It's a solved problem - torrent your damn entertainment. Movies are just basic files sitting in a directory. Files that can be rewatched whenever you'd like. If you are traveling, copy to your laptop. If you move and haven't quite set up your entertainment center or Internet connection, watch it on a computer. If your friend is interested in something, copy it to their USB drive. No fucking nonsense of some third party capriciously disrupting your life precisely when you're trying to relax.

Any business trying to sell me some productized solution needs to beat torrenting for ease of use. So far none of them have even attempted, because they all end up warping the user experience to appease Hollywood's delusion of control. Just say no.

For movies that YIFY doesn't have, 1337x is a good site.

Also, I'll just note that we never got the 21st century we were promised, which was this - any movie ever made: https://www.youtube.com/watch?v=xAxtxPAUcwQ The fundamental problem is that the people who own movie IP rights are truly evil.

That is wildly overstated. The worst they can do to you is not let you watch their entertaining movie. Possibly even after you've paid for it (the subject of the OP) - and in that case the culprit isn't even the IP owner, it's the distributor! And honestly it sounds like the problem with digital ownership is simple fraud that is a) covered by existing law and b) too expensive for anyone to litigate. Maybe a class action could do it.

Here's the really interesting part - you're railing against artificial scarcity. Someone has a good that they could give away, and they aren't, and you're calling them evil for doing that, but I ask you, in all honesty, how else do you make money from movies? If you can't make money from it, how will you convince investors to fund your next movie? (Now substitute "album book software" for movie and ask the same question.)

That's not to say that IP owners can't be "evil". George Lucas believed it was his right to keep changing Star Wars over time, and it's impossible to find a legit copy of Star Wars that is the original theatrical release. That's some 1984-level memory hole bullshit and although the stakes are low, it's evil. Disney is arguably quite evil for a variety of reasons, e.g. it's unholy influence over Congress, it's unhealthy consolidation of huge chunks of the American movie market. But neither of them are evil for using artificial scarcity to profit from their work, because that's the only way to profit from data goods.

(Professional open source tries to square the circle by giving away the data goods but charging for (actually scarce) knowledge. It's a good model but cannot apply to entertainment goods, since viewers don't need scarce knowledge to enjoy a movie.)

Is it, in absolute terms, still a very important concern worth addressing at some point? Sure.

We should start addressing it now. We can work on more than one problem as a species, and moreover, we should. The big problems never stop coming, so focusing 100% of our energies on that means that the smaller problems won’t get solved.

Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians.

Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place.

"Turns" don't generalize to higher dimensions either. With radians you can calculate arc length on a circle by multiplying with the radius. This extends naturally to higher dimensions: a solid angle measured in steradians lets you calculate surface area on a sphere by multiplying with the radius. How do you do the same with "turns" on a sphere? You can't in any meaningful way.

That's nice, but as the article points out most implementations of trig functions on computers don't use things like Taylor series.

Another terrific use of turns is in calculating angle differences, where you take a difference and just use the fractional part of the result. No bother with wrap around at some arbitrary 2*pi value. Since it wraps at integer values we simply discard the integer part. This can even be for free when using fixed-point math.

Implementations which are accurate in terms of turns even for values close to half a turn can be useful for avoiding numerical issues that sometimes pop up because π is not exactly expressible as a floating point number. These functions usually names like sinpi, cospi, etc. It would be nice if they were provided more often in standard libraries.

In the next installment, maybe he'll propose that turns can be limiting because diving up a circle requires the use of fractions, and suggest instead of 1 turn per circle, we make a number that's easily divisible into many integer factors. Maybe 216, or I don't know, 360?

And using fraction of a turn is also a very good option, much better than radians in many cases, especially if you chose a power of two fraction (e.g. 1/256), in this case all the modular arithmetic needed for angles comes for free as simple integer overflow, and lookup tables became a simple array access.

It is much computationally cheaper and more robust (and easier to reason about) to use vector algebra throughout. Then you have no transcendental functions, just basic arithmetic and the occasional square root. You need the dot product and the wedge product (or combined, the geometric product), and derived concepts like vector projection and rejection.

If you need to store a rotation, you can use a unit-magnitude complex number z = x + iy, where x = cos θ, y = sin θ, without ever needing to calculate the quantity θ directly. If you need to compress it down to one parameter for whatever reason, use the stereographic projection s = y / (1 + x) = (1 – x) / y. Reverse that by x = (1 – s²) / (1 + s²), y = 2s / (1 + s²). [If starting from angle measure for whatever reason s = tan ½θ, sometimes called the "half-tangent".]

The angle measure is the logarithm of the rotation, θi = log z. In some contexts logarithms can be very convenient, but it’s not the simplest or most fundamental representation.

With units of "radians" angle measure is the logarithm of base exp(i) [related to the natural logarithm], and with units of "turns" it is the logarithm of base 1 (sort of).

If you use angle measures (of whatever units), when you say "rotate by an eighth of a turn" you are instead going to end up with something internally like: multiply some vector by the matrix

[cos ¼π, –sin ¼π ; sin ¼π, cos ¼π]

which is ultimately the same arithmetic, except you had to compute more intermediate transcendental functions to get there.

If you just have to do this a few times, angle measures (in degrees or whatever) are a convenient human interface because most people are very familiar with it from high school. You can have your code ingest the occasional angle measure and turn it into a vector-relevant internal representation immediately.

P.S. If you write 1/√2 in your code compilers are smart enough to turn that into a floating point number at compile time. :-)

(Game engine developers are smart people and have lots of practical experience with the benefits of avoiding angle measures, as do developers of computer vision, computer graphics, robotics, physical simulations, aerospace flight control, GIS, etc. etc. tools.)

The Taylor expansion works out like

   sin θ = θ - θ³/₆ + θ⁵/₁₂₀ - θ⁷/₅₀₄₀ + ⋯

https://en.wikipedia.org/wiki/Chebyshev_polynomials

which are optimized across the range. You could rewrite these just as easily to work in degrees as radians.

One of the best ways to calculate sin and cos is CORDIC,

https://en.wikipedia.org/wiki/CORDIC

which is really based on turns, half-turns, quarter-turns and so forth.

phi_n = atan(2^-n)

and then using an abbreviated sum formula where computing cos(theta + phi_n) depends only on sums and bitshifts.

The small-angle approximations sin(x) ≈ x and cos(x) ≈ 1-x^2/2 are the real killer feature of radians, though, because when you can deal with the loss of accuracy you get to avoid using any loops whatsoever. They're also fundamental to understanding simple physical systems like a pendulum.

On the other hand with CORDIC it is extremely easy to reach any desired precision, and in cheap hardware it does not require multipliers.

So CORDIC may be considered as "one of the best ways" depending on how "best" is defined.

Even when developing a polynomial approximation for the fast evaluation of a trigonometric function, it may be useful to use an alternative evaluation method by CORDIC, in order to check that the accuracy of the polynomial approximation is indeed that expected, because for CORDIC it is easier to be certain that it computes what it is intended.

The article actually argues the opposite: that the common implementations of sine and cosine start by converting their radian based arguments to turns or halfturns by dividing by pi.

For example, e^x can be implemented by handling the integer and fractional parts separately, for similar reasons. But no one really cares about the functions e^floor(y) and e^(y-floor(y)). They are only useful as part of an implementation trick.

Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighted, I'd say, it doesn't "scale out" to the myriad of other applications of angles.

Regarding long variable names: I'd rather have long variable names, than a mathematician using some greek symbol in formulas without telling what the meaning of it is (and it could be different depending on their background). But I have no issues with the single letter variables if they're specified properly.

And proposing to write equations in books and articles with long variable names... Well, Algebra was invented for a reason.

https://blogs.scientificamerican.com/observations/the-tao-of...

There was an earlier effort that used a new "two pi" symbol consisting of a "π" with an extra leg in the middle: https://www.math.utah.edu/~palais/pi.pdf.

That could never work. If anything the words comprising mathematical texts should be defined once and thereafter truncated to their first letter to reduce cognitive burden and facilitate greater comprehension.

c = "could"; d = "don't"; f = "for"; g1 = "go"; g2 = "great"; i = "it"; i2 = "i"; m = "me"; s = "see"; w = "works"; w2 = "what"; w3 = "wrong"

i w g2 f m; i2 d s w2 c g1 w3.

That said, I would like for my compiler to combine any multiplications involved down to one factor for input to the fastest sin/cos operations the machine has. And, to treat resulting multipliers close enough to 1, 1/2, and 1/4 as exact, and then skip the multiplication entirely.

But the second part is a hard thing to ask of a compiler.

I wish I had done CS, those kinds of compiler optimization sounds so fun. I'd love to work on that

You can do your own scribbles with single letters, so do I, it works fine.

But when you present maths in a scientific article, maths book, Wikipedia article or similar, your convenience as a writer should be secondary. Your task is to present information to someone who does not already know the subject. Presenting an equation as six different Greek letters mashed together means that the equation itself convey almost no information. You need a wall of text to make sense of it anyway.

It boggles the mind, truly!

From an algebra perspective (the one given in the blog post) it may be fine, but from a calculus perspective it's really not.

The lack of continuity really hurts when you add floating points shenanigans into the mix, just a fun example:

When you have 1/0 = 0 but 1/(0.3 - 0.2 - 0.1) = 36028797018963970. Oopsie, that's must be the biggest floating point approximation ever made.

  zero_point equals negative prefactor divided_by two plus_or_minus square_root_of( square_of(prefactor divided_by two) minus absolute_term )

  zero_point = -prefactor/2 ± √((prefactor/2)² - absolute_term)

  x = -p/2 ± √((p/2)² - q)

And for pedagogic purposes, I do would like more meaningful names at times.

A good compromise might be to put the equation in the comments in symbol-heavy form, and use the spelled out names in code.

I'm not talking about using it in code, I'm talking about someone arguing that books and articles should do it as well.

Most importantly the flat dismissal and horror that many express when someone brings up adjusting the symbolic traditions of Maths should be investigated. Engage with why you feel so strongly that anything other than rigid adherence to tradition is sacrilege. Based on what I've heard, in order to be a great Mathematician, you need to hold onto tradition lightly and think outside the box. Rigid adherence to tradition doesn't sound like that to me.

Who’s saying that? Inventing good notation is a big part of mathematics (and that also frequently gets criticized on HN because it may introduce ambiguities)

Also, there’s nothing wrong with texts that target an audience with a certain level of understanding.

It’s not as if adding, for example, “By Hermetian matrix we mean a complex square matrix that is equal to its own conjugate transpose” will make a paper much easier to understand, just as adding a comment “this is where the program starts running” doesn’t help much in understanding your average C program, or adding a definition of “monarchy” to a history paper.

In the end, any scientific paper has to be read critically, and that means making a serious effort in understanding it. A history paper, for example, may claim that Foo wrote “bar” but implied “baz”. A critical reader will have read thousands of pages, and (especially if they disagree with the claim) then think about that for a while, and may even walk to their bookshelf or the library to consult other sources before continuing reading.

> Ummm, actually it did.

No, it didn't. Some specific uses looking better with radians does not mean you have to use radians always.

When I first learned sine and cosine, we used degrees, and that worked fine. Later we switched to radians, but there's no reason why you shouldn't use turns, and the article gives a very good argument why in some cases you definitely should.

It's not just some specific use cases, it's the majority of cases if you look across all of math and science. Switching to turns would be stupid, especially once you start doing differentiation and integration. The fact that we use radians almost across the board isn't some accident.

There is only one consequence of those series that matters in practice, which is that when the angles are expressed in radians, for very small angles the angle, its sinus and its tangent are approximately equal.

While this relationship between small angles, sinuses and tangents looks like an argument pro radians, in practice it isn't. There are no precise methods for measuring an angle in radians. All angle measurements are done using an unit that is an integer divisor of a right angle, and then the angles in radian are computed using a multiplication with a number proportional with the reciprocal of Pi.

So the rule about the approximate equality of angles, sinuses and tangents is at best a mnemonic rule, because to apply the rule one must convert the measured angles into radians, so no arithmetic operations can be saved.

"Turns" generalize perfectly to higher dimensions.

To the 3 important units for the plane angle, i.e. right angle, cycle and radian, there are 3 corresponding units for the solid angle, i.e. the right trihedron (i.e. an octant of a sphere), the sphere and the steradian.

The ratio between the right trihedron and the steradian is the same as between the right angle and the radian, i.e. (Pi / 2).

The ratio between the sphere and the right trihedron is 2^3, while that between cycle and right angle is 2^2. In N dimensions the ratio between the corresponding angle units becomes 2^N.

Moreover, while in 2 dimensions there are a few cases when the radian is useful, in 3 dimensions the steradian is really useless. Its use in photometry causes a lot of multiplications or divisions by Pi that have no useful effect.

There is only one significant advantage of the radian, which is the same as for using the Neper as a logarithmic unit, the derivative of the exponential with the logarithms measured in Nepers is the same function as the primitive, and that has as a consequence similarly simple relationships between the trigonometric functions with arguments measured in radians and their derivatives.

Everywhere else where the radian is convenient is a consequence of the invariance of the exponential function under derivation, when the Neper and radian units are used.

This invariance is very convenient in the symbolic manipulation of differential equations, but it does not translate into simpler computations when numeric methods are used.

So the use of the radian can simplify a lot many pen and paper symbolic transformations, but it is rarely, if ever, beneficial in numeric algorithms.

The addition theorems for trigonometric functions can easily be shown by the multiplication theorem for Taylor series (and adding two Taylor series). This proof would be more convoluted if the Taylor series were not so easy.

Also, because of the simplicity of their Taylor series, one immediately sees that sin and cos are solutions of the ODE y'' = -y.

Another application of the Taylor series is that by their mere existence, sin and cos (as real functions) have a holomorphic extension.

Any proof that uses the expansion in the Taylor series is a serious overkill.

Moreover, those proofs become even a little simpler when the right angle is used as the angle unit, instead of the radian.

In this case, the sine and the cosine can be defined as the odd and even parts of the function i ^ x.

Excuse me? Have you done any computation in Physics? Have a look at the pendulum equation, for a start...

In practical physics computations, the solution of differential equations requires numerical methods that do not use the Taylor series of specific functions, even if the theory used for developing the algorithms may use the Taylor series development of arbitrary functions.

For accurate prediction, the simple pendulum equation also requires in practice such numerical methods, which do not rely on the small-angle approximation that enables the use of the Taylor series of the trigonometric functions, for didactic purposes.

> In practical physics computations, the solution of differential equations requires numerical methods that do not use the Taylor series of specific functions, even if the theory used for developing the algorithms may use the Taylor series development of arbitrary functions.

I'm sorry, but you have no idea what you're talking about. Series expansions is one of the most widely used techniques in Physics. Obviously some equations require full blown numerical methods to be solved, but one can do a whole lot with analytical techniques by doing series expansions and using perturbation theory.

Saying that this is only used "in school exercises" shows that you're completely out of touch with reality.

I have said that the Taylor series of arbitrary functions have various uses, but there is no benefit in knowing which are the specific Taylor expansions of the trigonometric functions, with the exception of knowing that the first term of the sine and tangent expansions when the argument is in radians is just X.

Solving physics problems using the expansion of an unknown function in the Taylor series has nothing to do with knowing which is the Taylor series of the sine function.

There are more terms in the expansion that you can use, that's the whole point of using an expansion...

> Solving physics problems using the expansion of an unknown function in the Taylor series has nothing to do with knowing which is the Taylor series of the sine function.

I hope you're aware that the sine function appears quite often in Physics problems.

I cannot believe I just read this.

When you approximate functions by polynomials, including the trigonometric functions, the Taylor series are never used, because they are inefficient (too much computation for a given error). Other kinds of polynomials are used for function approximations.

The Taylor series are a tool used in some symbolic computations, e.g. for symbolic derivation or symbolic integration, but even in that case it is extremely unlikely for the Taylor series of the trigonometric functions to be ever used. What may be used are the derivative formulas for trigonometric functions, in order to expand an input function into its Taylor series.

The Taylor series of arbitrary functions (more precisely, the first few terms) may be used in the conception of various numeric algorithms, but here there are also no opportunities to need the Taylor series of specific functions, like the trigonometric functions.

The Taylor series obviously have uses, but the specific Taylor series for the trigonometric functions do not have practical applications, even if they are interesting in mathematical theory.

Can you point me to some implementation of sin that’s not actually using Taylor expansion in some form? Because most that I am aware of do in fact use Taylor series (others are just table lookup). See glibc for example:

https://github.com/bminor/glibc/blob/release/2.34/master/sys...

And here is musl

https://git.musl-libc.org/cgit/musl/tree/src/math/__sin.c

(The constants are easily checked to be -1/3!, 1/5! Etc)

This might have something to do with the Taylor’s theorem. You know, that the Taylor’s polynomial of the order n is the only polynomial of order n that satisfies |f(x)-T(x)|/(x-a)^(n+1) -> 0 as x -> a. In other words, the Taylor polynomial of order n is the unique polynomial approximation to f around a to the order n+1. This means you cannot get any better than Taylor close to the origin of the expansion. This causes implementers to focus on argument reductions instead of selecting polynomials.

What is much more likely is that if you will carefully compare the polynomial coefficients with those of the Taylor series, you will see that the last decimals are different and the difference from the Taylor series increases towards the coefficients corresponding to higher degrees.

Towards zero, any approximation polynomial begins to resemble the Taylor series in the low-degree terms, because the high-degree terms become negligible and the limit of the Taylor series and of the approximation polynomial is the same.

So when looking at the polynomial coefficients, they should resemble those of the Taylor series in the low-degree coefficients, but an accurate coefficient computation should demonstrate that the polynomials are different.

  >>> print("{:.30f}".format(sin(pi/4, 0, 0, *standard_coeff)))
  0.707106781186567889818661569734
  >>> print("{:.30f}".format(sin(pi/4, 0, 0, *musl_coeff)))
  0.707106781186547461715008466854

I've used them a few times, mostly in the embedded space, and mostly in conjunction with lookup tables and/or Newton's method, but yes I've absolutely used them outside school (years ago, I forget the exact details).

- implementing my own trig functions for embedded applications where I wanted fine control over the computation-vs-precision tradeoff

- implementing my own functions for hypercomplex numbers (quaternions, duals, dual quaternions, and friends).

- automatic differentiation

Does the Taylor series form survive to the final application? Usually not, usually it gets optimized to something else, but "start with Taylor series and get back to basics to get a slow but accurate function" has gotten me out of several pickles. And the final form usually has some chunks of the Taylor series.

However, the performance when using Taylor series is guaranteed to be worse than when using optimal approximation polynomials, according to appropriate criteria.

Still I cannot see when you would want to use the Taylor series of the trigonometric functions, even if for less usual functions it could be handy.

There are plenty of open-source libraries with good approximations of the trigonometric functions, so there is no need to develop one's own.

In the case of a very weak embedded CPU there is the alternative to use CORDIC for the trigonometric functions, instead of polynomial approximations. CORDIC can be very accurate, even if on CPUs with fast multipliers it is slower than polynomial approximation.

If only computers could do a bit of symbolic algebraic manipulations before issuing the machine code.

Wait, isn't that what optimizing compilers can do? That requires an optimization across library calls and thus a form of inlining, which doesn't see far fetched for a math library call. Or some optimizations can't be done due to floating point error propagation (which could be relaxed)?

Out of interest, when did you go to school in Bavaria and in which grade did you learn about turns? I was in school in Bavaria a long time ago and I don't remember learning about turns there. Could very well be that I forgot or our teacher forgot to teach it.

I feel a bit of humility would have helped the author and perhaps they would have considered the possibility that they didn’t think of the problem deep enough rather than hastily write a blog post about it.

It speaks to the hubris and the superficiality of thinking for some authors.

Casey didn't say the world would be better with turns instead of radiants. He said that game engine code would be better with turns instead of radiants. Be more charitable.

Reading the submission and the comments here, I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA. While I’m slightly envious you might not have to suffer developing powers of cosine and sine but that would explain the lack of familiarity with radian I see here. Am I wrong?

Yes. Trigonometry is extensively taught in the US. People forget this stuff if they don’t use it.

Ask some 30 year old chef in whatever country you fantasize teaches properly to compare and contrast turns vs radians and you’ll get similar responses.

Where this idea comes from I have no idea. Personally looking around in school itself it was plainly obvious this was all going in one ear and out the other for the majority of students even at the time. The better students retained it long enough to spew it out on the test but that was already above average performance. That doesn't mean there isn't still a certain amount of value in that in terms of what that knowledge may do to their brain during the brief period of time it is lodged in there. (I think there's a lot of value in just learning the "shape" of all this stuff, and perhaps having some index of what might be valuable to know.) But the idea that we can spend 15 minutes and a one-page homework assignment on something and expect that to last 60+ years is just nonsensical.

I mean, honestly, anyone over the age of 22 or so ought to be able to notice a distinctly sub-100% retention rate simply by looking inside themselves.

Yes, to a first approximation everyone with a normal education in the US has been present while some sort of trig was discussed. Not all of them, but still quite a lot of them, were present for the Taylor expansion discussion. The vast bulk of them have had it decay by 25, and there simply isn't anything to be done about that if you're talking about humans and not some homo educationous who mythically retain all knowledge they were exposed to even for 30 seconds just as the mythical as homo economicus perfectly rationally conducts all their economic business at all times. Perhaps they're actually the same species.

To me, even though I was barely out of high-school at the time, this was obviously absurd - expecting especially someone who wants to drop out of high-school early to retain any notion of organic chemistry taught in a school year, that they couldn't learn on the job if it was really required, seems so obviously nonsense that I couldn't help but laugh. Especially since the same thing was done to basically every other subject as well, with the same intentions.

One note: in my country, the curriculum is completely centralized; there is some small amount of choice, but it amounts to, at most, 1-2 classes per semester; everything else is fixed.

Yeah they belong to the genus homo mythicus

There’s just an absolute ton of math being taught that’s going completely to waste, and it’s at the expense of the humanities.

And then we learned a bunch of algorithms that spit out approximate answers to almost anything. And a bunch of ways to verify that the algorithm doesn’t have a bug and spat out an approximately correct answer. It was amazing.

But the most long-term useful math class (beyond arithmetic and percentages) has been the semester on probabilities and the semester on stats. I don’t remember the formulae anymore, but it gave me a great “feel” for thinking about the real world. We should be teaching that earlier.

The upper level Real Analysis made us bring some rigor as to what an interval on the real line actually meant going from raw points and sets to topological spaces to metric spaces, then compactness, continuity, etc. all with fun and crazy counterexamples.

People love to hate on their school curriculum and all the useless knowledge they had to acquire but I'm positive it makes you a smarter person overall, and the body of high school knowledge makes learning more specialized knowledge easier (even if that's baking bread or whatever)

(People also love to talk about how little they remember from school, yes the brain is a muscle and you stopped working out, congratulations.)

Came to use trig functions quite frequently while playing video games, and that was a big surprise to me. Not to assume you've played it, but I've recently discovered that Stormworks is a programmer's game - you can write microcontroller code in LUA for your vehicle designs. And, wow, does it ever use my trig knowledge everywhere.

Realized the transponder beeps can be triangulated, tick being 1/60th of a sec and that's a distance estimate resolution of up to 5-10 km. And that's when cos and sin came back to be useful because you can do intersection of circles and figure out where to do a sea rescue more precisely. So video games, trig. Who would've thought?

And I even remember what it means:

SOS: sine = opposing side divided by diagonal (schuine) side

CAS: cosine = adjacent divided by diagonal

TOA: tan = opposing divided by adjacent.

I don't think I've ever used it for anything practical, but I can still reproduce it after all this time (I'm 47 now).

Just the other day I wanted to compute viewing angle and did it by hand even though there are plenty of calculators like this out there:

http://www.hometheaterengineering.com/viewingdistancecalcula...

I’d say I use it for something practical/random like that a few times a year?

Another example was placing some ceiling speakers whose tweeters had a 15° angle so that they were pointed directly at a seating position below. How far did I need to place them in front of the seating position from directly overhead.

I would guess any sort of construction you’re using it fairly often.

What I remember from trig is to draw a unit circle. Most of the rest falls out of that.

Statistics is also useful and applicable to everyday life, but I didn’t learn that till college as best I can recall.

I don’t regret having spent time learning calc, or physics or chemistry or biology for that matter. If you asked me to come up with a curriculum I’d have a really hard time prioritizing. Maybe the one thing I’d like to see kids learn better is how to be self-directed learners. I’m still fairly surprised at the number of colleagues I have who seem unable to problem solve and figure something the fuck out. Even knowing when and how to ask for help.

Perpendicular/hypotanuse = Sin

Base / Hypotanuse: CoSin

Perpendicular / Base: Tan

edit: try to fix the HN god awful formatting

Sine: Opposite over Hypotenuse

Cosine: Adjacent over Hypotenuse

Tangent: Opposite over Adjacent

There are even other bits that I was shown, never incorporated into my brain at all, but later recognized as truly important (Taylor series expansions, the central limit theorem, the prime number theorem, etc).

I do recall taking a "probability in CS" as an electrical engineering student -- it was pretty mind-blowing to me the extent to which the CS students did not like to talk about any continuous math. It makes sense, though, these are different specialties after all.

> Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

> Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

and so on. However, in practice, this means that students need to be able to answer "C" when presented with the question:

> One radian is:

> A) Another word for degree.

> B) Half the diameter.

> C) The angle subtended on a unit circle by an arc of length 1.

> D) Equal to the square root of 2.

A surprising number of students can get through without ever really comprehending what a radian is. They might just choose the longest answer (which works way too often), identify trick answers and obviously wrong answers, and eventually guess the teacher's password from a lineup by association of the word salad of "radians" and "subtended."

They might not even have a clue what the word "subtended" means, but they know it's got something to do with "radian" and that's enough. It is more important for the school that the students answer (C) than that they understand what a radian is.

[0] https://web.archive.org/web/20220112000314/http://www.corest...

This happens with very basic things like human languages. Bilingual people can forget an entire secondary language if they don’t use it for a decade+.

Derivative of sin(x) is cos(x). Many people probably think this works for degrees, but it's actually some abomination like pi cos(pi x/180)/180.

Of course, turns are very reasonable units sometimes for sure.

That's what it would be if you are using sin in degrees and cos in radians. But if you are using degrees for both then the derivative of sin(x) is pi/180 cos(x).

However, I still remember that "eureka!" moment of realizing that radians were special, that the small angle approximation of sin(x) = x, and many related math rules, work only when x is expressed in radians. I guess that's a credit to my math teacher, who basically led the class in deriving mathematical formulas rather than just presenting them to us.

I think the article is still valid and interesting, as "turns" in some use cases might improve performance and accuracy. But radians aren't at all "arbitrary" - if we ever encounter technologically advanced aliens, they certainly won't use degrees, but they will understand radians.

First, I would be cautious about suspecting someone of Casey Muratori's calibre didn't consider something just because he didn't directly addressed it.

Second, the choice of unit is kind of arbitrary, even if the unit itself is not. Radiants are nice because the length of a 1 radiant arc is the same as the length of the radius. But turns are also nice because angles expressed in turns are congruent modulo 1 instead of modulo 2π.

Third, he talks in the context of video games. Such games use code, that have to be read by humans and executed by the CPU. And that's the main point of his article: in this context, expressing stuff in terms of (half) turns reduces the amount of code you have to write & read, reduces the number of multiplications & divisions the CPU has to make, and makes some common operations exact where they were previously approximated.

Do we even care at this point whether the definition of radians is arbitrary or not? I love the elegance of radiants, but for game engine code I'm willing to accept they're just the wrong unit for the job.

This is by convention, but has been and is still being debated because calling it dimensionless causes some problems. https://en.wikipedia.org/wiki/Radian#Dimensional_analysis

Furthermore, the whole reason to treat radians as dimensionless, the problem, is with angles, not with radians specifically. Degrees are also considered dimensionless. So, a turn could be treated as dimensionless too, with a conversion constant to radians & degrees, just like between degrees and radians.

Of course, the declared dimensionlessness of angles like radians isn’t something generally discussed in pre-college trig courses, that’s a subtle subject that matters more in physics. In my high school trig, we all understood radians to be a unit of angle and never pondered whether angles had dimension.

Also subtle point, but dimensionless doesn’t mean unitless. It’s another separate convention to drop the units when working with radians.

It varies by school, but overall I think this prediction is incorrect. Trigonometry was an important subject in high school — for all of the math, physics, and possibly chemistry courses — and then if you take calculus in university, it's very, very important to learn trigonometry well (or you'll really struggle as a student).

So, even on the off-chance that trigonometry is not taught in high school (which I predict is rare), a first-year student taking calculus in university must learn it on their own time. Good calculus textbooks (e.g. Thomas Calculus) even account for this, having fairly comprehensive textbook sections on what you need to know about trigonometry to succeed in the calculus course.

Most students who therefore took math to pre-calculus or calculus (or physics and possibly chemistry), should therefore have a good exposure to the definition of the radian.

Every couple of years i try to get some higher math education, but nothing makes sense. It's one of the reasons i [think] i suck at programming - i should note that another reason is i first learned BASIC, then qbasic, then fortran, and then C never made sense to me. At least i can putter around with python and R.

however i can do "basic" math things that generally everyone else has to dig out a calculator app for in my head, percentages, fractions, moving decimals, "making change". Since i suck at higher math, i'm only able to help my kids with basic math, and i try to ensure that they know it fairly well.

I had my fair share of higher math, but C never made really sense to me either. It's not us, it's C that's to blame.

- '=' is for equality only

- assignment is ':=' which is the next best symbol you can find in math for that purpose

- numeric data types are 'integer' and 'real', no single/double nonsense

- 'functions' are for returning values, 'procedures' for side effects

- Function and procedure definitions can be nested. I can't tell you what shock it was for me to discover that's not a thing in C.

- There is a native 'set' type

- It has product types (records) and sum types (variants).

- Range Types! Love'em! You need a number between 0 and 360? You can easily express that in Pascal's type system.

- Array indexing is your choice. Start at 0? Start at 1? Start at 100? It's up to you.

- To switch between call-by-value and call-by -reference all you have to do is change your function/procedure signature. No changes at the call sites or inside the function/procedure body. Another bummer for me when I learned C.

Pascal wasn't perfect but I really wish modern languages had syntax based on Wirth's languages instead of being based on BCPL, B and C.

I learned a lot from the Art of Problem Solving book series because they're highly focused on the reader solving problems to learn, versus giving explanations. Even if you don't finish all of it, you can strengthen any problem areas.

For a less-comprehensive but still great introduction to precalculus (with a great section on trigonometry in particular from memory), Simmons' Precalculus in a Nutshell has a great introduction to this. Then you can read a book like Thomas Calculus, which has a great introduction to trigonometry in the first review chapter.

I would even say that you would be better off working through the books above than if you had the high school classes; the best math students probably took the same approach too (working through books instead of focusing just on the class material). The main obstacle is time, because it's hard to find time when you have work and children to take care of.

Basic trig is taught in middle school, but exclusively using degrees. Advanced trig is optional in high school if you take the "hard math" track.

This depends on the state. NYS absolutely requires everyone to learn "advanced trig" in high school.

I don't think every high school student can master advanced trig.

It's been a while, but I used to have an argument that rad should be a unit. This even plays well in physics where it allows torque to not have the same units as a joule.

That discussion has to do with the failure of SI to notate the directions of vectors. When it's torque, the N and the m are at a right angle. When it's work, they are both in the same direction.

Well if radian is a unit then torque become Nm/r and is no longer Nm like energy. Then when multiplied by an angle in radians you get energy. It was *something like that*.

Though most people haven't used any of that since college and so don't know it very well anymore. I smelled BS when I read the blog, but couldn't put my finger on why - the comment you replied to explained what I knew was the case but couldn't remember.

    s/per-calculus/pre-calculus/g

Education quality and quantity vary greatly across the country. Many schools don't require trig at all or lump it in with other classes. I memorized SOH CAH TOA and brute forced a CLEP test (the state of MN is required to allow you to test out of classes and to write a test if one doesn't exist; usually AP and CLEP tests are accepted, and they don't count for/against your GPA).

It's also culturally accepted to "be bad at math," with undertones of defeat and that it's the world doing that to you and not something you can change (maybe the blame lies elsewhere like with how math is taught as a sequence of dependencies and bombing one course makes the rest substantially more difficult). I don't know how many people scrape by a D in trig and subsequently forget it all, but I'd wager it's a lot.

A base measurement unit is a unit that is chosen arbitrarily.

A derived measurement unit is one that is determined from the base units by using some relationship between the physical quantity that is measured and the physical quantities for which base units have been chosen.

While there are constraints for the possible choices, the division of the units into base units and derived units is a matter of convention.

Whenever there are relationships between physical quantities where so-called universal constants appear, you can decide that the universal constant must be equal to one and that it shall be no longer written, in which case some base unit becomes a derived unit by using that relationship.

The reverse is also possible, by adding a constant to a relationship, you can then modify its value from 1 to an arbitrary value, which will cause a derived unit to become a base unit for which you can choose whatever unit you like, e.g. a foot or a gallon, adjusting correspondingly the constant from the relationship.

There are 3 mathematical quantities that appear frequently in physics, logarithms, plane angles and solid angles (corresponding to the 1-dimensional space, 2-dimensional space and 3-dimensional space). All 3 enter in a large number of relationships between physical quantities, exactly like any physical quantity.

For each of these 3 quantities it is possible to choose a completely arbitrary measurement unit. Like for any other quantities, the value of a logarithm, plane angle or solid angle will be a multiple of the chosen base unit.

For logarithms, the 3 main choices for a measurement unit are the Neper (corresponding to the hyperbolic a.k.a. natural logarithms), the octave (corresponding to the binary logarithms) and the decade (corrsponding to decimal logarithms).

Like for any physical quantities, converting between logarithms expressed in different measurement units, e.g. between natural logarithms and binary logarithms is done by a multiplication or division with the ratio between their measurement units.

The same happens for the plane angle and the solid angle, for which arbitrary base units can be chosen.

What has confused the physicists is that while for physical quantities like the length, choosing a base unit was done by choosing a physical object, e.g. a platinum ruler, and declaring its length as the unit, for the 3 mathematical quantities the choice of a unit is made by a convention unrelated to a physical artifact.

Nevertheless, the choices of base units for these 3 quantities have the same consequences as the choices of any other base quantities for the values of any other quantities.

Whenever you change the value of a measurement unit you obtain a new system of units and all the values of the quantities expressed in the old system of units must be converted to be correct in the new system of units.

The fact that the plane angle is not usually written in the dimensional equations of the physical quantities in the International System of Units, because of the wrong claim that it is an "adimensional" quantity, is extremely unfortunate.

(To say that the plane angle is adimensional because it is a ratio between arc length and radius length is a serious logical error. You can equally well define the plane angle to be the ratio between the arc length and the length of the arc corresponding to a right angle, which results in a different plane angle unit. In reality the value of a plane angle expressed in radians is the ratio between the measured angle and the unit angle. The radian unit angle is defined as an angle where the corresponding arc length equals the radius length. In general, the values of any physical quantity are adimensional, because they are the ratio between 2 quantities of the same kind, the measured quantity and its unit of measurement. The physical quantities themselves and their units are dimensional.)

In reality, the correct dimensional equations for a very large number of physical quantities, much larger than expected at the first glance, contain the plane angle. If the unit for the plane angle is changed, then a lot of kinds of physical quantity values must be converted.

To add to the confusion, in practice several base units of the 3 mathematical quantities are used simultaneously, so the International System of Units as actually used is not coherent. E.g. the frequency and the angular velocity are measured in both Hertz and radian per second, the rate of an exponential decay can be expressed using the decay constant (corresponding to Nepers) or by the half-life (corresponding to octaves), and so on.

A clear sign of how wrong people can be about the naturalness of units is Avogadro's constant which was recently demoted from a measured value to an exact arbitrary value. Chemists often believe that N_A, moles, atomic mass units, etc. are all somehow important or fundamental and don't realize that it's all based on a needlessly complicated constant with an (until 2019) needlessly complicated definition that could have just been a simple power of 10 if history had gone differently. Luckily the people defining SI have finally moved away from the old two independent mass units to just the kg that can now be exactly converted to atomic mass units by definition.

While it might be something you are now realizing, the US is not a single entity in many ways. Rather, it's some 50 states that form a country. Each state has it's own laws and ways of doing things. While there are many similar ways of doing things, none are exactly the same. On top of that, even within the state you'll have different school systems with different policies.

And we aren't even going to discuss going to American schools in Europe.

I never really learned trigonometry until I started doing game programming in my spare time when suddenly that knowledge and linear algebra became necessary to understand. They only way I learned it was by needing to know it.

In fact, I regularly forget knowledge I don’t need to know. The stuff I do need to know remains fresh in my mind.

Which is why it's important to add the unit after the measurement. If someone tells you an angle measures 1, can you tell whether it's 1/360 of a revolution or the angle that would be formed by traveling along a circumference a distance equal to the radius of the circle?

Is the equivalent set available for Martian Nav (MCMF/MCI, I guess), or do you have different/specialized/etc. frames based on something unique to Mars.

For the rover, we're pretty much always dealing in local coordinate systems based on reference frames defined using the rover's observations of the sun and alignment of local imagery with orbital imagery. The two frames used most frequently are called RNAV (centered on the rover) and SITE (centered on where we last did a sun observation). But then there is a tree of frame transformations for knowing the location and orientation of each part of the rover with a lot of named frames (especially important for operating the robotic arm, which I also do).

To sum it up: "Computer Science has nothing to do with Math!" ;)

A unit that could be inherently defined by math itself and not a farmer looking at their hands and feet? Preposterous!

Where do you think base ten comes from?

Imagine using an angle measurement for physical things such that angles with rational measurements stack to a full circle. Preposterous.

You probably take out more scaling factors than you introduce.

> Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians.

Euler’s still simple:

e^(2 i pi y) = cosy + isiny

Or if you start noticing c = e^(2 pi) showing up all over the place:

c^iy = cosy + isiny

> How do you do the same with "turns" on a sphere?… You can't in any meaningful way.

Why not do the same thing? One steradian is 1/(4 pi) of a sphere’s solid angle. What if one “steturn” or whatever just covered a full solid angle? And similarly for higher dimensions?

Neither definition seems more natural to me, especially being used to all the factors of 2 and pi that pop over all over the place in the status quo.

The former are R->R functions, while the latter are defined on Angles (Angle is unfortunately not an SI physical dimension yet, but I expect it soon to change), and they don't care about the measurement unit.

I have no idea what you mean by radians generalizing for higher dimensions, but not turns.

I guess that turns interpreted as parts of whole circles generalize to parts of whole spheres, and you should divide by 4pi instead of 2pi???

sin(x) is precisely the unique function f(x) such that f''(x) = -f(x). Similar to how exp(x) is the unique function g(x) such that g'(x) = g(x).

Sine does not operate on 'angles measured in radians'. It operates on real numbers. It is zero whenever the real number passed in is a multiple of pi. It happens to have applications in relating angles to distances in circles and triangles, and in order to use sine in that context it is useful to introduce the concept of a 'radian' as a specific, constructed angle of a particular size, such that when you express an angle in terms of multiples of a radian, you can just use the sine function to generate useful values.

If we used turns for cos and sin we could redefine what e^ix means so it works without radians. From the other answer I guess this is completely wrong...

(I do understand it is nuts to redefine, i'm just interested as a theoretical thought)

Now, how is Eulers formula is deduced? How did we figure out what e^ix means?

There are a couple other "paths" to this result, and the choice we have is by far the most elegant.

Step 2: sell out your country to wealthy foreigners to "stimulate the economy"

Step 3: act surprised that you can't afford living in your country, that got sold out to wealthy foreigners

Could you explain to me why I'm unhinged? I didn't feel like I had posted anything controversial or whatever.

I believe I understand why you feel I am unhinged.

I said, "Sure sounds like iphone will get there first, pine could beat them to it."

Obviously I said iphone will win, I optimistically or enthusiastically said pine could put in the work and get there first.

You feel I am unhinged because of optimism?

For me. My love for webOS is so great it’s all I want

I feel like I wasn't being unrealistic at all.

solid state batteries exist today and can be bought. Obviously early in the commercialization but they do exist. Still unclear how safe they are.

88mp camera is less than what I can go buy from costco right now.

3d scanners totally fit in your pocket. I don't mean photogrammetry neither. The newer iphones have lidar that can scan. Newer androids have depth sensors or TOF sensors.

https://www.youtube.com/watch?v=r26OhSxBUXM

https://www.youtube.com/watch?v=_3UXeJWmEn8