ARM1/ARM2 Alternative? (20/20 Hindsight)

By: Adrian (a.delete@this.acm.org), July 3, 2019 2:45 am
Room: Moderated Discussions
Michael S (already5chosen.delete@this.yahoo.com) on July 3, 2019 1:37 am wrote:

> >
>
> IMHO, inclusion of sinPi/cosPi into official language standards is long overdue.
> They are so much less controversial w.r.t. to range reduction, make explaining GIGO so much easier!
> They are parts of IEEE standard for a long time (I don't know exactly
> for how long. 15 years?), but still ignored by language committees.
>


Yes, it is also my opinion that anyone who needs to compute trigonometric functions of large arguments should rewrite his model to use something like sinPi & cosPi.


Actually I do not like sinPi & cosPi, but I think that sin & cos of 2*Pi*x are better and the functions of (Pi/2)*x are the best.

Another way to put it is that if you need to compute trigonometric functions of large arguments, you should not use the radian as the angle unit, but you should use either the cycle (previously used, e.g. in megacycles per second) or the right angle (the angle unit used already by Euclid) as the plane angle unit.


While the standard relationships between physical quantities in the International System of Units are based on the use of the radian as the plane angle unit (with the exception of the inconsistent use in SI of some doublet quantities, e.g. frequency & angular velocity), it is possible to change the proportionality factors in some relationships between the physical quantities to obtain a consistent system of physical quantities and units where the angles are measured in right angles and all the computations of trigonometric functions need just trivial argument reductions.


Normally this unit change in the plane angle should require just obvious conversions, but there is a catch. All the available literature shows dimensional equations for the physical quantities of SI that omit the plane angle unit from the equations.


This it due to an unbelievable stupidity of a large number of conferences where they were not able to decide whether the units for plane angle and for solid angle are base units or derived units.

The correct answer is that they both are base units and if you change either the unit for plane angle or the unit for solid angle, then, exactly like when you change the length unit, you must apply conversion formulas to all the physical quantities that are derived from angles.


The lack of acknowledgment of the fact that the angle units are base units resulted in wrong dimensional relations in all the books. For example both the units for frequency and for angular velocity are not reciprocal time unit, but angle unit divided by time unit.


More tricky are the physical quantities derived from the angular momentum, because students are taught the BS that the pseudovector that is the moment of a vector with respect to an axis has a magnitude that is the product of the vector magnitude with the radius.

This is a wrong definition, which is accidentally correct when the angles are measured in radians. The correct definition must replace the "radius" with the ratio between the length of a circular movement and the central angle between the initial and final position.

So the formulas for angular momentum, torque, moment of inertia and so on, whose dimensional formulas include in reality the plane angle unit, are different when the angles are measured in right angles.



Nevertheless, even if you need to be a little careful when writing the formulas for a physical model where the angles are measured in right angles or in cycles, there are accuracy advantages.

The simpler argument reduction is only one of the advantages. The other is that the input and output quantities correspond to the actually measured quantities, so you avoid loss of precision due to conversions to radian-based units, which might enable the use of number formats with lower precision.


Almost all angle measurement devices (or devices that measure various physical quantities derived from angles, which include less obvious things, such as electronic counters), do not measure angle in radians but in fractions of a cycle or of a right angle.

Those that really measure angle in radians, e.g. using a flexible ruler to measure both the diameter and the length of a circular arc, have very low precision, so they do not matter.




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