[metapost] arclength() tolerances

[Hans Hagen]

On 5/24/2014 4:54 PM, Hartmut Henkel wrote:
> Hi Taco,
>
> On Thu, 3 Apr 2014, Taco Hoekwater wrote:
>> On 02 Apr 2014, at 21:10, Hartmut Henkel <hartmut_henkel at gmx.de> wrote:
>>> in metapost there seem to be absolute tolerance values for iterative
>>> algorithms like arclength() even in the double numbersystem:
>>
>> Yes, the arc_test() routine (the core or arclength/arctime) uses a
>> predefined tolerance of unity/4096. While it is clear to me why it
>> needs a tolerance setting, it is not clear why that precise value, nor
>> does lowering it actually help, it just makes mp run slower:
>
>> Maybe there is some intrinsic error in the approximation code?
>
> for the "double" number system the cause is in mp.w:
>
>    number_add_scaled (tmp, 2);
>
> This adds scaled 2, which still is a fixed value (1/(2^16)), perhaps for
> better rounding (?) in the original fixed mpost number system, and it
> limits the precision, even when math->arc_tol_k.data.dval is reduced
> considerably.
>
> When this line is just removed for "double", the arclength calculation
> seems to scale nicely with smaller tolerance settings. Then there is no
> problem to get an arclength exact to e. g., 1e-12. It gives deeper
> mp_arc_test() recursions, but not excessively. For 1e-12 some test arc
> gave approx. total 1800 instead of originally 13 calls to mp_arc_test(),
> so it gets slower, but not much. For the checked example arc the method
> by Gravesen needed about 18000 steps, but with another arc shape this
> can be an inverse ratio as well, seems with rather straight curves the
> Gravesen method is faster than the method with Simpson rule.
>
> A similar improvement is possible for arctime(), since there is another
> such number_add_scaled() to be removed. It looks as if for "double" all
> (?) these number_add_scaled() give more errors than they help. Should
> such a tolerance be presettable and how?

So, do any of these _scaled thingies make sense in non scaled mode? I 
guess only in some kind 'equal' test, where we could check for 
overlapping +/- eps instead.

Hans

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