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Author Topic: Geometry Problem  (Read 2660 times)

Offline Howard Rush

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Geometry Problem
« on: November 17, 2011, 03:34:22 PM »
I was trying to figure out the headwind component on an airplane at the outside upper corner of a square eight.  I am insufficiently smart to do this.  Anybody else want to do it?  Here are some simplifying assumptions:

Wind is uniform, and parallel to the ground.

Lines are straight, and the wing is aligned with them.

The airplane's path is parallel to the ground, and angle of attack is zero. 

The airplane is at 45 degrees elevation and at 45 degrees around the circle from downwind, as it would be just before starting a zero-radius corner. 

I think the pitch angle will be nonzero.  A wind component something like .707 Vw or (.707)^2 Vw or (.707)^3 Vw gets added vectorially to ground speed.
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Offline Tim Wescott

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Re: Geometry Problem
« Reply #1 on: November 17, 2011, 04:42:45 PM »
Why Howard, this is easy, if only you use quaternions*

Define the windward direction to be y and down to be z (use your favorite version of the right hand rule to find x  : http://xkcd.com/199/).  So the half-angle rotation vector of your personal pointing vector to the vector referenced by this projection is

r_g = cos(22.5) + k * sin(22.5)

These will be raised up by 45 degrees from the vertical, so in a reference frame that uses r_va as it's x direction, the half-angle rotation vector from you to the plane will be

r_u = cos(22.5) + j * cos(22.5)

To find the vector from one to one's plane, one need merely multiply r_g * r_u * i * r_u' * r_g' (using the well-known** rules of quaternion multiplication), to get

i / 2 - j / 2 - k / sqrt(2)

That's a nice check, and gives us the expected results (note that the first time I did this I inexplicable forgot the order of multiplication for the rotations -- and of course, since quaternions are not commutative, the answer to r_u * r_g * i * r_g' * r_u' is different).

Of course, the plane's direction of flight is in a frame of reference that's rotated 90 degrees from the lines:

r_p = cos(45) - k * sin(45)

(as a check, in level flight the plane's 'nose' vector is r_p * i * r_p' = -j -- that's exactly as expected).

So the airplane's pointing vector with respect to the earth is simply

r_g * r_u * r_p * i * r_p' * r_u' * r_g' = (-i - j)/sqrt(2)

Dang.  Well, that's a really good check, but you wanted the wind's pointing vector with respect to the plane, didn't you.  Oh well -- just do the rotations in the reverse direction and the reverse order, then, keeping in mind that the wind's vector is i:

r_p' * r_u' * r_g' * i * r_g * r_u * r_p = -i / sqrt(2) + j / 2 + k / 2.

And we know this is right, because we have used math, and math is never wrong!!

It's also intuitively about right: there's a good amount of wind against the plane's nose (-i), a good amount of wind blowing in the starboard direction across the plane (+j), and a good amount of wind down (+z), which is your "downward component".

There.  Another problem in 3D geometry solved easily and plainly by use of simple arithmetic.

* Note: sane people*** need not read this post.  However, if you are sane and you take the trouble to wrap your brain around it, you can read it again later in comfort, secure in the knowledge that you won't have any remaining sanity to lose!

** heh heh

*** "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." Lord Kelvin, 1892.  (from http://en.wikipedia.org/wiki/Quaternions#Quotes)

« Last Edit: November 17, 2011, 05:13:46 PM by Tim Wescott »
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Offline Howard Rush

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Re: Geometry Problem
« Reply #2 on: November 17, 2011, 05:06:37 PM »
Thanks, Tim.  I'll pore over this later, but it reminds me of some stories, as many things do to old men.  I undertook a study of quaternions a couple of years ago, but got sidetracked by having to go back to fill in other stuff I'd forgotten, and never got that filled in.  I do remember something about Hamilton figuring it out and writing it on a bridge on his way to the pub, because he knew he would be cognitively impaired later and would forget.  As for the right-hand rule, I remember stopping to talk to a teacher about something involving vectors, meanwhile mentally using my free hand to perform the right-hand rule to support the conversation.  Trouble is, I was holding some books in my right hand, so I was attempting to use my left hand for the right-hand rule. 
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Offline Tim Wescott

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Re: Geometry Problem
« Reply #3 on: November 17, 2011, 05:18:45 PM »
I wouldn't recommend a study of quaternions to anyone who isn't doing a lot of 3D rotations, and even then they're not to be embraced lightly.  I just know them because I had a two-year long project for a customer doing a bunch of inertial navigation stuff and while they certainly make the math hard, the arithmetic ends up being easy (and fast for the computer).  Your loved ones just have to accept that you're going to be wandering around the house mumbling to yourself and making strange gestures with the fingers of your right hand.

In fact, the reason I could do those calculations so fast is because I have what amounts to a four-function calculator for quaternions written for Scilab, as a consequence of the contract.

I guess I just need to thank my lucky stars every day that there are no practical applications for octonions.
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Offline Chris Wilson

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Re: Geometry Problem
« Reply #4 on: November 17, 2011, 05:30:07 PM »
What ............. the heck ................. is an 'octonion?'
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Offline Tim Wescott

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Re: Geometry Problem
« Reply #5 on: November 17, 2011, 06:25:49 PM »
What ............. the heck ................. is an 'octonion?'

As near as anyone can tell they're a cruel joke played upon the universe by the Gods of Mathematics, although they do have some use in leading edge (or perhaps bleeding edge) physics.

"There are exactly four normed division algebras: the real numbers, complex numbers, quaternions, and octonions. The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. "

http://math.ucr.edu/home/baez/octonions/node1.html
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Offline Chris Wilson

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Re: Geometry Problem
« Reply #6 on: November 17, 2011, 06:31:10 PM »
Ow, my head hurts!
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Offline Tim Wescott

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Re: Geometry Problem
« Reply #7 on: November 17, 2011, 06:52:50 PM »
Y'know, I really chose to use quaternions for this because they're about the most esoteric math that I'm actually facile with, and I figured it'd be a good way to tweak Howard.  (And if you have gone and inflicted brain damage upon yourself by learning how to use 'em, you may as well get some mileage out of the experience).

I didn't mean to cause any collateral damage.

Sorry.
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Offline phil c

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Re: Geometry Problem
« Reply #8 on: November 18, 2011, 04:17:42 PM »
Howard, you probably should figure on the corner being 60 deg. otherwise it won't be close to what everybody is flying and give you not so useful numbers.
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Offline Howard Rush

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Re: Geometry Problem
« Reply #9 on: November 19, 2011, 01:33:25 AM »
Howard, you probably should figure on the corner being 60 deg. otherwise it won't be close to what everybody is flying and give you not so useful numbers.

Maybe in your neighborhood.
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Offline Richard Entwhistle 823412

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Re: Geometry Problem
« Reply #10 on: November 23, 2011, 01:43:46 PM »
Howard

Perhaps your brain went into "Gimbal Lock".  I tried using Del but quickly came to the conclusion that I would have to go back school.

Richard
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Offline Howard Rush

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Re: Geometry Problem
« Reply #11 on: November 23, 2011, 03:29:37 PM »
Head up and locked.
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Offline Tim Wescott

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Re: Geometry Problem
« Reply #12 on: November 23, 2011, 04:37:18 PM »
Y'know, I really chose to use quaternions for this because they're about the most esoteric math that I'm actually facile with, and I figured it'd be a good way to tweak Howard.

And, since writing the above I've been trying to think of whether I know any less esoteric ways to arrive at the same answer; the only ones I know are somewhat less oddball (direction cosine matrices), but are only popular in applications where the number crunching isn't the prime mover for the product performance (3D graphics, for instance, use quaternions).  I don't know if there is anything that works and doesn't require at least some weird science; I know from investigating this that there isn't anything that requires fewer operations to do the computation.

If you had a handy globe, a felt pen, and a protractor you could do this good enough for a seat of the pants calculation, but that doesn't help if you want to get an answer for a whole bunch of arbitrary plane angles.

So from the "you only know it if you can teach it" standpoint, I'm not doing so well!
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Offline Mike Scholtes

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Re: Geometry Problem
« Reply #13 on: November 24, 2011, 06:30:05 PM »
I'm not even remotely qualified to understand the math but I am curious as to what makes Howard want an answer to this question, and also whether it makes any difference what maneuver we are talking about so long as the plane is oriented in the same direction relative to the (assumed constant and parallel) wind. Or is this just something that afflicts 64-year-olds, which I became one of recently? Should I be nervous?

Offline Howard Rush

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Re: Geometry Problem
« Reply #14 on: November 24, 2011, 07:19:56 PM »
I've been trying to figure out the relationship between handle input and control surface position.  I remember square eights being particularly difficult to get right in the wind, so I was going to calculate the situation at the upper, outer corners.
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Offline Tim Wescott

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Re: Geometry Problem
« Reply #15 on: November 24, 2011, 10:47:08 PM »
So to answer Mike's question, I guess that you could use this particular answer for any maneuver that put a square corner right there -- which is only the square eights...
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Offline Mike Scholtes

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Re: Geometry Problem
« Reply #16 on: November 24, 2011, 11:34:25 PM »
My brain is still trying to recover from Thanksgiving dinner, but if I get the drift of this, there might be some calculable reason to not center the maneuver downwind, but rather biased to one side or the other? I would have thought this was settled by empirical evidence from a zillion guys flying a zillion square 8's for the last 60 years. Still, I would be interested in what the guys who DO understand the math think about this.

Offline Brett Buck

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Re: Geometry Problem
« Reply #17 on: November 25, 2011, 12:49:37 PM »
I wouldn't recommend a study of quaternions to anyone who isn't doing a lot of 3D rotations, and even then they're not to be embraced lightly.  I just know them because I had a two-year long project for a customer doing a bunch of inertial navigation stuff and while they certainly make the math hard, the arithmetic ends up being easy (and fast for the computer).  Your loved ones just have to accept that you're going to be wandering around the house mumbling to yourself and making strange gestures with the fingers of your right hand.

     If you *are* doing any sort of generalized inertial navigation, or anything beyond a fairly trivial generalized rotation, quaternions are really the only way to go. And the math is far simpler than any of the alternatives. The other obvious solution, Euler angles, have so many disadvantages that they can only be used in limited situations or with extreme complexity to deal with the singularities. As long as you understand the concept (even without the math) of the eigenvector, quaternions are simple to grasp. For some cases you can easily use the definition directly to calculate your rotation direction and magnitude. The only significant difficulty comes in how it is implemented - you can have it defined relative to the destination frame, the source frame (the correct Lockheed way), and you can have the magnitude component first (wrong, even though that's how the NASA paper does it) or the last (the correct Lockheed way). Even then, as long as you understand who chose what, it's trivial to convert one type to the other.

    I offered Howard some some of my references on the topic, but I couldn't give him my best stuff because it was, arguably, proprietary. And at the time (and to this day) we are severely restricted and scrutinized with any contact we have with Boeing or ex-Boeing employees. Due to the some legal unpleasantness arising from the booster world, and even more egregious issues with FIA (and not the one in France). The latter did eventually work out and the Govt. backed up tractor-trailers full of money to our doors as a result.

     Brett

Offline Tim Wescott

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Re: Geometry Problem
« Reply #18 on: November 25, 2011, 01:28:53 PM »
     If you *are* doing any sort of generalized inertial navigation, or anything beyond a fairly trivial generalized rotation, quaternions are really the only way to go. And the math is far simpler than any of the alternatives. The other obvious solution, Euler angles, have so many disadvantages that they can only be used in limited situations or with extreme complexity to deal with the singularities.

That's certainly the conclusion that I came to, and that the 3D graphics world appears to have come to, although I'm sure I had tons less evidence to base it on than you (or they) do.  I had seen material both pro and con on using quaternions (there is some guy who used to travel around offering strapdown inertial navigation seminars; one of his selling points was "you don't have to use those nasty old quaternions!"), so I dug up just about every alternative method I could find.  After several attempts to comb a hairy ball flat, quaternions won out.  The Euler angles (and all other methods that use a 3-vector to represent angles) lost out for exactly the reason you mention. 

(I could have told my customer "just don't use this at the North or South poles" and they would have accepted it, albeit with puzzlement.  I still didn't want to go there).

Quote
The only significant difficulty comes in how it is implemented - you can have it defined relative to the destination frame, the source frame (the correct Lockheed way), and you can have the magnitude component first (wrong, even though that's how the NASA paper does it) or the last (the correct Lockheed way).

Well, it's good to know that you're unbiased about the notational conventions  :).
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Offline Target

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Re: Geometry Problem
« Reply #19 on: October 02, 2017, 02:56:40 PM »
What ............. the heck ................. is an 'octonion?'
That is where extreme math nerds fight in a cage,  armed with only a calculator and a note pad of quadril paper.
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