infinite or none? does a circle has infinite corners or none? 5

[Spoonful]

Hi All:

infinite or none? does a circle has infinite corners or none?

I guess this could be a interesting or pointless discussion?

Can we say a shape with infinite number of corners, if it is not infinitely large, it has to be a circle? Then it becomes have no corners?

If true, how can one from linearly increasing number of certain property (in this case corners) to become none of that property?

 

[IDS]

As this is a philosophical question, NOT an engineering one, it would be infinitely more appropriate to post it in a philosophy based site not an engineering one.

There are no philosophical questions, only philosophical responses.

Personally, I'll stick to the engineering response; what we deal with are appoximations to circles, and you can approximate a circle to any precision you like with a finite number of straight lines.

As for a corner having 90 degrees, I don't know where that definition comes from, but the OP was clearly not talking about any such corners, and Google tells me that a corner is "a place or angle where two or more sides or edges meet".

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[SomptingGuy]

In our madcap UK road network, corners are often 3D as well as not 90 degrees. Roundabouts are round but can be taken as corners (even though my old driving instructor told me that there is no "racing line" through a roundabout).

- Steve

 

[benta]

Here is a description of how the "infinite corner" circle works:

http://www.physicsforums.com/showthread.php?t=452917

Benta.

 

[zekeman]

Of all these answers, I think THE TICK got it exactly right.
A corner is a discontinuity of the first derivative of the curve. If that curve is a circle, it is clear its curve is an analytic function which means that it has one and only one first derivative at each point on the curve and therefore has NO corners.
To say it has an infinite number of corners would be to say it has an infinite number of discontinuities which is total nonsense.

 

[IRstuff]

The same argument can be applied to the derivative of any curve. The derivative is approximated by a series of piecewise linear approximations taken to the limit of a infinitely small span.

In the limit, the "corners" become straight angles, and there are no longer any discontinuities, and there are no longer "corners."

TTFN
faq731-376

 

[rb1957]

"The derivative is approximated by a series of piecewise linear approximations ..."

i don't think you're going to win this one ... a derivative is often a well behaved continuous mathematical function, the piece-wise approximation is just that, an approximation, and has no standing in this argument discussion

"In the limit, the "corners" become straight angles, and there are no longer any discontinuities ..."

another IMHO lost cause ... though the idea is better expressed (IMHO) by saying the corner initially has a radius (so, yes, the derivative is continuous) and then reduce that radius to be infinitismally small (at which time the derivative is not continuous).

 

[telecomguy]

"Rats, the boss has us cornered. Wheel have to circle back to this topic later."

btrueblood, your humor did not go unnoticed.

 

[WolfHR]

Rb1957, I concede defeat in providing google support of my definition... FWIW, I asked on a mathematical board where one member called it 'a good intuitive definition' and even took trouble to post a proof that it's valid (which I will not post here because I don't understand the method of proof myself). And I'd offer a quote from german book 'Matematische Formelsammlung' (1957) which claims:

Die Gerade schneidet die Ellipse in zwei reellen Punkten (Sekante), wenn b^2+m^2*a^2>n^2
die Gerade berührt die Ellipse in zwei zusammenfallenden Punkten (Tangente), wenn wenn b^2+m^2*a^2=n^2
&c

BTW, the similar formulation (tangent touching the curve in two points lying together) was used for circle, but it did not explicitly specify the line being a tangent, so I chose the quote from chapter on ellipses where it does. HTH

 

[IRstuff]

The tangent is the limiting case of the secant tangent, where the two intersections meld into a single point. The limit is arrived at the conclusion of moving the points closer together, and they cannot be any closer together than if they are the same point.

TTFN
faq731-376

 

[rb1957]

a "secant" i understand, and a "tangent", but a "secant tangent" ?

i can see that a tangent could be viewed as a special secant, i still wouldn't say that a tangent intercepts the circle at two points (like a secant) and qualify the two points as being infintely close to each other.

 

[IDS]

i don't think you're going to win this one ... a derivative is often a well behaved continuous mathematical function, the piece-wise approximation is just that, an approximation, and has no standing in this argument discussion

Sure it does. As engineers, the approximation is what we are concerned with.

In fact I'd go as far as saying that treating the mathematical idealisation as being more "exact" than a good approximation is a frequent cause of error.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[JohnRBaker]

A Circle is an example of a 'periodic curve' and thus has, by definition, NO corners:

http://docs.mcneel.com/rhino/5/help/en-us/popup_moreinformation/periodic.htm

John R. Baker, P.E.
Product 'Evangelist'
Product Engineering Software
Siemens PLM Software Inc.
Industry Sector
Cypress, CA
Siemens PLM:
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To an Engineer, the glass is twice as big as it needs to be.

 

[patprimmer]

Further to IDS, I have to say that a good approximation of the construction material for this thread would indicate it is mostly Male Bovine excrement.

Regards
Pat
See FAQ731-376 for tips on use of eng-tips by professional engineers &

http://eng-tips.com/market.cfm

for site rules

 

[racookpe1978]

Nah.

I "know" a circle has 4x corners 'cause I drew one in AutomatiCAD the other day using a square and 4 fillets. Yah can't argue with success! (And duct tape.)

And, in the plant, I done did use a two bladed drill to drill me a hole with no corners!

 

[Cockroach]

Not corners, Spoonful, but tangents. A circle is composed of infinitely many tangent lines. This is how a computer draws them on screen.

But you raise a more interesting mathematical problem. You can determine the constant "pi" using this method. Infact, this is a typical college type computing project, although I ran into it in mathematical physics. Essentially you begin with an equilateral triangle and determine the perimeter. For ease of the problem, you have the three apex of the triangle subtended (i.e. inscribed) in a circle of unit one. From the centroid of this equilateral triangle, you determine the distance to any apex, obviously 1/2 units. Then divide the perimeter of the triangle by twice the distance to the apex. This is the first approximation to "pi", our constant.

Now add one "side" to the figure. So from an equilateral triangle, you get a square. Repeat the process. Find the perimeter and divide by twice the distance from it's centre to any corner. You get a number which is slightly less than the first, a better estimate to "pi".

Repeat the process of another side added to the square, a pentagon. Repeat the process. You keep repeating for an added side to the figure at hand, hexagon...septagon...octagon...nonagon...decagon....eleven sided figure (whatever)....dodecagon.....and so on....

So eventually the computer program kicks out a constant for a figure of X sides, say X=250,000. Noting the number of sides of the polygon begin to approach the circumference of a circle. So at infinite number of sides, the constant begins to emerge as 3.1415692....whatever your level of significance may be.

Answering your question, a circle is a polygon of infinite sides, i.e. number of tangents is astronomically high. The constant of perimeter around the figure divided by twice the distance of centre to any corner is "pi".

And now you know.

Regards,
Cockroach

 

[Sparweb]

I just drew this in AutoCAD, where I found that the number of sides needed to make a polygon become indistinguishable from a circle is 42.
Hope you can all accept that as the ultimate answer.


STF

 

[IRstuff]

How coincidental that 42 is also the meaning of life...

TTFN
faq731-376

 

[zekeman]

Ho-hum
Watching grass grow is more interesting than this oversubscribed boring exercise in futility.
Why don't we let it go?

 

[CorBlimeyLimey]

Because there is no end to a circle.

 

[JohnRBaker]

Or for ANY Periodic curve for that matter...

John R. Baker, P.E.
Product 'Evangelist'
Product Engineering Software
Siemens PLM Software Inc.
Industry Sector
Cypress, CA
Siemens PLM:
UG/NX Museum:

To an Engineer, the glass is twice as big as it needs to be.