>> Nobody ever needs to compute the exact quotient of two 4-figure numbers
>> mentally. It's not necessary. You just need to be able to estimate the
>> answer with sufficient accuracy.
>
> True.
>
>> (Something which apparently a great
>> many people can't do for some reason...)
>
> Again true.

I hypothesize that this is due to their being far too much emphasis on 
memorising the algorithms for (say) computing a quotient, and far too 
little emphasis on getting a feel for what all the symbols *mean*, and 
*why* you manipulate them in this way.

>> Practising something you're going to need to do every single day of your
>> life is worth while. Practising something which you will never, ever
>> need to actually do is pointless.
>
> Well it surprises me that you never need to calculate something when you
> are away from some form of calculator.

It's not that I never need to calculate something. It's that I never 
need to calculate something to 6 figures. Usually I just need to 
estimate the answer. If I really do need an *exact* answer, I can use a 
calculator (given that there is *always* one in my pocket), or use 
pencil and paper if it matters that much.

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>> You just need to have a firm grasp of /how/ it works and /why/ it works.
>> Once you've got that, practising it on endless question sheets is just
>> an utter waste of time.
>
> Yes, but instead of a firm grasp of /how/ and /why/ it works, some
> people (and I guess they're the majority) need training to get a firm
> grasp of /how to/ actually do it - either because they never will get a
> firm enough grasp of the /how/ and /why/, or because they just don't
> give a shit.

It's a well-established fact that meaningful things are far easier to 
remember than meaningless things. Hence, if you understand why a 
procedure works, it's much easier to recall it.

This doesn't change the fact that some people don't give a damn about 
maths. Then again, when "maths" just means memorising a bunch of stuff 
for no other reason than "it's in the test", who can blame them?

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Am 22.08.2011 10:14, schrieb Invisible:
> On 20/08/2011 03:43 PM, Darren New wrote:
>> On 8/20/2011 4:11, Warp wrote:
>>> Drawing an accurate antialiased line (of certain width) is not a trivial
>>> problem. Basically for each pixel you need to calculate how much of
>>> it is
>>> covered by the line. Doing this accurately with integer math only can be
>>> complicated.
>>
>> I am not sure you can do it accurately with integer math at all, given
>> that there's a "portion of a pixel" involved in there somewhere. At best
>> you'd be working in scaled fixed-point.
>
> You realise that all of the quantities involved are rational, right? And
> that a rational number is just two integers?

But in the end you want to paint a pixel a particular color, which is a 
/single/ integer. That's where the precision you might be able to keep 
with rational representation ultimately breaks down.

Post a reply to this message

>>> I am not sure you can do it accurately with integer math at all, given
>>> that there's a "portion of a pixel" involved in there somewhere. At best
>>> you'd be working in scaled fixed-point.
>>
>> You realise that all of the quantities involved are rational, right? And
>> that a rational number is just two integers?
>
> But in the end you want to paint a pixel a particular color, which is a
> /single/ integer. That's where the precision you might be able to keep
> with rational representation ultimately breaks down.

That's where the precision with ANY POSSIBLE REPRESENTATION will break 
down. :-P

Post a reply to this message

>>> (Come to think of it, *all* of the networking stuff at college was done
>>> by Novel Netware. Now there's a name I haven't heard in a long time...)
>>
>> They're still around.
>
> Novel is clearly still around. I meant Netware.

Netware, the OS, is pretty much gone.  Novell still sells its NDS 
(Novell Directory Services) and file servers, but they're now using 
Linux for the OS (SuSE, if I recall)


-- 
/*Francois Labreque*/#local a=x+y;#local b=x+a;#local c=a+b;#macro P(F//
/*    flabreque    */L)polygon{5,F,F+z,L+z,L,F pigment{rgb 9}}#end union
/*        @        */{P(0,a)P(a,b)P(b,c)P(2*a,2*b)P(2*b,b+c)P(b+c,<2,3>)
/*   gmail.com     */}camera{orthographic location<6,1.25,-6>look_at a }

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On 22/08/2011 01:35 PM, Francois Labreque wrote:

>>>> (Come to think of it, *all* of the networking stuff at college was done
>>>> by Novel Netware. Now there's a name I haven't heard in a long time...)
>>>
>>> They're still around.
>>
>> Novel is clearly still around. I meant Netware.
>
> Netware, the OS, is pretty much gone.

That's what I suspected.

> Novell still sells its NDS (Novell Directory Services) and file servers

Interesting...

> but they're now using Linux for the OS (SuSE, if I recall)

Yeah, that's the one.

Post a reply to this message

Darren New <dne### [at] sanrrcom> wrote:
> On 8/20/2011 11:55, Warp wrote:
> >    So no, you don't get it "almost for free". Not if you want a high degree
> > of accuracy.

> Actually, I think you do, if you keep the slope of the line positive and 
> below 45-degrees. Then you use basic rotations of the algorithm to handle 
> other cases. The trick is the line will never go through more than 2 pixels 
> in the same column, so whatever doesn't go into the first pixel does go into 
> the second picture.

  If the line is anything else than 0 or 90 degrees, it can cover a
triangular area of a pixel, not just a trapezoidal one.

  Also, likewise, both edges of the line can go through the same pixel
(unless the width of the line is at least sqrt(2).)

  Symmetries do not help here at all.

-- 
                                                          - Warp

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Invisible <voi### [at] devnull> wrote:
> You realise that all of the quantities involved are rational, right? And 
> that a rational number is just two integers?

  In practice the integers have a maximum value they can hold, so too much
precision in your rational value can easily overflow either (or both) of
the integers.

  (If you were to use integers of unbounded size, you might just as well
use floating point values. They would be faster too.)

-- 
                                                          - Warp

Post a reply to this message

On 22/08/2011 04:09 PM, Warp wrote:
> Invisible<voi### [at] devnull>  wrote:
>> You realise that all of the quantities involved are rational, right? And
>> that a rational number is just two integers?
>
>    In practice the integers have a maximum value they can hold, so too much
> precision in your rational value can easily overflow either (or both) of
> the integers.

Yes. But if the quantities you're calculating with are pixel 
coordinates, overflow is highly unlikely to be a problem.

Post a reply to this message

Invisible <voi### [at] devnull> wrote:
> On 22/08/2011 04:09 PM, Warp wrote:
> > Invisible<voi### [at] devnull>  wrote:
> >> You realise that all of the quantities involved are rational, right? And
> >> that a rational number is just two integers?
> >
> >    In practice the integers have a maximum value they can hold, so too much
> > precision in your rational value can easily overflow either (or both) of
> > the integers.

> Yes. But if the quantities you're calculating with are pixel 
> coordinates, overflow is highly unlikely to be a problem.

  Except that we are talking about antialiasing here, which by definition
is calculating at subpixel accuracy.

-- 
                                                          - Warp

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