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Attached is a real worked example of the need for the Angle unit, extracted from todays' work.
I was computing the effect amount of extra angle of attack that will occur when an aircraft starts a "rate one turn", as this extra angle could affect where my camera is pointing. We can see the amount of banking that the plane does for a ROT, and the calculated extra angle of attack. There are references to the relevant wiki pages.
I have attached a V14 xmcdz file and the pdf of the same for all to consider.
Philip
Interesting! In the last line you "inadvertently" multiplied by CI instead of dividing by CI, but Mathcad didn't give any indication of this because it thought the result was dimensionless. If it had a special unit for angle the user would have been aware of the mistake immediately. As it was, "the user" inserted 'deg', thinking that the dimensionless number displayed was an angle. I now understand why you are so keen on a special unit for angle!
Alan
Alan Stevens wrote:
Interesting! In the last line you "inadvertently" multiplied by CI instead of dividing by CI, but Mathcad didn't give any indication of this because it thought the result was dimensionless. If it had a special unit for angle the user would have been aware of the mistake immediately. As it was, "the user" inserted 'deg', thinking that the dimensionless number displayed was an angle. I now understand why you are so keen on a special unit for angle!
Alan
The example is ceratinly a good one. I hadn't noticed my mistake and all afternoon I'd been saying that the bank angle and the extra angle of attack were similar, then, late in the day, I got to thinking again and felt that then AoA was a bit larger than experience showed, so I looked again and spotted the correction (as Alan correctly describes) and a far smaller 0.5 degrees comes out as the answer. It just shows what happens wihen 'believable numbers" turn up.
Doing a lot of electro optics one finds many equations that have per radian or even per steradian [with its units of Angle^2 ! ]
I have seen quite senior folk simply add in a "divide by 2pi" just to make the numbers feel right, when they had completely forgotten some part of the (angular) inputs.
The best story is of a rotary encoder with let's say 12 bit encoding. The software engineers read the number as 0 to 1 and claimed it was in radians because it was an angle (well that is what the documentation said). The system engineer then had to add a magic 2pi to the system equations to take out the conversion error. It was simpler than trying to change the documenation!
Philip,
You present a valid (and interesting) point. One of my first reactions to this, however, was that many equations use angles as dimensionless variables. For instance, the length of a circular arc is calculated by the radius of the circle multiplied by the angle through which the arc sweeps. You would not want this equation to produce a result with units of Angle * Length, you would just want a unit of Length. Therefore, this equation would need to be written as ArcLength = Radius * (Angle / rad). I could see this as being very confusing for many users (it's certainly not how I would expect to have to set up this equation).
Not making any conclusions, just poiting out that if an Angle unit is added, it would require some good planning on PTC's part, and likely some adjustment for many users.
-Mark
Mark,
If you have a look at the proposal I made in the document thread http://communities.ptc.com/message/152971 (Sep 23, 2010 1:02 PM) and its attachment, you will see that I have carefully considered the various cases where these ignored (hidden) mathematical subtleties should be covered with just 4 extra functions, such as "arc".
It is a worthwhile read. Some of the stuff about complex exponentials and branch cuts and stuff may be too much for some, but all the basics are covered.
Philip
PS I've copied the document below, just in case (I hope its the same version ;-).
Tom writes some good stuff, but in general I disagree with his final points about needing two different sets of functions, with a special one for mathematicians....
However from a pragmatic point I had already proposed a set of options for such deserving cases and actually ended up with four special cases (that is, four options for use / non-use of angle unit enforcement).
So while I disagree with Tom wanting two variants, I (hypocritically) allow four, but only as options.
Philip
I think two options are needed, from a purely practical point of view. We agree that for some applications keeping track of units for angles and solid angles is necessary. But people are sometimes forced to submit reports that adhere to a standard, so it needs to be possible to turn them off.
I regret that I see little to change to my stance on this issue. The angle should be a formal part of the SI, unit radian, but it should be recognized that this will necessitate significant revision of standard standard equations and ways of conducting dimensional analysis. The primary objections to having angle as a unit are based on the definition of angle as a ratio of 2 lengths and the problem with handling anything involving mixed powers of angle or non-linear functions of angles.
Message was edited by: Stuart Bruff, twice to try and put space between paragraphs but they keep getting deleted!!! The attached rtf file is slight better laid out
Stuart,
Yes the editor interface is diabolical. I think SI will have an Angle 'dimension' before PTC manages to correct it (at this rate )
I personally disagree that Angle is defined as a ratio of two lengths. A right angle is well defined without any lengths..... The choice of the unit of angle is more of the problem than is discussed. That is, it is because every body appears to assume that the radian defines angle, rather than the the causality being other way around, we have Angular measure as the clear independent dimension, and then we argue about what base unit to use, just like imperial vs metric. In this case we have that the metricators would like to use radians, but all the imperialists still want to use degrees and right angles (90 degrees) as their base unit. And what a mess it causes, with every calculator having different defaults and different methods of changing between them.
Everyone knows sin(30)=1/2 !!! even I can draw that equilateral triangle
There is a potential problem with lengths of arc. The problem being you can't measure them with a ruler !! but no one mentions that in the formulae, which Is why I proposed, for mathcad the arc(r,theta) function. That is, there is no "transformation constant", rather there is a "transformation function" (using your phrase for 'k')
the y = r.sin(theta); theta in angle, r and y in units of length, has no problems at all in my way of thinking. When theta arrives at the sin function, it has a number and a unit quantity. The number is divided by the unity quantity, measured in an appropriate scale (e.g. fraction of a right angle), and then numerically calculated by say the CORDIC algorithm to produce a numeric value for sin(theta), which is then multiplied into r to get the length y.
I've deliberately avoided both the classic sin(x)=x-x^3/3!+.. formula and the use of radians, just to show that they aren't necessary.
Another diversion is that most pictures of the sin() triangle I would argue as figuratively "wrong", in that the common divisor should be along the x-axis, that is the Hypotenuse of the constructed triangle (yes the right triangle is constructed after the angle : see causality) is along the x-axis. At this point you get to choose how to do the sin(2.theta) vs sin(theta/2), because one can construct the semicircle over the angle to get all sorts of angular doubling formulae..
Have fun.
Philip
Philip Oakley wrote:
I personally disagree that Angle is defined as a ratio of two lengths. A right angle is well defined without any lengths..... The choice of the unit of angle is more of the problem than is discussed. That is, it is because every body appears to assume that the radian defines angle, rather than the the causality being other way around, we have Angular measure as the clear independent dimension, and then we argue about what base unit to use, just like imperial vs metric. In this case we have that the metricators would like to use radians, but all the imperialists still want to use degrees and right angles (90 degrees) as their base unit. And what a mess it causes, with every calculator having different defaults and different methods of changing between them.
I don't have a problem with that; my proposal handles it easily.
There is a potential problem with lengths of arc. The problem being you can't measure them with a ruler !! but no one mentions that in the formulae, which Is why I proposed, for mathcad the arc(r,theta) function. That is, there is no "transformation constant", rather there is a "transformation function" (using your phrase for 'k')
Well, you can measure them with a ruler ... if you cheat and use a bendy ruler sideways
However, the problem I have with most other proposals (sorry) is that they end up doing something 'special' to handle different circumstances, thus reducing the consistency and presenting potential problems with the theoretical underpinning. My proposal, AFAICT, applies consistently across all dimensions. If you want to define an arc function, fine - I often find it useful to define functions for things that can be done otherwise. In this instance, I believe that one should be able to simply write c = k . r . θ.
the y = r.sin(theta); theta in angle, r and y in units of length, has no problems at all in my way of thinking. When theta arrives at the sin function, it has a number and a unit quantity. The number is divided by the unity quantity, measured in an appropriate scale (e.g. fraction of a right angle), and then numerically calculated by say the CORDIC algorithm to produce a numeric value for sin(theta), which is then multiplied into r to get the length y.
Once again, I wouldn't treat angle any differently to any other dimension. sin(θ), sin(x) and sin(t), where θ=angle, x=length and t=time, are all handled exactly the same, giving results of angle, length and time respectively. My proposal avoids treating angle as special and doesn't require division by anything to get a dimensionless number. y = r . sin(θ) gets rewritten as y = k . r . sin(θ) as above and gives a dimensionally consistent answer, et sim for y = k . a . sin(t), except that k now has dimension 1/time (or, as stated, it might be conceptually more accurate to let k = length/time and a = length/length).
I've deliberately avoided both the classic sin(x)=x-x^3/3!+.. formula and the use of radians, just to show that they aren't necessary.
Tom and I have addressed this - the proper derivation of such formulae shows that there is a conversion constant of appropriate dimension for each terms that makes the expression dimensionally consistent - think of the standard expression for distance covered under constant acceleration, s = s0 + v0.t +1/2 a. t2, where s0=length, v0=length/time and a=length/time2; this gives additional justification for allowing sin(t) without any attempt to provide spurious conversions from time to angle.
These are interesting reads. I've only skimmed them so far, but I think you may be on to something here. I'd also be interested to see if anyone has given a coherent argument against the addition of an angle unit. Any takers out there?
I'd also be interested to see if anyone has given a coherent argument against the addition of an angle unit.
You have no idea what you just asked!
We have had many heated discussions about units, including angular units. I'll refrain from calling them arguments, although a lot could accurately be described that way. I'll leave you to decide if they are coherent! Here's a nice juicy one to start you off:
http://communities.ptc.com/message/52193#52193
Once you have plowed through the 370 posts in that thread (no, that number was not a typo) you can find lots more material by searching these forums for combinations of "angle" "unit","Hz", "radian".
I'd also be interested to see if anyone has given a coherent argument against the addition of an angle unit. Any takers out there?
Jean would have had you for asking that!!!!!
Mike
I don't think there ever was a coherent argument against, since everyone is free to not use any units at all, as Jean espoused. Unfortunately, he was adamant about using ANY units at all, and very vociferous about it, which detracted from his otherwise mostly helpful persona.
See http://stacks.iop.org/Met/47/R41 available for review (pdf) for a few more days! It covers many of the viewpoints, and names most.
Also http://www.ifi.unicamp.br/physicae/ojs-2.1.1/index.php/physicae/article/viewFile/64/46 that shows that theta is an ANGLE inside sin(theta), and that the sine function (non-linear) converts an input of ANGLE to an output that Dimensionless.
Philip
Philip Oakley wrote:
See http://stacks.iop.org/Met/47/R41 available for review (pdf) for a few more days! It covers many of the viewpoints, and names most.
Also http://www.ifi.unicamp.br/physicae/ojs-2.1.1/index.php/physicae/article/viewFile/64/46 that shows that theta is an ANGLE inside sin(theta), and that the sine function (non-linear) converts an input of ANGLE to an output that Dimensionless.
This is one of the aspects of handling angles that I disagree with. Quantities should be handled consistently and as independently of their exact nature as possible. Expressions involving trig functions and other quantities are very common and it is not always obvious why angles have to be introduced. The physics of a given situation often doesn't directly support the use of angles, and the flow of quantities through a set of calculations can be broken by the need to make expressions dimensionless. As I've said before, sin(time) should be handled in the same way as sin(angle), and this necessitates doing away with special treatment of angles - they're just this, like, quantity, man.
Mark Gase wrote:
These are interesting reads. I've only skimmed them so far, but I think you may be on to something here. I'd also be interested to see if anyone has given a coherent argument against the addition of an angle unit. Any takers out there?
Ok then.
If there is an angle unit, and exp(i.theta) is valid for cos(theta)+i.sin(theta), then theta is an angle.
Now we can also have exp(sigma+i.theta) = exp(sigma).exp(i.theta) = Amplitude.exp(i.theta).
But sigma = Neper(Amplitude) i.e. the Neper is logarithm of the Amplitude (ratio).
However we need units consitency. so the units of sigma and of theta must be the same.
That is the unit and dimensions of the Neper MUST BE in Angle and radians.
JDGT [Just don't go there, rather than QED...]
What it actually shows is that we have truly changed from one measurement space to another, from the cartesian world to the helix world. Just like shifting from mass to time to ...
Philip
Ther are other arguments from the "realists" that say no to units[dimensions], whilst we "Systematists" want the consistency of dimensions, Mathcad being a key exemplar of a software tool the uses them.
Philip Oakley wrote:
Mark Gase wrote:
These are interesting reads. I've only skimmed them so far, but I think you may be on to something here. I'd also be interested to see if anyone has given a coherent argument against the addition of an angle unit. Any takers out there?
Ok then.
If there is an angle unit, and exp(i.theta) is valid for cos(theta)+i.sin(theta), then theta is an angle.
Now we can also have exp(sigma+i.theta) = exp(sigma).exp(i.theta) = Amplitude.exp(i.theta).
But sigma = Neper(Amplitude) i.e. the Neper is logarithm of the Amplitude (ratio).
However we need units consitency. so the units of sigma and of theta must be the same.
That is the unit and dimensions of the Neper MUST BE in Angle and radians.
JDGT [Just don't go there, rather than QED...]
What it actually shows is that we have truly changed from one measurement space to another, from the cartesian world to the helix world. Just like shifting from mass to time to ...
Philip
Ther are other arguments from the "realists" that say no to units[dimensions], whilst we "Systematists" want the consistency of dimensions, Mathcad being a key exemplar of a software tool the uses them.
I kind of disagree with this. An amplitude is not measured in nepers; specifically, a neper is the ratio of two field levels - as you've mentioned parenthetically, it is a ratio. As you say, if theta is an angle, then sigma must be an angle for dimensional consistency. The 'correct' expansion, I hypothesize, is:
exp(sigma+j.theta) = exp({sigma}[angle]+j.{theta}[angle])
= exp({sigma}+j.{theta})[angle]
= (exp({sigma}).exp(j.{theta})[angle]
where {} & [] indicates the numerical value & unit of a quantity, respectively.
The above allows replacement of angle by any arbitrary quantity and does not require the neper. Any quantity conversions should be handled explicitly.
Hi Stuart,
Yep, the Neper is a ratio measure, so I was being a bit naughty, but then I did hint that in the updated title..
I disagree that one can pass a unit/dimenion through the equation to the outside. One can't take the sin(3.ft) and say it is sin(3).ft Even sin(3) isn't actually well defined (except when in a cliche of mathematicians..). The modern convention has become to assume that it should be in radians, but that isn't necessary. It must simply be an angle.
The review paper by Marcus Foster is pretty good in covering many areas, and highlighting the different approaches to the problems.
Philip
Stuart_Bruff wrote:
I kind of disagree with this. An amplitude is not measured in nepers; specifically, a neper is the ratio of two field levels - as you've mentioned parenthetically, it is a ratio. As you say, if theta is an angle, then sigma must be an angle for dimensional consistency. The 'correct' expansion, I hypothesize, is:exp(sigma+j.theta) = exp({sigma}[angle]+j.{theta}[angle])
= exp({sigma}+j.{theta})[angle]
= (exp({sigma}).exp(j.{theta})[angle]
where {} & [] indicates the numerical value & unit of a quantity, respectively.
The above allows replacement of angle by any arbitrary quantity and does not require the neper. Any quantity conversions should be handled explicitly.
Philip Oakley wrote:
Yep, the Neper is a ratio measure, so I was being a bit naughty, but then I did hint that in the updated title..
I disagree that one can pass a unit/dimenion through the equation to the outside. One can't take the sin(3.ft) and say it is sin(3).ft Even sin(3) isn't actually well defined (except when in a cliche of mathematicians..). The modern convention has become to assume that it should be in radians, but that isn't necessary. It must simply be an angle.
Stuart_Bruff wrote:
I kind of disagree with this. An amplitude is not measured in nepers; specifically, a neper is the ratio of two field levels - as you've mentioned parenthetically, it is a ratio. As you say, if theta is an angle, then sigma must be an angle for dimensional consistency. The 'correct' expansion, I hypothesize, is:exp(sigma+j.theta) = exp({sigma}[angle]+j.{theta}[angle])
= exp({sigma}+j.{theta})[angle]
= (exp({sigma}).exp(j.{theta})[angle]
where {} & [] indicates the numerical value & unit of a quantity, respectively.
The above allows replacement of angle by any arbitrary quantity and does not require the neper. Any quantity conversions should be handled explicitly.
As stated, I see no theoretical reason to disallow sin(3.ft) - it merely needs care in defining exactly what the expression means. I'm not sure what you mean by sin(3) not being well defined; AFAICT, as a non-quantified number, it is well defined and evaluates according to the standard expansion. Tom has drawn the distinction between the 'mathematical' function and the 'physical' one, but I believe that this doesn't really address the core of the issue, which is that there should be a consistent mathematical structure that allows continuous flow of numerical quantities through equations - after all, nature manages it 🙂
Stuart
Hi Stuart,
Stuart_Bruff wrote:
As stated, I see no theoretical reason to disallow sin(3.ft) - it merely needs care in defining exactly what the expression means. I'm not sure what you mean by sin(3) not being well defined;
AFAICT, as a non-quantified number, it is well defined and evaluates according to the standard expansion.
Tom has drawn the distinction between the 'mathematical' function and the 'physical' one, but I believe that this doesn't really address the core of the issue, which is
that there should be a consistent mathematical structure that allows continuous flow of numerical quantities through equations - after all, nature manages it 🙂
Stuart
I've split your response into what I think are the four relevant parts, so we can be clear about each.
1. It is my contention that "sin" is a mechanism for taking in an angle and from it determining the ratio now attributed to it. In this(my) context, it must have an angle given to it. It has a non-linear response, that is if the angle is doubled, the response isn't twice the previous value.
2. The problem (I see) is the association of that "standard expression" (which is OK as a stand alone mathematical expression) with my "sin of angle" context above(#1). [Note that the standard expression is non-linear]
3. I understood that Tom had the two separate function, which he treated as distinct and separate. This side stepped the difficulties - one foot in each camp.
4. "Nature abhors mathematics, simply getting on with it" rather mathematics is our abstraction of the situation. For the units& dimensions arguments to work as we systematists desire, we need the summed dimensional components to have a common dimensional power (Buckingham's Pi theorem). This makes the "standard expression" unsuitable for the purpose of evaluating sin(X), rather the standard expression needs to take x=X/rad (I used big and little X for the distinction between with and without the Angle dimension)
Philip
Philip Oakley wrote:
Hi Stuart,
Stuart_Bruff wrote:
As stated, I see no theoretical reason to disallow sin(3.ft) - it merely needs care in defining exactly what the expression means. I'm not sure what you mean by sin(3) not being well defined;
AFAICT, as a non-quantified number, it is well defined and evaluates according to the standard expansion.
Tom has drawn the distinction between the 'mathematical' function and the 'physical' one, but I believe that this doesn't really address the core of the issue, which is
that there should be a consistent mathematical structure that allows continuous flow of numerical quantities through equations - after all, nature manages it 🙂
Stuart
I've split your response into what I think are the four relevant parts, so we can be clear about each.
1. It is my contention that "sin" is a mechanism for taking in an angle and from it determining the ratio now attributed to it. In this(my) context, it must have an angle given to it. It has a non-linear response, that is if the angle is doubled, the response isn't twice the previous value.
2. The problem (I see) is the association of that "standard expression" (which is OK as a stand alone mathematical expression) with my "sin of angle" context above(#1). [Note that the standard expression is non-linear]
3. I understood that Tom had the two separate function, which he treated as distinct and separate. This side stepped the difficulties - one foot in each camp.
4. "Nature abhors mathematics, simply getting on with it"
rather mathematics is our abstraction of the situation. For the units& dimensions arguments to work as we systematists desire, we need the summed dimensional components to have a common dimensional power (Buckingham's Pi theorem). This makes the "standard expression" unsuitable for the purpose of evaluating sin(X), rather the standard expression needs to take x=X/rad (I used big and little X for the distinction between with and without the Angle dimension)
Philip
Unfortunately, I don't believe that state and religion (physics and mathematics) can be separated so readily. As I've stated somewhere else, I think that the 'necessity' to have distinct 'physical' and 'mathematical' functions to deal with units, is more an indication that something is rotten in Blegdamsvej 17,
2100 Copenhagen than a reflection of the 'reality' (at least, that's my interpretation).
Consider the following (see attached mcad file).
Take a pair of variables, x & y say, in any old units you like, so I'll go for dimension length with unit the metre. Find a blackboard (if you can), and draw a line y = a.x, where a is a constant. Most people, I think, would be prepared to admit that a was either dimensionless or had units m/m. This is interesting, but a little bit predictable. Let's make it slightly more interesting by plotting y = a.x - b.x3/6. Clearly, to be dimensionally consistent, b must have units of 1/m2 (or m/m3). Now, I go into a wild frenzy of series extension, plotting y = a.x - b.x3/6 + c.x5/120 (c having units of 1/m4 (or m/m5)) and drafting in the infinity of myselves in the parallel universes to create an infinite dimensionally-consistent series. As it happens, I set each constant to have numerical value 1; a few seconds cogitation reveals that the resultant curve looks suspiciously like a sine, and a little mathematical prestidigitation allows me to rearrange the final equation as y = sin({x}) m (where {x} is the numerical value of the quantity x). Further reflection, indicates that as my choice of unit, [x], was arbitrary, I could have chosen [x] = V, s, or even rad.
But here's an interesting thing, if [x] was rad then, using your's and Tom's choice, I have sin({x}) rad = sin({x} rad) => sin(x).
However, given that the choice of unit is arbitrary, I have to ask what's so special about the radian that it is allowed as a unit in sin(x) whereas the metre, second, volt, etc aren't? And the answer,AFAICT, is nothing. The nature of our universe is such that physically rotating an object through 360 degrees in 2D results in a physically superposition of that object with itself. This might imply that the radian is special because it's effective domain is 2.pi. However, the total angle is important in our 3D universe, eg if you start turning round on one spot, the fluid in your ears doesn't stop when you pass through 0 degrees again and one complete turn of a winch doesn't unwrap the rope. This makes it physically meaningful to talk about angles in excess of 2.pi.
The apparent problem with attaching a meaning to sin(3 m) is dealt with by noting that the bald statement implies a cyclic period of 2.pi m in the same way that sin(3 rad) implies a period of 2.pi rad. A more general solution for any periodic function allows the user to specify the periodicity given any set of units. Indeed, many of us are introduced to sin (x) where [x]=degree long before we meet the radian. See the attached worksheet for examples, which includes sin (x degrees). Note (changing notation slightly) that this means a rewrite of the standard equations to obtain a 'correct' dimensional understanding, eg, from y=r.sin(theta) to be y=k.r.s where k=1/rad and s=(rad/deg).sin({theta} deg), in a similar way to the redefinition of the arc equation.
I don't see any need to have a foot in each camp if it means bifurcating my body or pretending that the other half doesn't really exist. Mathematics is the handmaiden of physics and if it won't let physics handle units in a consistent fashion that doesn't involve invoking mysterious 'angles' where none exist, then it should be sentenced to bread and water rations for a week.
Message was edited by: Stuart_Bruff SB: The terms x3 and x5 are supposed to be the respective powers of x and show up as such in the editor; however, I can't see them as superscripts when looking at the thread.
A littel detail:
Mathcad <12: rad + sr = error - OK
Mathcad >11: rad + sr = 2 ???
A degradation?