [Parabola] g^2 = 1 a+c = 1 A+C = k k = A+C [but if 0, then A=B=C=0, so Dx+Ey+F = 0, giving line unless D=E=0, in which case plane if F = 0, else nothing] <a,b,c,d,e,f> = <A,B,C,D,E,F>/k Case 1: B^2-4AC <> 0 S := (A+C)^2/(B^2-4AC) + 1 = (B^2+(A-C)^2)/(B^2-4AC) = (b^2+(a-c)^2)/(b^2-4ac) = g^4 / (4g^2-4) g^2 = 2S +- 2(S^2-S)^.5 = 2(S +- (S(S-1))^.5) 2-g^2 = -2(S-1 +- (S(S-1))^.5) with following choice of sign: B^2-4AC < 0 [g^2 < 1] [Ellipse] S < 0 Since g^2 >= 0, must use + B^2-4AC > 0 [g^2 > 1] [Hyperbola] S > 1 Now, if A+C = 0, S(S-1)=0 so sign is irrelevant. Otherwise (Case a below), sgn(k) = sgn(A+C) sgn(2-g^2), while (from Case 1 below) sgn(k) = sgn((B^2-4AC)F+AE^2+CD^2-BDE), so sgn(2-g^2) = sgn(A+C) sgn((B^2-4AC)F+AE^2+CD^2-BDE). So, must use same sign as -(A+C)[(B^2-4AC)F+AE^2+CD^2-BDE] Case a: A+C <> 0 k = (A+C)/(a+c) = (A+C)/(2-g^2) <a,b,c,d,e,f> = <A,B,C,D,E,F>/k Case b: A+C = 0 0 = a+c = 2-g^2, so g^2 = 2 and b^2-4ac = 4g^2-4 = 4 k^2 = (B^2-4AC)/(b^2-4ac) = (B^2-4AC)/4 k = +-(B^2-4AC)^.5 / 2 By Case 1 below, must use same sign as [(B^2-4AC)F+AE^2+CD^2-BDE] <a,b,c,d,e,f> = <A,B,C,D,E,F>/k P = (1-a)^.5 Q = -b/(2P) = -b/(2(1-a)^.5) if P <> 0 = (1-c)^.5 if P = 0 u = -d/2-PR v = -e/2-QR f = u^2 + v^2 - R^2, while P^2+Q^2 = g^2, so (g^2-1)R^2 + (dP+eQ)R + (d^2+e^2)/4 - f = 0 Case 0: R = (f - (d^2+e^2)/4) / (dP+eQ) Case 1: R = (-(dP+eQ)+-((dP+eQ)^2-(g^2-1)(d^2+e^2-4f))^.5)/(2(g^2-1)) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Note: Radicand = ae^2 + cd^2 + (b^2-4ac)f - bde Note: | a b/2 d/2 | det := | b/2 c e/2 | = -(ae^2 + cd^2 + (b^2-4ac)f - bde)/4 | d/2 e/2 f | This determinant is invariant under skew, rotation, reflection, and translation, as can be seen using homogeneous coordinates: [ a b/2 d/2 ] [x] [x y 1] | b/2 c e/2 | |y| = ax^2 + bxy + cy^2 + dx + ey + f [ d/2 e/2 f ] [1] Funny Cases: Ellipse: det = 0: single point, given by "Center" det > 0: empty Parabola: det = 0: A=B=C=0: D=E=0: F=0: plane F<>0: empty else: single line (Dx+Ey+F = 0) else [k = A+C] (mx+ny+o)^2+p, where m = a^.5, n = c^.5, if a<>0, o = d/2m else c<>0 and o = e/2n p = f-o^2 [f-d^2/4a or f-e^2/4c] p>0: empty p=0: double single line (mx+ny+o = 0) p<0: parallel lines (mx+ny+o+-(-p)^.5) Hyperbola: det = 0: pair of crossing lines, given by "Asymptotes" Finding the Center: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 d/dx = 2Ax + By + D = 0 d/dy = Bx + 2Cy + E = 0 *******Solvable only in Case 1: [2A B] [s] = [D] [s] = ____1____ [2C -B] [D] = ____1____ [2CD-BE] [B 2C] [t] [E] [t] B^2 - 4AC [-B 2A] [E] B^2 - 4AC [2AE-BD] Asymptotes of Hyperbola: (A(x-s)+(B+-((B^2-4AC)^.5)/2)(y-t))(((B+-(B^2-4AC)^.5)/2)(x-s)+C(y-t)) where we choose signs to be + if B >= 0, - otherwise, to ensure nonzeroness. On 08-Nov-18 08:43, Cris Moore wrote:
I’m going to talk to my daughter’s 8th grade class about conic sections. I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples. Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4. Using Pythagoras’ theorem produces an equation with a bunch of square roots. Squaring both sides eventually turns this into
3x^2 + 4y^2 = 12
but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation
xy = 2.
Am I doing something wrong? Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to another… I’m not sure how to justify this without a little calculus.
- Cris _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
You should definitely show them the "Dandelion" (actually, Dandelin!) Theorem, which I call the *ice cream cone* theorem. It proves the sum-of-the-distances theorem about ellipses w/o requiring analytic geometry. When my plane geometry teacher showed it to us, it completely blew my mind! In particular, I couldn't get over the fact that the ellipse was symmetrical, even though the spheres were of different sizes! Even if the kids haven't yet had 3D geometry, they can still follow this easy theorem. https://en.wikipedia.org/wiki/Dandelin_spheres At 05:43 AM 11/8/2018, Cris Moore wrote:
I'm going to talk to my daughter's 8th grade class about conic sections.
I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples.
Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4.
Using Pythagoras' theorem produces an equation with a bunch of square roots.
Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2.
Am I doing something wrong?
Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to anotherÂ
I'm not sure how to justify this without a little calculus.
- Cris
You could come back to your daughter's class in a couple of years and present Tristan Needham's very cool demonstration that the focus-centered elliptical Newtonian orbit is merely the (complex) "square" of an origin-centered simple harmonic motion ellipse. z = p*exp(i*t)+q*exp(-i*t), p>q>0 real, foci at +-2*sqrt(p*q). (z = a*cos(t)+i*b*sin(t), a=p+q, b=p-q.) z -> z^2 = (p*exp(i*t)+q*exp(-i*t))^2 = p^2*exp(i*2*t)+q^2*exp(-i*2*t)+2*p*q, foci at 0, 4*p*q. "Bohlin's Theorem". Newton and the Transmutation of Force, Tristan Needham The American Mathematical Monthly, Vol. 100, No. 2. (Feb., 1993), pp. 119-137. 484KB. http://users.uoa.gr/~pjioannou/mech1/READING/newton20force.pdf This work is a portion of a larger book: Visual Complex Analysis, 612 pages. 9.2MB. http://umv.science.upjs.sk/hutnik/NeedhamVCA.pdf The laws of planetary motion, derived from those of a harmonic oscillator (following Arnold) 472KB. https://arxiv.org/pdf/1404.2265.pdf At 10:35 AM 11/8/2018, Henry Baker wrote:
You should definitely show them the "Dandelion" (actually, Dandelin!) Theorem, which I call the *ice cream cone* theorem. It proves the sum-of-the-distances theorem about ellipses w/o requiring analytic geometry.
When my plane geometry teacher showed it to us, it completely blew my mind!
In particular, I couldn't get over the fact that the ellipse was symmetrical, even though the spheres were of different sizes!
Even if the kids haven't yet had 3D geometry, they can still follow this easy theorem.
https://en.wikipedia.org/wiki/Dandelin_spheres
At 05:43 AM 11/8/2018, Cris Moore wrote:
I'm going to talk to my daughter's 8th grade class about conic sections.
I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples.
Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4.
Using Pythagoras' theorem produces an equation with a bunch of square roots.
Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2.
Am I doing something wrong?
Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to anotherÂ
I'm not sure how to justify this without a little calculus.
- Cris
I really like Dan’s question about playing billiards with a hyperbola rather than an ellipse as the curve that the balls bounce off of. My offhand guess would be that when a ball goes off to infinity along a ray, you should draw the line that contains that ray and imagine the ball coming in from infinity along that line from the other side, until it hits the curve, at which point the usual angle-of-reflection rule applies. If we do this, does a billiard shot from one focus of a hyperbola hit the other focus (after passing through infinity)? Jim Propp On Friday, November 9, 2018, Henry Baker <hbaker1@pipeline.com> wrote:
You could come back to your daughter's class in a couple of years and present Tristan Needham's very cool demonstration that the focus-centered elliptical Newtonian orbit is merely the (complex) "square" of an origin-centered simple harmonic motion ellipse.
z = p*exp(i*t)+q*exp(-i*t), p>q>0 real, foci at +-2*sqrt(p*q).
(z = a*cos(t)+i*b*sin(t), a=p+q, b=p-q.)
z -> z^2 = (p*exp(i*t)+q*exp(-i*t))^2 = p^2*exp(i*2*t)+q^2*exp(-i*2*t)+2*p*q, foci at 0, 4*p*q.
"Bohlin's Theorem".
Newton and the Transmutation of Force, Tristan Needham The American Mathematical Monthly, Vol. 100, No. 2. (Feb., 1993), pp. 119-137. 484KB.
http://users.uoa.gr/~pjioannou/mech1/READING/newton20force.pdf
This work is a portion of a larger book:
Visual Complex Analysis, 612 pages. 9.2MB.
http://umv.science.upjs.sk/hutnik/NeedhamVCA.pdf
The laws of planetary motion, derived from those of a harmonic oscillator (following Arnold) 472KB.
https://arxiv.org/pdf/1404.2265.pdf
At 10:35 AM 11/8/2018, Henry Baker wrote:
You should definitely show them the "Dandelion" (actually, Dandelin!) Theorem, which I call the *ice cream cone* theorem. It proves the sum-of-the-distances theorem about ellipses w/o requiring analytic geometry.
When my plane geometry teacher showed it to us, it completely blew my mind!
In particular, I couldn't get over the fact that the ellipse was symmetrical, even though the spheres were of different sizes!
Even if the kids haven't yet had 3D geometry, they can still follow this easy theorem.
https://en.wikipedia.org/wiki/Dandelin_spheres
At 05:43 AM 11/8/2018, Cris Moore wrote:
I'm going to talk to my daughter's 8th grade class about conic sections.
I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples.
Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4.
Using Pythagoras' theorem produces an equation with a bunch of square roots.
Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2.
Am I doing something wrong?
Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to another…
I'm not sure how to justify this without a little calculus.
- Cris
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participants (5)
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Adam P. Goucher -
Cris Moore -
Henry Baker -
James Propp -
Mike Speciner