Re: atan2_sharp() -- a Motion?
John Pryce wrote:
> If people have no objection to such a function, let's have a motion.
I was going to propose a simple motion that sort of side-steps the issues,
namely to suggest adding atan2m() and atan2p() to the list of recommended
operations in 10.7 (Table 10.6), with Note g referring to a new subsection,
either 10.7.2 (shifting others) or 10.7.1.1 (insertion only):
10.7.1.1. atan2_sharp()
One issue with the interval extension of the standard atan2() function,
with range (-pi,+pi], is that a tight input box that straddles the x axis
on the negative side would like to return a tight result that includes
either -pi or +pi. But with the atan2() range this is not possible, and
atan2() would have to return a maximally-wide result of [-pi,+pi].
By allowing a choice of branch cut, either atan2m() with range (-2pi,0]
or atan2p() with range [0,+2pi), a programmer can construct a sharp
version of atan2. (Nevertheless an actual atan2_sharp() function is not
recommended, because (a) such a function is technically not an interval
extension of one point function, and (b) there is no generally-useful
fixed rule for choosing a branch cut based on a given input box.
But then it occurred to me that atan2m() and atan2p() are not the only
choices. Any rule that selects among two or three fixed branch cuts
suffers from the problem that, as a tight input box slides across the
point where the branch cut choice is made, the result jumps by 2pi.
What we want is a branch cut choice where the cross-over point is far
from the expected result.
So perhaps what we need is atan3(yy,xx,m) where m is an estimate of the
expected result midpoint (think of it as a parameter rather than as a
third interval input). The programmer could then implement atan2_sharp()
by using context information about the expected result, and pick the
appropriate m for atan2_sharp(yy,xx) = atan3(yy,xx,m). This will pick
a branch cut such that the range of atan3(yy,xx,m) is (m-pi,m+pi].
Is that reasonable? People who regularly program with branch cuts,
please chime in, because I don't.
Michel.
---Sent: 2014-02-25 15:59:46 UTC