Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

A trick to eliminate 2π (sometimes)

marci.gunyho.com

Post reply on HN
Print view View original

171–179 of 179 posts

Re: A trick to eliminate 2π (sometimes)

#171

Post by Misdicorl »

jacobolus wrote: post #158
Misdicorl wrote: post #151

Earlier quoted context omitted.

I think this will devolve at this point. But you should consider that you are perhaps far too confident in your position here. My last attempt is due you to consider what happens when you A) multiply two degrees together B) multiply two dBm values together The output "units" in A change but do not in B. dB and angles are very different. Edit: the units in B do change, but the dB part doesn't. Tired.

Multiplying degrees together is not a meaningful concept. But you can compose rotations. Multiplying decibels together is also not meaningful, but you can compose scaling.

> Multiplying degrees together is not a meaningful concept.

Curious what you think of the d_theta * d_phi term in a spherical coordinates integral...

Re: A trick to eliminate 2π (sometimes)

#173

Post by Misdicorl »

movpasd wrote: post #170
Misdicorl wrote: post #36

Earlier quoted context omitted.

This is not true. Angles very much have units and it's why you can express the same concept with different numbers. Pi equals 180 degrees equals 0.5 turns. 1 radian has different units than 1 steradian and if they didn't there wouldn't be a need for two different words to denote them. The quantity is a ratio of two lengths, and the length measure does "drop out". But it's not just any ratio, it's a very particular ra…

Jeez, I've really kicked off quite the heated discussion in the replies... I don't want to get bogged down in lengthy arguments, but I feel I should explain my reasoning in more detail. Essentially, I think that whatever angles are, they are not like other dimensionful physical quantities. I have two arguments. The first: Someone mentioned symmetries in a reply. I wanted to mention them too but didn't have time to st…

The solution to your quandary is to realize the division operator is massively overloaded in your expression. What you actually want is to specify s and r as true line segments and define a new operator which takes two segments and outputs the angle between them. This operator happens to reduce to division of magnitudes in certain circumstances.

Edit: in other words, you've encoded tons of information in the problem statement about the relationship between r and s and you aren't properly encoding that in your type system allowing s/r to output an angle

Re: A trick to eliminate 2π (sometimes)

#174

Post by paulddraper »

1-more wrote: post #132
RogerL wrote: post #39

Earlier quoted context omitted.

But what is the answer? What should I do differently? I get don't use images, but then what? I can't imagine all screen readers have the same capabilities, or that there is a base common ability, so what should we do? Googling says MathML is the answer (e.g. https://www.washington.edu/doit/how-do-i-create-online-math-... this site uses MathML and your reader isn't handling it. So now what? (alt-tags? something else?)

When I view source (not the DOM) I can see that all of your math has aria-hidden="true". That seems to carry over into the parsed/generated math markup: run `document.body.innerHTML += ' [aria-hidden="true"]{ outline: 3px solid red; } '` to see all the hidden-from-screenreader things highlighted. That may be enough. For content that really cannot be written in a way that screen readers can handle, there is always the…

How does this interact with Find? Select text?

Re: A trick to eliminate 2π (sometimes)

#175

Post by 1-more »

paulddraper wrote: post #174
1-more wrote: post #132

Earlier quoted context omitted.

When I view source (not the DOM) I can see that all of your math has aria-hidden="true". That seems to carry over into the parsed/generated math markup: run `document.body.innerHTML += ' [aria-hidden="true"]{ outline: 3px solid red; } '` to see all the hidden-from-screenreader things highlighted. That may be enough. For content that really cannot be written in a way that screen readers can handle, there is always the…

How does this interact with Find? Select text?

solid point: it kind of wrecks them. I just checked in a codepen. Copying the text around the hidden paragraph will also copy the hidden paragraph. Maybe that's a feature? Searching for the content will also show that it's on the page but that you cannot meaningfully get to it, which kind of stinks.

These two issues get solved by aria-description where the description exists entirely in an attribute on the element rather than as a reference to another node in the DOM tree. But it's still in the draft for ARIA 1.3 so it's not fully adopted yet.

about the attribute https://developer.mozilla.org/en-US/docs/Web/Accessibility/A...

CanIUse info (tl;dr only in chromesque browsers) in the property info https://developer.mozilla.org/en-US/docs/Web/API/Element/ari...

Re: A trick to eliminate 2π (sometimes)

#176

Post by cycomanic »

mgunyho wrote: post #163
cycomanic wrote: post #153

Earlier quoted context omitted.

he writes: > where ln(e^2πi)=ln(1)=0, that is incorrect because the logarithm of a complex number is multi-valued. He even cites the correct source on wikipedia, but his argument is incorrect (and because the exponent is 2πi he actually would get a meaningful results I believe).

Hmm, this indeed seems to be the case! I think I was confused by ln(1), since 1 is a real number, but the multi-valued complex logarithm of 1 is indeed k 2pi (and now I see the appropriate notation should be "log" instead of "ln"). Perhaps the whole thing could indeed be reduced to "1^x" with an asterisk that 1^x means complex exponentiation. I'll have to update this section in the post.

See my other reply down below. That said I agree with the final conclusion that including the i is not a good idea, although I'm a bit on the fence on if the whole thing (I'm convinced by the tau=2pi arguments on the other hand).

I think this really is just a redefinition of sin and cosine (and consequentially the exponential). My feeling is that we now have to deal with different derivative operators for periodic and non-periodic functions and a lot of other weird disconnects, e.g. we don't have a relation between a cycle and the radius (diameter, circumference ...) anymore. I'm not convinced that the (minor IMO) conveniences that gives us is worth it.

Re: A trick to eliminate 2π (sometimes)

#177

Post by jacobolus »

Misdicorl wrote: post #171
jacobolus wrote: post #158

Earlier quoted context omitted.

Multiplying degrees together is not a meaningful concept. But you can compose rotations. Multiplying decibels together is also not meaningful, but you can compose scaling.

> Multiplying degrees together is not a meaningful concept. Curious what you think of the d_theta * d_phi term in a spherical coordinates integral...

You can compute solid angle (a.k.a. spherical excess, normalized spherical surface area) by taking a surface integral, but the units are not "square degrees" or "square radians", but instead an entirely new type, usually just measured in radians ("steradians"). Some people have defined https://en.wikipedia.org/wiki/Square_degree but that is a stupid unit.

While rotation is naturally oriented like a bivector (plane), solid angle is naturally oriented like a trivector (3-space).

The natural representation is as a kind of (unitless) ratio formed from 3 vectors, not the product of two vector–vector ratios.

Re: A trick to eliminate 2π (sometimes)

#178

Post by downvotetruth »

adrian_b wrote: post #64
movpasd wrote: post #12

Earlier quoted context omitted.

Thing is, angles don't really have units (the technical term is they are dimensionless). They are a length (the subtended arc of a circle) divided by a length (the radius of the circle). When you want to do something like get a sine wave of period T, you inevitably have to include a 2π somewhere. Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular fr…

The claim that plane angle, solid angle and logarithms are dimensionless quantities is a horrendous mistake and many generations of physicists have been brainwashed by being taught this aberration without ever stopping to think whether this claim can be proved. I will discuss only the plane angle, because it is the most important, but the situation is the same for solid angle and logarithms. The justification commonl…

> is a horrendous mistake and many generations of physicists have been brainwashed by being taught this aberration

The synathroesmic writing style and inapt analogies obfuscates the claims along with distracting and detering others from refuting them, but does not make them true.

Angles are defined in the abstract realm of mathematics not physics, which use them to describe physical phenomenon. In Cartesian space, the basis dimensions are defined using fixed perpendicular oriented lines, called coordinate lines on coordinate axes. Angles are not a basis dimension and as defined in Cartesian space are dimensionless. Angles are a dependent measure. Arc length is measured based on the units of the system. x radians or x degrees cannot be physically measured in and of themselves. In coordinate systems using an angle as a basis like spherical coordinates, angles exist as a dimension and the unit for them is free to chosen. Once the units are chosen, then the angles can be physically measured.

Re: A trick to eliminate 2π (sometimes)

#179

Post by broses »

dan_fornika wrote: post #61
broses wrote: post #56

People have proposed introducing a symbol for 2π before, most often τ. I like to go a step further and introduce a symbol for 2πi. I use pi with a dot above it, pronounced "pi dot". Pi dot can be defined as the period of the exponential function (which can be defined in terms of its Taylor series). Then 2π is pi dot / i, and π is pi dot / 2i. Of π, 2π, and 2πi, 2πi is probably the most natural, even though it's imagi…

Would it be more appropriate to call 2πi "tau dot"?

The problem with Tau is that it's already used for a lot of other things. I like to use π with a line through it for 2π ("pi cross"). But that's less likely to catch on since Tau for 2π is already well known.
Post reply on HN