Well, it has turned out to be an interesting week. Somehow I found enough time to do some meta-mathematics of my own.

## Logic

Or to stumble across some by someone else. I was sorting through a pile of books that had been sitting on the floor of my library for too long. One of the books I found was called “Cognition” by Dodd & White. In one section it talks about why some syllogisms seem harder than others for people to get right. I haven’t taken notes on this yet, but I’m looking forward to it. It should serve as spice when I get to talking about syllogisms. Real Soon Now. Probably this weekend.

## Color

In color, I spent too much time trying to do something that was impossible. At least, I’m pretty sure the fundamental geometries of two particular descriptions of color are just too different from each other. I have shown you how to get and to lay out tints, tones, and shades from HSB. The resulting layout is rectangular or square. In particular, it has a line which is black.

On the other hand, the original representation of tint-tone-shade uses a triangle whose three vertices are labeled color, white, and black. (I think I have never displayed one of those, but here’s something. Not exactly a triangle, but use your imagination.) In particular, let me emphasize that it has only a vertex — not a line — which is black.

I do not believe these can be reconciled. To be specific, I do not believe that HSB can be reconciled with color-white-black specifications of color. Incidentally, I didn’t set out to reconcile them — I was just trying to get unique color-white-black specifications for Birren’s color triangles.

If it’s any consolation, let me quote Fairchild:

“It is not possible to say that one is better than the other [speaking of the Munsell, rather than HSB, and NCS color order systems — but NCS is a specification in terms of white-black-color], it can only be stated that the two are different. This was recently reaffirmed in the report of CIE TC1-31…, which was requested by ISO to recommend a single color order system as an international standard along with techniques to convert from one to another. This international committee of experts concluded that such a task is impossible.”

In color, I also learned something new… and I will probably be adding a comment to the post about from tristimulus to spectrum.

Here it is. We know how to take an illumination spectrum and a reflectance spectrum, and compute tristimulus values. What happens if we divide the illumination spectrum by two?

That is, what happens if we reduce the absolute level of illumination?

Answer: we will get exactly the same tristimulus values (because Y is scaled by the illuminant).

This means that unless we have an absolute measure of the illumination level — or at least a relative measure of “full illumination”, and I was assuming we did — then there are still an infinite number of reflectance spectra (hence illumination spectra) which will reproduce the same tristimulus values.

We cannot recover the absolute intensity from XYZ alone.

## Projectiles

Meanwhile, my kid (me doing whatever I want) has been playing with the simplest models of projectile motion (constant gravity and no air resistance), and it has been fascinating to look back at something I understand.

It emphasizes, once again, that it is far, far easier for me to supply my own meta-mathematical commentary for mathematics I understand than for mathematics I’m trying to figure out.

I set out to do as many calculations as I could from first principles — more precisely, perhaps second principles. That is, I didn’t start from the differential equations (first principles) over and over again — but my initial approach was simply to use the general solutions (second principles? I’m being foolishly pedantic.), i.e. four equations: horizontal and vertical position, and horizontal and vertical velocity, as functions of time.

I’ve already discovered — been reminded of — three significant caveats… that is, it is extremely worthwhile to use additional information.

For one thing, it is extremely worthwhile to use magnitude and direction for the initial velocity — much as I hate to use magnitude and direction for vectors! You see, the angle of launch completely dictates the form, the shape, of the trajectory; while the initial speed — the magnitude of the initial velocity — dictates the scale of the trajectory. That correspondence can be very useful; more importantly, the independence of the form and the scale can be very useful.

For another thing, conservation of energy makes it trivial to compute the maximum altitude as a function of the initial velocity vector (strictly, of the initial vertical component of velocity).

Third, once we have the maximum altitude, it is almost trivial to get the time at which it occurs… and by symmetry, twice that time is the time to impact for equal initial and final altitudes… and then it is trivial to get the range. And having the range, even for a problem which does not have equal initial and final altitudes, can be a useful sanity check on the answer to the given, albeit different, problem.

To put that another way: the sequence of calculations

initial vertical speed -> maximum altitude -> time to maximum altitude -> time to impact (at final altitude = initial altitude) -> distance to impact (i.e. range)

is easy and informative. (It also might answer the given question, but that’s another issue.)

Fourth, in the modern world the sensible way to check your answer is to graph the entire solution… and if your solution was algebraic, the sensible way to check it is to graph it for a selection of numerical values of the parameters.

Part of what I’m saying there is that even though a question may ask for one specific characteristic of some trajectory, a sensible real world solution would end up with a complete description, in order to draw at least one picture, if not more.

April 24, 2010 at 2:42 pm

Rip

About Color…

Quote “That is, what happens if we reduce the absolute level of illumination?

Answer: we will get exactly the same tristimulus values (because Y is scaled by the illuminant)”

I think that these actions 1) and 2) should be equivalent :

1) To reduce level of illumination, mantaining object reflectance curve constant

2) To reduce reflectance curve, mantaining level of illumination constant

These actions should modify XYZ tristimulus values, just because color will decrease in value and chroma when it will be converted by the mind into color, because it means a different sensation caused by different stimuli

Could you explain how you have arrived to that conclusion?

Thanks

April 24, 2010 at 4:21 pm

Hi Daniel,

It’s always a good idea to guess first and then work it out. I really believe that. (If your guess is right, you reinforce your intuition; if it’s wrong, you correct your intuition.)

How do we get XYZ from an illuminant spectrum C and a reflectance spectrum R?

We compute the pointwise product of S = C R to get the reflected spectrum…. We apply the AT matrix (the color matching functions) to the S vector, and get three numbers — which are the components of the fundamental of S with respect to the basis E dual to A. But those three numbers — call them comp — are not XYZ, not yet.

We have to apply the middle row of AT to the illuminant spectrum S to get a scale factor, scale = AT(2) . C. We divide comp by scale to get XYZ.

If we now divide the illuminant spectrum C by 2, so C2 = C/2, then S is also cut in half, S2 = S/2. Apply the AT matrix to S2 and we get three components, half of what we had before: comp2 = comp/2.

We now get the scale actor by applying the middle row of AT to C2 — that’s the new illuminant, half the old one, so our new scale factor is half the old one: scale2 = scale / 2.

Our components have been halved, but so has our scale factor. The resulting XYZ are the same.

The key, of course, is that I am using the actual (halved) illuminant spectrum.

(If you run though the same computation, but halve the reflectance spectrum, you will get different tristimulus values. In fact, they will all be halved.

I’m glad you asked.

April 25, 2010 at 4:09 am

Rip,

In the CIE Model, you can’t vary the Illuminant Spectrum, because it is defined as a constant into the system.

In such a way it is not an arbitrary “S”, is an Illuminant defined by its blackbody equivalent temperature, so there are a set of illuminants you can preset (4000 K <= T <= 25000 K)

What you can do is to choose it at first by defining the CCT (Correlated Color Temperature) for the Illuminant and calculating it through the CIE Daylight Canonical Components, S0-S1-S2, in this way you define “S” as the Illuminant (as you explained it in your blog)

Obviously you can pass from one Illuminant (S1) to other (S2) for the same reflectance curve to analyze how it affects stimuli (XYZ)

It can't be varied as if it were with a potentiometer adjusting the illumination level, so you cannot divide the Spectrum by 2 (i.e.)

To emulate this behavior acting as a stimulus, you should apply “the potentiometer” on the reflectance curve, at the end what it matters is the reflected spectrum that affects our system vision as a primary stimulus, then when we apply the CIE Tristimulus Model we get XYZ, this is the stimulus that our mind will convert into a color sensation (out of the CIE scope)

The reason of the Illuminant Spectrum constancy into the CIE model is that it is defined relative to the power at lambda = 560 nm, so we can’t adjust its level into the model

April 27, 2010 at 6:05 pm

Here’s what works for me: I believe the mathematics is just fine — and the prohibition against arbitrary illuminant spectra is intended to make things nice.

I’ll keep an open mind on this issue, but I really like turning it down “as if it were with a potentiometer”.

And it is precisely the independence of XYZ with illuminant level that let me ignore whether the illuminant ranged from 0 to 1 or from 0 to 100. I’ve been using that independence for most of these posts.

Once again, thanks for widening my perspective.