OK, so I'm currently failing to figure this out...

How do you compute the smallest possible (axis-aligned) bounding box for 
an ellipse (which is not necessarily axis-aligned)?

Apparently such an ellipse can be represented as

   X(t) = Xc + A cos t cos K - B sin t sin K
   Y(t) = Yc + A cos t sin K + B sin t cos K

Clearly you need to find the minimum and maximum values for X(t) and 
Y(t). But I am apparently too stupid to do this. (And even Wolfram Alpha 
can't work it out. Oh, it'll give me the answer if I remove all the 
symbolic constants with actual numbers, but that's no help at all...)

It strikes me that the sum of two sine waves of identical frequency is 
another sine wave, but I can't seem to apply this fact to obtain a result.

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Invisible wrote:
> OK, so I'm currently failing to figure this out...
> 
> How do you compute the smallest possible (axis-aligned) bounding box for 
> an ellipse (which is not necessarily axis-aligned)?
> 
> Apparently such an ellipse can be represented as
> 
>   X(t) = Xc + A cos t cos K - B sin t sin K
>   Y(t) = Yc + A cos t sin K + B sin t cos K
> 
> Clearly you need to find the minimum and maximum values for X(t) and 
> Y(t). But I am apparently too stupid to do this.

When you're looking for the maximum or minimum of a function the first 
thing to look for is in general a zero in the derivative.

For example for X:

dX/dt = - A cos K sin t - B sin K cos t
so dX/dt = 0 => A cos K sin t = - B sin K cos t
or assuming A cos K  and cos t are not null (you could make special 
cases if needed)
tan t = -B/A tan K

so the bounds for X are
X0 = X( atan(-B/A tan K) )
X1 = X( pi + atan(-B/A tan K) )

Sort and do the same for Y...

-- 
Vincent

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Vincent Le Chevalier wrote:

> so the bounds for X are
> X0 = X( atan(-B/A tan K) )
> X1 = X( pi + atan(-B/A tan K) )

My God, man, IT WORKS!

> Sort and do the same for Y...

Heh, that should be amusing...

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Invisible wrote:
> Vincent Le Chevalier wrote:
>> so the bounds for X are
>> X0 = X( atan(-B/A tan K) )
>> X1 = X( pi + atan(-B/A tan K) )
> 
> My God, man, IT WORKS!

That's the good thing about math, you know :-)
More than the result you should keep the method in mind, it's useful in 
plenty of problems...

-- 
Vincent

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>> My God, man, IT WORKS!
> 
> That's the good thing about math, you know :-)
> More than the result you should keep the method in mind, it's useful in 
> plenty of problems...

I was attempting to work out what the result of

   A sin t + B cos t

is. I was convinced there was a standard identity for this, but I can't 
find it. In the end I came up with

   Sqrt(A^2 + B^2) sin (t + atan(B/A))

but I don't even know if that's correct. And I still have to apply it to 
my original formula to figure out the result.

Man, who knew a simple ellipse was this complicated?!

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"Invisible" <voi### [at] devnull> wrote in message 
news:4b7548f1$1@news.povray.org...

> Man, who knew a simple ellipse was this complicated?!

This is an incredibly important life lesson: the complexity of a phenomenon 
is utterly disconnected from the number of syllables needed to describe it.

Sometimes I feel the desire to emphasize this concept to my co-workers with 
a heavy, blunt object. Then I take a deep breath, get more coffee, and try 
to convince myself that homicide is a detriment to career advancement (well, 
in my field, anyway).

--
Jack

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Captain Jack wrote:

> This is an incredibly important life lesson: the complexity of a phenomenon 
> is utterly disconnected from the number of syllables needed to describe it.

Exhibit A:

http://www.youtube.com/watch?v=58_s6r7PaKo

It's just a swinging pendulum, after all...

Exhibit B: Rule 30 can be stated in 8 equations. Oh, did I mention? It's 
Turing-complete. It can compute any computable function.

Note also that simple /= easy.

x^5 - x + 1 = 0 is "simple". Now try solving it...

Similarly, the Waltz is "simple". Try dancing it sometime; you'll 
discover that it's surprisingly difficult, despite being so simple.

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Invisible wrote:
> Exhibit B: Rule 30 can be stated in 8 equations. Oh, did I mention? It's 
> Turing-complete. It can compute any computable function.

You know, he says that, but I'm pretty sure a CA with an infinite amount of 
initialization counts as Turing-equivalent.

-- 
Darren New, San Diego CA, USA (PST)
   Forget "focus follows mouse." When do
   I get "focus follows gaze"?

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>> Exhibit B: Rule 30 can be stated in 8 equations. Oh, did I mention? 
>> It's Turing-complete. It can compute any computable function.
> 
> You know, he says that, but I'm pretty sure a CA with an infinite amount 
> of initialization counts as Turing-equivalent.

The point being that some rules can't emulate a Turing machine, while 
others can. E.g., rule 0 maps every possible input to 0, so you can't do 
very much with that. Rule 2 maps each 1-bit input to the same output. 
And so on. Only a few of the possible rules are Turing-complete.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 wrote:
> Only a few of the possible rules are Turing-complete.

I fully understand exactly what he's saying. I'm simply denying that he's 
correct, on the grounds that he does an infinite amount of setup before he 
even starts running the CA.  If he came up with a small rule that would 
create the initial pattern in a finite number of steps, then maybe it would 
make more sense. But as he described it, he needs to set up an infinite 
number of clock pulses before starting the emulation in order to clock the 
emulation. I'm not sure that's allowed, any more than it's allowed to set up 
an infinite pattern of states on a turing machine before starting.

-- 
Darren New, San Diego CA, USA (PST)
   Forget "focus follows mouse." When do
   I get "focus follows gaze"?

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