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
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