[x3d-public] Sequencer interval definition

[Andreas Plesch]

Thanks, this is very reasonable. For the less mathematically fluent, it may
still be helpful to add an explanation, perhaps in the tooltips.

If there aren't any more suggestions with regards to Vince's proposal for a
corrected function definition, I could submit the function as a standards
comment.

Andreas


On Oct 2, 2017 1:16 AM, "Richard F. Puk" <puk at igraphics.com> wrote:

Hi –



As editor of these standards, we used normal mathematical notation. We
consider infinity as a number that cannot be expressed. This is why there
is always an indication of an open interval  whenever infinity (+ or -) is
at one end of the interval.



n  Dick



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*From:* x3d-public [mailto:x3d-public-bounces at web3d.org] *On Behalf Of *Andreas
Plesch
*Sent:* Saturday, September 30, 2017 10:47 PM
*To:* vmarchetti at kshell.com
*Cc:* X3D Graphics public mailing list
*Subject:* Re: [x3d-public] Sequencer interval definition







On Sep 30, 2017 7:49 PM, "vmarchetti at kshell.com" <vmarchetti at kshell.com>
wrote:


> On Sep 4, 2017, at 11:45 AM, Andreas Plesch <andreasplesch at gmail.com>
wrote:
>

> Resending this without html formatting to ensure proper transmission:
>
> 30.2.4 defines how sequencer values are assigned to key intervals:
>
> http://www.web3d.org/documents/specifications/19775-1/V3.3/Part01/
components/utils.html#SequencingEvents
>
> (−infinity, t0), [t0, t1), [t1, t2), ... , [tn-1, +infinity)
>
> are the intervals.
>
> The square bracket [ seems to indicate that the limiting value does not
belong to the interval. Is that the case ?

I'm more used to the square bracket indicating a closed boundary, so that
the for the interval denoted [a,b) ; x is in the interval a <= x < b
so a is an element of the interval but b is not. Take the view that neither
+infinity nor -infinity is a number; so (-infinity,a) is just shorthand for
the set x < a, while [a,+infinity) is x >= a.



VRML97 seems to be when the bracket for the interval was introduced:



http://tecfa.unige.ch/guides/vrml/vrml97/spec/part1/concepts.html#4.6.8



But it is not accompanied by an explanation. For interpolators it may not
matter if the limit is included in the interval since the function is
continuous.



For sequencers, the function produces different values depending on from
which side an interval limit is approached.



I guess it is time to look up how various browsers behave at sequencer
interval limits.





>

> For example, in the second interval [t0, t1) t0 does not belong to the
interval but t1 does.
> I have seen other notations where this interval would be described as
]t0, t1] which I would offer as a suggestion for an improvement.
>
> The next question is about the piecewise function definition:
>
> f(t) = v_n, if t_n <= t < t_(n-1)
>
> Given that the keys t_i are given in a monotonically non-decreasing
order, t_n is always larger (or equal to) than t_(n-1). Therefore, a t
larger than t_n can never be smaller than t_(n-1). But this is the
condition described in the function for value v_n. There is no possible t
which can fulfill this condition.
> Presumably this condition was designed as:
>
> f(t) = v_n, if t_n >= t > t_(n-1)
>
> If this is the case I can submit a spec. comment but I very much would
appreciate additional eyes on that since it seems like a rather fundamental
oversight. Did I overlook something ?
>

My set of eyes agrees with yours that the formula form in the v 3.3 version
of section 30.2.4 is incorrect, I would propose a corrected version:

f(t)    =  v_i, if t_{i-1}   ≤ t < t_{i}  for  1 <= i < n-1
        = v_0, if t < t_0,
        = v_{n−1}, if t ≥ t_{n−1}



Looks good to me if [ is meant to include the limit in the interval.



Andreas






> The Interpolator component spec. may provide additional insight:
>
> http://www.web3d.org/documents/specifications/19775-1/V3.3/Part01/
components/interp.html#Linearinterpolation
>
> Here the equivalent piecewise function definition is:
>
> f(t) = linterp(t, v_i, v_(i+1)),    if t_i <= t <= t_(i+1),
>
> Note that the interval condition here is between t_i and t_(i+1) and not
t_(i-1). So this is correct. But why change for Sequencer to t_(n-1) ?
Perhaps there is a reason ?
>
> Also note that i is used rather than n which is also more correct since n
is reserved for the fixed, total number of keys, whereas i is the variable
index depending on the whatever time t is considered.
>
> -Andreas
>
> --
> Andreas Plesch
> Waltham, MA 02453

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