Happenings Feb 28

So much math, so little time. Another week has gone by, and I am at least as ignorant as I was last Friday.

“The known is finite, the unknown infinite; intellectually
we stand on an islet in the midst of an illimitable
ocean of inexplicability. Our business in every
generation is to reclaim a little more land.”
–T. H. Huxley

The problem is, as I step further into the ocean, I see more of it. As I learn more, I see more that I don’t know.
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Enough about different colors, for now. What about different forms of one color? Artists have some simple terminology for that. If you take some color of paint and mix it with white, you get a lighter form of that color; this is called a tint of that color. If you take some color of paint and mix it with black, you get a darker form of that color: this is called a shade. In other words, light red is not — technically speaking — a shade of red, but a tint of red.

If, on the other hand, you take some color of paint and mix it with both white and black — that is, you mix it with gray, whether light or dark — this is called a tone.

If your memory is as bad as mine, you might want to note the corresponding vowels: white and tint, black and shade, both and tone. (And that correspondence is precisely why I say “both” instead of “gray”,)

One of the reasons I like the HSB color wheel is that its colors can be interpreted as tints, tones, and shades. Oh, let me show you two different forms of it. The large disc at the beginning of the previous post was at full brightness. There was a slider to the right which would have given me a darker color wheel. For example:

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HSB: Hue, Saturation, Brightness

Now let us do something quantitative.

One of the terms I found most frustrating in books about color was “saturation”. And the first quantitative satisfaction I found was in the algorithm for converting from RGB to HSB. The S in all three of HSB, HSV, and HSI stands for “saturation”. HSB stands for hue, saturation, and brightness (V stands for value, I for intensity). Hue means color; but what are saturation and brightness?

I will take it for granted that we all understand that color on a television or computer monitor is specified by RGB: three numbers which specify amounts of red, green, and blue at each location on the screen.

Here is a picture of the HSB color picker on my computer.

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Happenings Feb 20

Of the questions I asked early this month, I really should be able to polish off the rotation equation, and the Basilevsky questions. (The computational failure last weekend was merely a typo, using the wrong variable.) I still have not touched simplicial coordinates at all.

The second color post is coming along nicely. As usual, I slow down when I remember that I can’t just copy something and put it out here. It might work for you, but it doesn’t work for me. I need to play with it, to even begin to understand it.

What am I going on about? The algorithm for converting RGB to HSV is very straightforward; the inverse algorithm is not. I was tempted to just put it out here (really), but then I came to my senses. And as usual, I saw something that I hadn’t noticed before.
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Color wheels, RGB, and CMY

This post has five goals: to describe two color wheels; to describe tints, tones, and shades; to describe the HSB (= HSV) color model; and to relate the HSB color model to tints, tones, and shades.

Correction: having written a first draft, I see that I will need three posts. The purpose of this post is to describe two color wheels.

Here they are. (Two of the greens in the 2nd color wheel look identical on my monitor – but they have different RGB specs. And I am used to putting red on top, but when I assembled them – below – I put red on the right. I had a reason. Next post!)

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Happenings and Q&A Feb 14

This is the post that I would like to have put out last night.

What am I hoping to do this weekend? Have I made any progress on the questions I asked earlier in February?

What am I hoping to do? Not a lot has changed. I keep reminding myself that it’s okay to work on B even though I can’t publish it until I publish A. (For example, B might be solar calculations, and A is explaining the equations.) As I have said before, doing mathematics and publishing mathematics are two different activities; I get to keep two different to-do lists running. Anyway, I haven’t added anything new to the mathematics I hope to do, but I have reminded myself that I can do some things I’m not yet ready to publish.
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There are three things about color which particularly interest me.

First, interesting color combinations, whether I find them on the exteriors of houses, or on the insides; on TV logos, or on advertisements; or in art.

Second, the mixing of colors, although my own interest is primarily in mixing them on a computer screen.

Third, the phenomenon called metamerism. We encounter it most often when two pieces of clothing appear to be the same color under one light (say, fluorescent), and appear to be different colors under some other light (incandescent, or daylight). I won’t say much more about metamerism today, because it is so much more complicated than combinations and mixing. It will require looking at the physiological basis of color perception. I might not be so interested in it had I not encountered an analysis of it using the pseudo-inverse of a matrix.
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What should I do this weekend? Feb 6

So, what mathematics am I working on?

I work on so many different things – okay, I play with so many different things – that I need to begin the weekend by sorting through my toybox. Here are the things I am playing with.

The control of aircraft. I have a book which uses classical methods to study the control of aircraft; this is where I want to begin, rather than in state-space. Nevertheless, I have found other books to be essential. There are two equations that gave me pause. What I call the magic $\omega$ equation (which I have not shown you), and our old friend the rotating coordinates equation (which I have shown you). I mentioned the latter in the questions last week, but I also indicated that I knew the answer. I need to add a mechanics book to the bibliography. I’ve probably got four posts to put together at this point, essentially just laying out equations. Then we can start on an example.
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Puzzle: splitting 8 gallons into halves

If nowhere else, you might have come across this in the third “Die Hard” movie. We have 3 containers, of volumes 8, 5, and 3 gallons. The 8-gallon container is full. Using only the 3- and 5-gallon containers, split the contents evenly: 4 gallons in the 8-gallon container, and 4 gallons in the 5-gallon container.

If you don’t want to see a solution, move along, quickly.
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Questions – early February

When I look back over the past year, I see that I do not have all that many posts about the doing of mathematics — almost none, in fact. If anything, the introductory material for my “books added” posts is the closest I have come to talking about the doing of mathematics. Even those are more about “here’s what I don’t know and where I think I might be able to learn about it”. Fair enough, they’re something like what I had hoped to do; I’ll try to keep on writing non-trivial introductions to the “books added” posts.

I can point out that many of my “Davis” posts on PCA / FA are, in retrospect, mostly about my learning to use the SVD (singular value decomposition). They were as much about learning mathematics as doing it, and are a rare example – in public, on the blog, I hope! – of my blundering through something. I daresay the calculations were correct, but my assessment of issues was not very clear. It took me a while to decide what was important in that work.

So let’s see if I can do something else, too, to show the process. Let me try posting the questions I’m working on. Now, not everything I am doing is the answering of a clearly stated question. All too often my question is the all too general “what the hell is going on?” In that situation, there isn’t much I can say until I make some progress. In fact, some of the following questions are still a bit vague.

And I will confess that I am a little afraid of making these questions public. For one thing, they may seem silly. I expect that most of them will certainly be simple. By publishing them, I may make myself look foolish and ignorant; and lazy, too, because surely I should have been able to answer the questions in less time than it took to publish them. (Yes and no.)

And how long will it take me to answer them? Will the list of questions grow so long as to become unmanageable? Will my readers start throwing the answers at me? (Hmm. Wouldn’t that be a good thing?)

Okay. Let me just dive in. I have three sets of questions: mechanics, principal components, and simplical coordinates.
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