rectangle geometry

[Grover Hughes]

Is there a name given to the problem of finding the maximum area-to-
perimeter ratio of a rectangle having radiused corners?

[Philippe 92]

Grover Hughes wrote :

> Is there a name given to the problem of finding the maximum area-to-
> perimeter ratio of a rectangle having radiused corners?

Hi,

Why do you restrict your problem of "radiused corners", if I correctly
understand what you mean by that, to rectangles ?

<http://mathworld.wolfram.com/IsoperimetricProblem.html>

--
Philippe C., mail : chephip, with domain free.fr
site : http://mathafou.free.fr/ (mathematical recreations)

[Grover Hughes]

Hi, Philippe, and thanks for your response. The answer to your
question is because that was the condition posed by a certain
mathematical problem I ran across on the sci.math site some time ago.
It had to do with a square, but I expanded it to include rectangles of
any aspect ratio. I solved the new problem to my satisfaction, but
I've wondered if there happens to be a name associated with it. It's
probably too trivial for many folks to be interested in it, I expect.

I do appreciate your response.

Best regards,

Grover Hughes

[Odysseus]

In article
<a461d103-4eaf-403d...@z41g2000yqz.googlegroups.com>,
Grover Hughes <ghu...@magtel.com> wrote:

<snip>
>
> [... T]hat was the condition posed by a certain mathematical problem

> I ran across on the sci.math site some time ago. It had to do with a
> square, but I expanded it to include rectangles of any aspect ratio.
> I solved the new problem to my satisfaction, but I've wondered if
> there happens to be a name associated with it. It's probably too
> trivial for many folks to be interested in it, I expect.

Not quite the same as a rectangle with radiused corners, but does the
"superellipse" relate?

<http://en.wikipedia.org/wiki/Superellipse>

(The article implies the term to be equivalent to "Lam� curve", but I
believe the term "subellipse" is more usual for an LC with n < 2 --
which the wiki calls a "hypoellipse", as opposed to a "hyperellipse".)

--
Odysseus

[Philippe 92]

Hi,

Odysseus wrote :

> Grover Hughes <ghu...@magtel.com> wrote:
>> Hi, Philippe, and thanks for your response.

>> <snip>
>> [...] That was the condition posed by a certain mathematical

>> problem I ran across on the sci.math site some time ago.

sci.math *site* ?? such a site doesn't exist, it is usenet.
You certainly mean "Google groups" web site, which is one of
the many servers that feed sci.math
BTW the worst server : doesn't respect the usenet rules about
quoting and character sets, and allows too much spam to be posted.

This also means that there is no "reference" server for sci.math
When one posts through one server, the message is fed to all
other *nntp* servers around the world. The same is true for alt.math
This parenthesis just to mention that I didn't get your message
because there has been a crash last weekend in the server I use,
precisely at the time you sent it !

I just saw the answer by Odysseus to you answer.

>> It had to do with a square, but I expanded it to include
>> rectangles of any aspect ratio.
>> I solved the new problem to my satisfaction, but I've wondered
>> if there happens to be a name associated with it.
>>
>

> Not quite the same as a rectangle with radiused corners, but
> does the "superellipse" relate?
>
> <http://en.wikipedia.org/wiki/Superellipse>

> The article implies the term to be equivalent to "Lamￜ curve"

The "rounded rectangle" has straight segments + circle arcs
The Lamￜ curve has a continuous curvature.
This is a question of "which additional constraints", see brelow.

However, the meaning of my answer is that for *ANY* curve, the
maximum area for a fixed given perimeter is for a *circle*.
And this is named the "isoperimetric problem".

So your /rounded-square/rectangle/lamￜ-curve/anything/ has a
maximal area when it _is_ a circle.
(that is your rectangle has identical null sides, connected by
four 1/4 circles, or a Lamￜ curve with m = n = 2 and a = b ;-)

So to get really "rounded rectangles/squares" you must explicitely
state some additional constraints than just the perimeter, otherwise
your rectangle/square _is_ a true circle.

Regards.

[Grover Hughes]

On Nov 24, 8:32 am, "Philippe 92" <nos...@free.invalid> wrote:
> Hi,
>
> Odysseus wrote :
>

> > Grover Hughes <ghug...@magtel.com> wrote:
> >> Hi, Philippe, and thanks for your response.
> >> <snip>
> >> [...] That was the condition posed by a certain mathematical
> >> problem I ran across on the sci.math site some time ago.
>
> sci.math *site* ?? such a site doesn't exist, it is usenet.
> You certainly mean "Google groups" web site, which is one of
> the many servers that feed sci.math
> BTW the worst server : doesn't respect the usenet rules about
> quoting and character sets, and allows too much spam to be posted.
>
> This also means that there is no "reference" server for sci.math
> When one posts through one server, the message is fed to all
> other *nntp* servers around the world. The same is true for alt.math
> This parenthesis just to mention that I didn't get your message
> because there has been a crash last weekend in the server I use,
> precisely at the time you sent it !
>
> I just saw the answer by Odysseus to you answer.
>
> >> It had to do with a square, but I expanded it to include
> >> rectangles of any aspect ratio.
> >> I solved the new problem to my satisfaction, but I've wondered
> >> if there happens to be a name associated with it.
>
> > Not quite the same as a rectangle with radiused corners, but
> > does the "superellipse" relate?
>
> > <http://en.wikipedia.org/wiki/Superellipse>

> > The article implies the term to be equivalent to "Lamé curve"

>
> The "rounded rectangle" has straight segments + circle arcs

> The Lamé curve has a continuous curvature.

> This is a question of "which additional constraints", see brelow.
>
> However, the meaning of my answer is that for *ANY* curve, the
> maximum area for a fixed given perimeter is for a *circle*.
> And this is named the "isoperimetric problem".
>

> So your /rounded-square/rectangle/lamé-curve/anything/ has a

> maximal area when it _is_ a circle.
> (that is your rectangle has identical null sides, connected by

> four 1/4 circles, or a Lamé curve with m = n = 2 and a = b ;-)

>
> So to get really "rounded rectangles/squares" you must explicitely
> state some additional constraints than just the perimeter, otherwise
> your rectangle/square _is_ a true circle.
>
> Regards.
>
> --
> Philippe C., mail : chephip, with domain  free.fr
> site :http://mathafou.free.fr/  (mathematical recreations)

Thanks to those who responded, but it seems that no one understood my
question, which was: is there a name given to the problem of finding
the maximum area-to-perimeter ratio for a rectangle with radiused
corners? I have the answer to the problem itself, and to superellipse
problems, etc., but all I am asking for is the answer to my question
as to the possible existance of a NAME.

Regards to all,

Grover Hughes