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]

Since this is an engineering site I'll say that a circle (or any other smooth curve) has a finite number of corners, for any desired level of precision.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IRstuff]

Your premise is not correct. A circle is approximated by a regular polygon with an infinite number of vertices, which are not necessarily "corners" since "corner" implies a right angle. As the number of vertices increase, their included angle increases, until every vertex becomes a straight angle, which is the limiting case for the circle, and why tangent lines intersect at only one point.

The interior angle of a vertex is given by 180 - 360/n = 180 * (n-2)/n where n is the number of vertices. The only regular polygon with true corners is a square. As n goes to infinity, the interior angle approaches 180.

TTFN
faq731-376

 

[WolfHR]

Sorry to be pedantic, IRStuff, but as far as I recall- as per definition, tangents intersect a circle at two infinitesimally close points.

 

[Walterke]

That's exactly the same thing.

Per definition: 0 angles
for any practical use: finite number of angels.

NX 7.5.5.4 with Teamcenter 8 on win7 64
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Nvidia Quadro 2000

 

[rb1957]

@wolf ... ? a tangent touches a circle at one point.

i guess if you constructed a cirlce from a single piece of stuff (from a single line element) by bending it ... then it'd have no corners (or verticies)

if you constructed it by increasing the number of corners (or verticies) (like strethcing a piece of string around pegs) then it'd have an infinite number.

now i'll ponder the imminent end of the world (as we know it)

 

[WolfHR]

Well, rb1957, I know a proper mathematical definition of a tangent, otherwise I wouldn't presume to correct as knowledgeable and respected member as IRStuff. You may choose to adopt more practical, if less accurate, interpretation but... The definition I gave is as far as I know the only correct one (for all curves, incl. a circle) and if you look at it more closely it will lead you to a more recognizable, derivative form.

 

[rb1957]

sorry wolf but every definition of tangent i know says a tangent touches a circle at a single point.

my memory thought of it that way.

a google search shows links to many definitions that say "touch" and "single point".

do you have a link that says "tangents intersect a circle at two infinitesimally close points" ?

 

[WolfHR]

OK, I'll look it up rb1957- but I'm positive it's the definition I was taught at high school (math oriented gymnasium) and at college, as well as my grandfather, who was college professor of descriptive geometry, using it...

 

[TheTick]

What is a corner? A corner is a C0 discontinuity. Circles are C2 continuous (C2 constant, for that matter). Mathematically, circles have no corners.

(x-a)^2 + (y-b)^2 = r^2

Anyone blathering about the approximation of a circle is simply not talking about a circle, but an entirely different shape with an entirely different definition, so their points are irrelevant.

 

[IRstuff]

Don't confuse the limit equation definition with the result. Per the definition of the derivative, which is a "slope", it does involve two points that approach each other, but, in the limit, the result is for a single, defined point. Otherwise, you could never properly quantify a numerical derivative. Refer to

http://en.wikipedia.org/wiki/Tangent_line#Intuitive_description

for the description.

TTFN
faq731-376

 

[btrueblood]

Reminds me of an old joke about driving a moron crazy (put him in a circular room and tell him to urinate in a corner). Never thought I'd see the online equivalent.

 

[KENAT]

I find this thread, like circles themselves, to be pointless.

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[btrueblood]

We'd better knock it off, Kenat, or we'll be told to go stand in the corner.

 

[rb1957]

it's not pointless (circles have an infinite number of points) ... but it may be corner-less

b'sides, it's silly season isn't it ? how is kate's pregancy ? (wtf cares ??)

 

[MiketheEngineer]

You guys must be bored --- Get back to work

 

[btrueblood]

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

 

[patprimmer]

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.

Regards
Pat
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[msquared48]

"corner" implies a right angle"

So neither Icosoles (sp?) or equilateral triangles have no corners?

"Corner" implies a shift in direction, not 90 degrees - any percieved angle change.

I guess I'm using circuitous logic here...

Mike McCann
MMC Engineering

http://mmcengineering.tripod.com

 

[ivymike]

A circle is made up of all the points a particular radius from the center - no lines nor line segments at all - and therefore has no corners. The individual points are not corners nor is there any space leftover between them.