Re: Clean Object Class Design -- Circle/Ellipse

Jan Hidders wrote:
> Vadim Tropashko wrote:
> > In article <7MUb7.1661$NJ6.5968_at_www.newsranger.com>, Mikito
> > Harakiri says...
> > >Jan Hidders wrote:
> > >>
> > >>Well, let's see, it should hold for all Reals x, y that,
> > >>
> > >> floor(x) I+ floor(y) = floor(x R+ y)
> > >>
> > >>Yes, it holds, so they could, if this is your only operator. :-)
> > >
> > >You mean '+' is my only operator? I see that I can't make it
> > >
> > >floor(x) I* floor(y) <==> floor(x R* y)
> > >
> > It doesn't hold for '+' either.
> >
> > floor(1.9) I+ floor(1.9) = 2
> >
> > while
> >
> > floor(1.9+1.9) = 3
>
> Oh dear, you are right. Thanks for the correction. It also does not
> work for round():
>
> round(1.5) I+ round(1.5) = 4, but
>
> round(1.5 R+ 1.5) = 3
>
> In fact, it can be shown that such a mapping cannot exist. It holds
> that the mapping maps the integer Reals to the corresponding Integer
> and it should also hold that
>
> f(1/2) I+ f(1/2) = f(1/2 R+ 1/2) = f(1) = 1.
>
> So f(1/2) should be an integer x such that x + x = 1.

Hm. This is not really correct because it is not necessarily true that f(1.0) = 1. But if you assume that f(1.0) = a then

  f(1/a) I+ ... I+ f(1/a) { a times } = f(1/a R+ ... R+ 1/a) = f(1) = a

from which it follows that f(1/a) = 1. So it holds that

  f(1/2a) I+ f(1/2a) = f(1/2a R+ 1/2a) = f(1/a) = 1

so f(1/2a) should be an integer x such that x + x = 1.

-- 
  Jan Hidders