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Re: asin2?
Posted:
Aug 17, 2000 9:59 PM
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In <Vmhjr-A54942.22424916082000@news.frii.com> Virgil <Vmhjr@frii.com> writes:
>In article <xrGm5.1082$9N1.18474@vixen.cso.uiuc.edu>,
>prussing@aae.uiuc.edu (John Prussing) wrote:
>> Apparently I have (surprisingly) confused everyone with the notation.
>> x and y were intended to be dummy arguments, not cartesian components
>> of some point.
>Since you r example did not make clear that your dummy arguments should
>not be coordinates, I am surprised that you were surprised.
And I am surprised that you were surprised that I was surprised.
Are we in an infinite loop here?
I assumed that readers of this newsgroup knew what the function
atan2(*,*) was.
>In your example, at least one of your dummy variables had to allow both
>positive and negative values to get the standard arcsine function. This
>certainly suggested that one of the dummy variables could be a
>coordinate, since for ordinary triangles there are no negatives needed.
> If either of the "dummy arguments" may be negative, why not both of
>them?
>And if both of them may take either positive or negative values, there
>is enough information available to construct an asin2 valid over all 4
>quadrants, since the two signs are enough to determine the quadrant
>uniquely and correctly.
Let me try to explain it again. And I'll try to respond to your
(different) email message also.
tan(t) = sin(t)/cos(t) = u/v and t = atan2(u,v). sin(t) and cos(t)
can independently be > 0 or < 0, There are two independent pieces of
information here: the value of sin(t) and cos(t). 2^2 = 4 possible
values of atan2 are available, covering all four quadrants.
On the other hand, what about sin(t) or cos(t)? As a ratio sin(t) =
sin(t)/sqrt(sin^2(t) + cos^2(t)). There is only one piece of independent
information here, namely sin(t). The denominator is > 0 by definition
(it is r in the x-y plane).
Any asin function cannot tell the difference between quadrants QI and QII
or between QIII and QIV. That is my contention, that an asin2 analogous
to atan2 does not exist. I said long back, that one is stuck with the
principal value of the ordinary asin being in QI or QIV.
In your (different) email message (why didn't you post it here?)
you ask whether, given tan(t) = 1/1 can atan2 tell the
difference between t = pi/4 and t = 5*pi/4? I note that this
is a question about atan2 and not the (mythical) asin2.
The answer is no. But that does contribute anything to our
discussion. atan2 assumes that sin(t) > 0 and cos(t) > 0
and returns t = pi/4, i.e., QI.
Next you ask that if you are given the standard form u/v,
how do you find the signs of u and v? You simply examine the
numerical values of u and v. If u = -1 the sign of u is < 0, etc.
You next note that:
If you _cannot_ determine the signs of u and v separately, then
atan2 is just as impossible as asin2.
As I've said for atan2 u = sin(t) and v = cos(t), so their
signs can be determined separately.
You next state that if you can determine the separate signs of u and
v, asin2 is just as possible as atan2. That statement is correct, but
as I have said, one cannot determine the sign of v for asin2
because v > 0.
Next, you address my test problem sin(t) = 1/sqrt(2).
You state that IF u > 0 and v > 0 then t = pi/4.
But, in fact, u = 1 > 0 and v = sqrt(2) > 0, so
THE answer is pi/4, as you say. But the sad fact is that
3*pi/4 has the same u and v. Hence the impossibility of asin2.
You then state that if u < 0 and v < 0 then t = 3*pi/4.
This is, unfortunately, incorrect for two reasons:
(1) u is not < 0 nor is v < 0, and (2) the value of t in this
case would be in QIII, not QII.
Next, you propose sin(t) = -1/sqrt(2).
You state that if u < 0 and v > 0 (which is correct)
then t = - pi/2. This is probably a typo and you meant
- pi/4. i.e., in QIV.
You then consider u > 0 (not true) and v < 0 (not true and impossible, as
I've stated), and come up with a typoed result based in false premises.
A negative argument for any asin function will return the principal
value in QIV. In simple terms, sin(t) = y/r, with r > 0 (the distance
from the origin in the fabled x-y plane).
Because v > 0, If u > 0 the value of t is in Q1, and if u < 0 the value
is in QIV (principal value). There's no way on God's Green Earth that an
asin function, given sin(t) = u/v, where u and v are general variables,
can reurn a value in QII or QIII.
So, to summarize, an asin function, because it has only
one independent piece of information, cannot do better than
the principal value. But atan2(u,v) has two independent
pieces of information and can therefore determine a unique
quadrant.
Virgil, if you respond, please stick with a single posted message.
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John E. Prussing
Dept. of Aeronautical & Astronautical Engineering
University of Illinois at Urbana-Champaign
http://www.uiuc.edu/~prussing
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