Conics and my state of mind

Every two months or so, I go kinda crazy--just a little, though. For about five days, I am restless, pouty, petty, dramatic, anxious, confused--all sorts of nasty things. An additional side-effect is that I can believe anything. I won't know waking reality from dreams; I won't know waking reality from things I read; I won't know waking reality from what I want to believe. Added to this, I have a mild fever that's been off and on for about a week.

For anyone who has an idea the state my mind's normally in, you may have an extra appreciation of the mess this stirs up.


Teaching

We've just "finished" teaching Conic Sections to the kids. I'm not sure what he taught, I was marking their papers so they'd know how they're doing before the exam comes.

In S4 Pre-Calculus, we made a flow chart for identifying types of conic sections. I want the students to have it, but I'm not sure what the teacher has in mind.

That being said...

Because Saturday, December 01 is my fitness test with the Army, I might miss part of Math school that day. That's a review class, and I'd be sorry to miss it. One student is even writing the exam that day because he won't be there next week.

Upcoming Saturdays:
December 01

09:00 - Army fitness test at base. Eep.

14:00 - Review class before exam. One student writing exam early.

December 08

14:00 - Exam on Trig and Conic Sections.

December 15

14:00 - Teacher is away and has not given me any specifics on what to do that day, so I have full control of what we do that day! I want to actually enrich* the students' understanding of Mathematics--because we're supposed to be an "enrichment program".

17:30 - Math School Christmas Dinner. Woots.

* Here is what I want to talk about on December 15...

Conic Sections

History

Way, way back, circa 200 BC, there was a Greek named Apollonius, and he wrote a book called On Conics. This earned him the title, "The Great Geometer". The study of conics has been around for a long time! (Will add more later.)

What are they?

Imagine two hollow cones placed together at their points, sort-of like an hour-glass. By cutting different 2D sections of this, you get "Conic Sections".

Taking a slice of a cone, parallel to an edge gives a parabola.
Taking a slice of a cone at an angle such that you slice through both halves, gives a hyperbola.
Taking a horizontal slice gives either a circle or a single point (if you cut at the joining point).
Tilting that circular slice gives an ellipse.

Definitions

A circle is the set of all points** equidistant from a single point. To draw one, wrap a loop of string around a pin and a pencil and draw as far from the pin as possible without tilting the pencil.


An ellipse is the set of all points** whose distance from both foci is constant (ie: the distance from one point to the first focus plus the distance from the same point to the other focus always adds up to the same number). To draw one, wrap a loop of string around two pins and a pencil and draw as far from the pins as possible.

A parabola is defined as the set of all points** equidistant from a line and a point F (the focus) not on the line (see the right-hand side of this image).

A hyperbola is the set of all points whose distance from one focus, minus the distance to the other focus, is constant.

**(in a plane).


Gravity

The path of a projectile thrown (ie: with another, smaller force in a perpendicular direction) near the surface of the Earth is a parabola.

The path the Earth travels around the Sun is an ellipse.

The path of an object (such as a rocket or comet) on an escape trajectory from a fixed mass (such as the Sun) is a hyperbola.


Reflection

The parabola, ellipse and hyperbola each have "focus points" or "foci". If you've ever seen a satellite dish, you have an idea what this means.

For parabolas, an incoming ray that is parallel to the axis of symmetry is reflected toward the focus.

For ellipses, any ray originating at one focus will reflect toward the other focus (this is still true for the special case of the circle, where the "other" focus is the same focus).

Hyperbolas are a bit more complicated. A ray originating from one focus will be reflected and look as though it originated from the other focus (see this image). (There are a few other cases that work out nicely, that I just can't remember right now.)


Light Cone
...Actually, maybe they won't be ready for this yet. But I'll mention it--briefly.


...I should take that book out of the library again...
--Charissa


Further reading:
http://fti.neep.wisc.edu/~jfs/neep602.lecture8.trajectories.97/neep602.lecture8.trajectories.97.html

"Spacecraft Trajectories"

http://mathworld.wolfram.com/Parabola.html

Mathworld: Parabola

http://mathworld.wolfram.com/Ellipse.html

Mathworld: Ellipse

http://mathworld.wolfram.com/Hyperbola.html

Mathworld: Hyperbola

http://www.practicalphysics.org/go/Experiment_386.html;jsessionid=alZLdQlAHb1

"Drawing" a parabola; teaching aid

Some Writing!

http://blog.wolfram.com/2007/05/the_mathematica_behind_televis.html

In my previous post, I linked to Mathworld because it's awesome. Well, I'm linking it again because it's awesome, AND has ties to the television series NUMB3RS, which is totally hoorays!

Also, I thought I would post this snippet from my collection of "smutty Math stories" that I write from time to time. This one is old, but I wuvs it dearly:

Short fiction: Models (Incomplete)

Oh, you don't need any help now, do you?, he asks mockingly. He's leaning low by my side; his face is close to mine, gazing across to the quickly scrawled initial value problem. I finger the corner of the paper, thinking distractedly. I can feel his warmth on my skin. He's closer than I thought, so I don't turn my head to reply.

I've got it, I tell him dismissingly, I've got it... In nervous habit, I press my lower lip to my teeth, biting my lip.

We've been here at least two hours, exchanging questions and explanations and waxing philosophical with our esoteric dialogue. We're indulging in our two greatest pleasures at the same time, and as an extra treat, I'm nibbling some dark chocolate.

He rights himself and walks across the kitchen. He's probably stretching himself out and getting snacks; I can hear him yawn.

I hear him open the fridge and see him, in my mind's eye, for I'll not indignify myself by staring. I wonder what he'll get. He'll always find a way to pleasantly surprise me. At least, until I can recognize his pattern.

I used to intimidate people with the way I predicted them. If I wanted a favour done, I wouldn't ask, but tell the person, "You're going to get me a large coffee from the cafeteria if I give you three dollars." Usually, the first reactions are to take offense and rebel against my suggestion, but I wouldn't have said it if I thought it would not work, and, with little effort on my part, she'll bring me my coffee, and my $1.50 change.

Wilbert was the first--and only--person I could not accurately predict. The first favour I told him was in Math class. "You'll bring my exercise book?" He looked at me as though trying to understand my own pattern, and then went off to fetch it, with a slight smile. But when he returned with the book, he did not give it to me. Instead, he stayed standing at my desk, reading the exercise book casually.

"'Exercise fifty-eight," he said, "question one."

He had a soft voice; soothing, really, but simultaneously evocative.

"'Find an acute angle x such that tan of ninety minus x equals cotangent of eight plus one-third x.' Hm, now that looks like a fun question. Let's see. Tan is sine over cos, and cotangent is cos over sin. Cross multiplying gets the sines and cosines on different sides, but if they're moved to one side, it makes a cos double-angle identity, and from there..."

And I just stared at him.

His mind was so analytical; always calculating--but in a haphazard and chaotic way, which still let him appear calm and intense. Finding his patterns would be the most captivating and satisfying task I'd set for myself yet. Not that I'm obsessed.

I have an unfinished model. It works, but it is flawed. It can predict 90% of his actions, but not thought process behind them. I think I can refine this model by the end of the month. It has taken too long already.

...to be finished at some later date.

...later
--Charissa

Quick Update

Origami

For my birthday, I received an Origam gift box and extra book from Mark! OMG! The gift box has printed paper (strange and elegant designs that remind me of wallpaper), gold and silver foil paper, a protractor, two 'bone' crease tools (one for making sharp creases, the other for "drawing" the crease patterns), a square board with grid and markings on it, and an origami instruction book. I'm only a little irritated that both books contain mostly diagrams that are reliant on cutting paper.

This is mostly an announcement that, as soon as I get my camera, I'll be posting picture of both the modular origami polyhedra and the new models--animals and flat geometric patterns. Man, I can't wait to get that camera!!


Teaching Math

I've been in correspondence with one of the students who had questions about exponents/roots, and after the two weeks between classes when I was helping her with that, she wrote back to say I'm a good explainer! That totally made my day!

We started conic sections on Saturday with the Circle and Ellipse. At the end of class, the teacher wrote this on the board:

"Next week: line, parabola and hyperbola."

The students had mostly never heard of hyperbola, and some never heard of parabola. The teacher said to look it up on the Internet, and I wanted to jump up and say, "MATHWORLD DOT WOLFRAM!!" but of course, that would have been inappropriate, so I didn't. Sigh.


Film Festival

One of my friends has a film in a Canadian film festival, and his film made it as far as a public showing! I'm so excited! I'm going to the showing, hopefully with a friend who is/was a major artistic-film-maker-type-person. In truth, I wonder if she'll like my friend's film, but I invited her mostly for the other films. Heh.


Army

My unit did a field exercise this past weekend. They slept outside with little form of heating for two nights. Yowza! I went on the last day to help with breakfast and tear-down. Man, I suck at moving things. I definitely need to work out. But with the army, I get a personal trainer and access to the gyms, so I'll have to make use of that. Sheesh, though, I wish I had more free time...


Art

I'm donating Boobfest III to the queer student group at the university I used to go to. At least, I'm donating a print--and I think I'll take it back at the end of the school year.

Also, that student group is planning on booking a small art gallery in one of the centres at the university--there's this glass room beside a major traffic junction and across from a Tim Horton's / food court, I think, which is totally a great place to put an art gallery.

If we end up booking it, I'll totally submit: my nude pieces; the Origami I'm proud of (goldfish in the bowl!!), including an enclosed chain of connected coloured boxes that are ordered to make a rainbow (sort of; I'm missing a real *purple* but I have this dull mauve thing); and any other artwork I deem public-worthy--which is not much, so I'll have to get busy making more, dammit! Mmm, I'm excited!


...Will update this entry with links and text formatting later.
--Charissa

Novemberance!

As a kid, I used to call today "Novemberance Day" and even wrote a terrible poem about it, once. I dunno, I never had a particular attachment to Remembrance Day. In school, one of my good friends, Diane, had/lost family in the World War/s, I can't quite recall.

But as a member of the Canadian Forces, I was at one of the many Remembrance Day ceremonies across the city. Of course, I have no training as of yet, so I was, at most, a spectator standing with some higher-ranked personnel who told us untrained Privates when to stand at attention, remove our headdress and so-on.

A few things were prominent today...

I know it shouldn't be funny--especially not today--but it can be rather hilarious when you actually see soldiers--mostly Cadets--pass out while standing on the parade ground. I was told that it's quite common, especially with the Cadets, for soldiers to get dizzy and pass out from standing for about two hours--especially the Cadets, who are often so nervous that they "thunder"; fall, stiff as a board, flat on their faces. One guy from our unit even fell on his face; and then began crawling about as though uncertain of where he was. I was told he lost a contact lens in the process.

My "boss" is disgusting. Really, she is. She's short, skinny, butch, loud, irritable, vulgar, and disgustingly, disgustingly hetero. I really did not want to hear her talk about condoms nor "sword-fighting" in her mouth. She reminds me of a girl I knew in high-school, except she's an adult. I want to believe that she's just disgusting when she's in uniform, it's her coping mechanism or something, but she makes me want to vomit--I can't believe she's a soldier! Does she have dignity? IT SURE DOES NOT SEEM SO!

I do not have to adopt a new lifestyle. Here is (sort-of) a summary of my thinking:

- I'm in the Army, hoorays! I'll have discipline!

- Oh, I'd better watch and learn from observation!

- Gee, this is a lifestyle thing!

- I'd better bring Army elements into my Civi lifestyle!

- Son-of-a-bitch, I'm getting a soldier's mouth!

- Hey, my "co-worker" has morals and refuses to profane!

- That is freaking awesome!

- I'm not going to profane! Nor drink--the way they do, at least.

- I don't have to be like anyone else! I'm special!

So, hoorays!

Remembrance Day is important! This I already knew! It just feels... more ceremonial--of course--now that I'm in the CF.

The Remembrance Day ceremony was Christian! Some of our speakers were very... "We are fighting Evil!" which made me a little uncomfortable. We fight Evil, yes, but it is not... We don't... We aren't to be "Holy Warriors", I think. I think we should never be "Holy Warriors". ... Sheesh, this is difficult to put into accurate words!!


In other news...

I found out something interesting the other day. I'm not yet going to say what, because it's not my story to tell; but it made me both excited and... Clingy. I mean, it's a bad thing to be excited over this. HONESTLY.

Also, at the Maths school, we gave out a rather fun-looking assignment. I do and don't look forward to marking these! They're all long-answer involving Trig Identities. Sigh. While it'll be exciting, it'll also be messy and headache-inducing, I think. Woots!

Also, I took out a book on Modular Origami from the library! I've already made a stellated icosahedron. Woots! I think, after I finish making some of the models in this book, I'll move onto this project! Bwahah!


That's all for now. I might update later with fun trig stuff.
--Charissa

Art and Stuff

I realized, the other day, that I'm not making as much art as I would like. And since Autumn passed me by uncaptured (by camera and canvass), I decided the next piece I do will be "epic" and about Autumn! Here's a "sketch" I've made of it in Flash MX.

(I still can't believe Adobe bought Flash... The version I use still says "Macromedia Flash MX" on it.)


New Loading Screen on Intro to Epsilons and Deltas

Also shown here.

Epsilons and Deltas is a little animation that I've been working on for about a year. It's a short story about Math and Love. Awwww. It also features two girls, a strip from xkcd, and La Campanella by Franz Liszt.

The old loading screen didn't have red-bowtie-girl running along the loading bar; she just stationary-ran beside the percent-loaded display, and it looked a little tacky. This one has a nicer "floaty" and "Ooh, numbers!" feel to it, which I don't mind at all! But, we'll see if I can even still improve it...


Other Stuff

My eldest brother went back to school in January, and has asked me to help him out with derivatives! I think I'll do another of those Flash tutorials like in the previous post.

He's also sending over a box of goodies from Germany! There'll be Lebkuchen, my favourite kind, with the candied orange peel inside and wafer on the bottom and either dusted lightly with powdered sugar or coated in chocolate! Mmmm! Also, a digital camera and computer gear!


Events

Nov 01

I'm Employee of the Month! Hoorays, I guess!

Nov 05 (today!)

Grocery shopping. Hells yeah.

Work at the Reserves tonight. Hells yeah...

Should call the bakery to pre-order special bagels.

Should drop by the library to return Dragons in the Water by Madeline l'Engle; and take out another Origami book and Math book.

Nov 06

Meet up with the teacher to lend him the entire S4 Pre-Calculus curriculum... I'm wary of lending that to anyone, and of all people, him...

Should swing by the bakery for more bagels... Mmm, pumpernickel bagels...

Also hoping to spend some good time with my best friend Eric!

Should also swing by the Rainbow Pride group at the university. I haven't been there in ages--mostly because I'm trying to avoid someone.

Nov 09

Kevin's birthday party, at last!

Nov 10

Homo Hop, a queer-oriented social. I'm not sure if I'll go... We'll see.

That's all for now. I should get some work done today.
--Charissa

018: Exponentiate!

Woo! Just a quick update to post this tutorial (Flash) on exponents/radicals. Check it out here, too!

Woots. Off to work now.
--Charissa

Issues

As much as I want to keep this from being angsty and personal, I sort-of want to write about my state-of-mind.

I've been so apologetic ever since getting the survey job; I haven't spoken fluently and with conviction, it seems, for ages. My sentences are full of "maybe"s and "I think"s and "I believe"; and "it might be"s; and more "like"s than ever before.

I wonder if it's mostly a self-confidence issue, or if it's something else?

I also wonder if doing more proofs will be helpful in gaining more self-confidence. Probably will!


Week so far:

Oct 29

I'm now 19. Hoorays, I guess. For my birthday, I paid $200 in Utilities.

Oct 30

Learned how to properly wear my Army uniform! Woo-hoo!

Had my first "class" in the Army (Dress and Deportment; and Rank Structure).

Oct 31

Started playing Brood War again. Sigh. What a wonderful way to waste time.

Received e-mail from the teacher, who said I should go over the Quiz in class with the students--WHICH IS OMG HOORAYS!!

O, Time...
--Charissa