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

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

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

(OP)
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?

 

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

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/

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

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

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

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

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
Intel Xeon @3.2GHz
8GB RAM
Nvidia Quadro 2000

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

@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)

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

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.

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

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" ?

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

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...

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

2
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.

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

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#Intuitiv... for the description.

TTFN
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RE: infinite or none? does a circle has infinite corners or none?

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. :)

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

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

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

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 ??)

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

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

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

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

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

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

"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

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

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.

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

Quote:

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/

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

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

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

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.

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

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

"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).

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

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

btrueblood, your humor did not go unnoticed. smile

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

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

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

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

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.

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


Quote:

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/

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

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_mo...

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.

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

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

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!

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

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

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


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

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


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

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

Because there is no end to a circle.

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

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.

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

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

Why do people who have no interest in a topic take the time to open it, read through it, and then post an insulting response?

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
 

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

Because we come here expecting a certain std of discussion. We often cannot tell if a thread is pointless, boring or inappropriate before we open it and read at least a little. By then some of our time has been wasted on stuff that should not really be here.

If this where in the pub it could be appropriate as the pub is kinda for this type of thread, but this philosophical non engineering really does not belong in a serious forum, well at least in my view, so people do have a right to complain and even red flag it.

Regards
Pat
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RE: infinite or none? does a circle has infinite corners or none?

(OP)
Hi All,

Thank for all the response.
I agree this question was a bit out of the scope of this forum, but I thought here I can get some intelligent response from some intelligent engineers. (even tho I do agree this topic is kinda of pointless of what we doing)

I thought my original question was more philosophic rather than mathematical. I was trying to emphasise on the question of how come when you increase something to infinite, at the end you could get zero(none).

When I say corner, I didn't mean corner by its narrow meaning, what I mean is more like a vertex formed by two (tangent lines). Starting with a basic shape with 3 corners(edges), a triangle, by increase the number of corners(edges) we get rectangle and so on. when your number of corners(edges) reaches infinite, geit becomes a circle, where all your increased number of your geometrical characteristic (corner,edges) has all gone.

Or perhaps you could never call a circle formed in such a way it a "true" circle?

To sum up, the question was, if something has a characteristic of A, and it has got so many of this characteristic, to an extend of infinite amount, can we say it has got none (or almost none) of that characteristic

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

OK, now that really sounds like a pointless exercise. Consider a tablespoon of salt; dissolve in a cup of water and the water is quite salty. Dissolve it in Lake Tahoe and you can't even tell if anything has changed. So what? Dilution is hardly a complicated or mysterious process.

Consider your many sided polygon with 10000 vertices; that makes the interior angles equal to 180-360/10000 = 179.964° Not much of "corner" already, and still a long ways from infinity. You'd be hard pressed to tell that it wasn't just a straight line. Again, so what? There is nothing magical or inherent about the "corners" in this problem. The "corners" of a triangle don't really have anything to do with the corners of a square. They didn't morph into the corners of the square. From a topological math perspective, all polygons are essentially the same closed shape, like a rubber band. I think you have too much time on your hands winky smile

TTFN
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RE: infinite or none? does a circle has infinite corners or none?

Quote (rb1957)

how is kate's pregancy ? (wtf cares ??)
The Australians

(too soon?)

NX 7.5.5.4 with Teamcenter 8 on win7 64
Intel Xeon @3.2GHz
8GB RAM
Nvidia Quadro 2000

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

not nice ! i'm sure the DJs thought they were having a bit of joke and didn't foresee the unintended.

as for "if something has a characteristic of A, and it has got so many of this characteristic, to an extend of infinite amount, can we say it has got none (or almost none) of that characteristic" i agree with IR ...

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

Circles are not polygons.

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

this is the song that never ends ...

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

A circle is a polygon of infinite sides.

Regards,
Cockroach

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

Okay, enough of this. Here is the definitive proof that a circle is a polygon of infinite sides.

I start with a equi-sided polygon, the distance from the centroid to the vertex being 0.50000 units. Clearly I am referring to a equilateral triangle which is inscribed within a circle of radius 0.50000 units. I ask the question, "What is the perimeter of this figure?". Using the Law of Sines and noting that the sum of angles equals 180 degrees in a triangle I get a side length of 0.86603 units. So the perimeter around the equilateral triangle is three times this amount or 3 X 0.86603 units = 2.59808 units. Because I know that the circumference of a circle is PI times diameter, the value of PI is simply the perimeter of a polygon divided by twice the radius which is 0.50000 units. In other words, the value of PI is the perimeter of the polygon divided by "1.0" or my polygon perimeter, 2.59808. I claim this is the first approximation to PI.

Increase the number of sides by one. I am now referring to a square which would be inscribed within a circle of radius 0.50000 units. I ask the question, "What is the perimeter of this figure?". Using Pythagorous Theorem and noting any two sides are of equal length, I get a side length of 0.70711 units. The perimeter of the square is therefore 4 X 0.70711 = 2.82843 units. Therefore following the logic above, PI equals the perimeter of the polygon divided by "1.0", the second approximation to PI is 2.82843.

Increase the number of sides by one. I am now referring to a pentagon inscribed inside a unit circle. Same question, find perimeter. Noting the composite triangle of the radius between any two successive verticies and a side, the central angle is 360/5 = 72 degrees. The other two angles noting they are equal and triangle interior angles sum to 180, I get 54 degrees. So Law of Sines using one of the known radii of 0.50000 units, side length is 0.58779 units, perimeter is 5 X 0.58779 units - 2.93893 units. This is the third approximation to PI, 2.93893.

Increase the number of sides by one. I am referring to a hexagon inscribed inside a unit circle. Noting the logic used in the pentagon, side length is 0.50000 units and the perimeter is 6 X 0.50000 units = 3.00000 units. This is the fourth approximation to PI, 3.00000. It also is the proof that a hexagon is the only regular polygon that can be inscribed inside a circle using a compass and straight edge because the sides of the hexagon equal the radius of the circle to which it is inscribed. The fourth approximation to PI is 3.00000.

Increase the number of sides by one. This is now a heptagon or seven sided polygon inscribed in a unit circle. I find the angle between two verticies subtended by a side with the centre of the figure to be 360/7 = 51.42857 degrees. Noting the angle sum to 180 degrees in this composite triangle, I get a half angle vertex to be 64.28572 degrees. Apply the Law of Sines, a side equals 0.43388 units. Therefore the perimeter for the heptagon is 7 X 0.43388 units or 3.03719 units. The fifth approximation to PI is 3.03719.

So in this fashion, I continue increasing the sides to the polygon inscribed in the unit circle. Finding the perimeter will converge to the value of PI. But we will never get to PI as 3.1415692...simply because I can never stop adding a side to the polygon. For an infinitely large number sided polygon, the value of PI would be just that, PI.

THEREFORE it comes to pass that a circle is a polygon of equal sides because the value of the perimeter of that infinitely many sided polygon is PI. In this special circumstance, the perimeter is called the circumference and the circumference of the circle is PI times diameter.

And now you have it.

Regards,
Cockroach

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

No, that does not "prove" that a circle is an infinite-sided polygon. It is only a method of calculating pi.

By definition, a polygon is composed of straight segments. Segments are of finite length (therefore not infinitely small).

Next batter...

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

The sides of the polygon form "tangents to the circle" when circumscribed to said circle. As usual, engineers muck up the mathematics and are poor at arriving at a proof. If I have a polygon of one million parts, surely this is much more infinite than a three sided polygon. Therefore increasing the population of sides to well past one million, I get a number approaching but never reaching Pi.

You can't see the forest because of the trees. The method of calculating Pi in this fashion is sufficient to understand a circle is an infinitely sided polygon. Your freedom to note otherwise is noted, and wrong.

Regards,
Cockroach

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

A segment is defined as a portion of a line between two points and all points between (therefore not "infinitely close" or coincident).

A polygon is a planar closed chain of segments.

A circle is defined as the set of points in a plane equidistant from a given point.

A segment can not be fully coincident with any portion of a circle's circumference, as it will always have points that are not equidistant from the circle's center.

So... no, a circle is not a polygon.

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

A circle is one continuous curved line where all points are equidistant from its centre.

If it where a collection of infinitely short straight lines, it would also have infinitely small variations in distance from the central point and would therefore fail the definition of a circle.

Regards
Pat
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RE: infinite or none? does a circle has infinite corners or none?

Infinite number of sides of a polygon = infinite number of points or verticies for that polygon = a circle on which it is inscribed. Perhaps it is a visual exercise and harder to describe lexigraphically.

But that's fine, you agree to disagree. I'm good with it.

Regards,
Cockroach

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

Actually, it is a very interesting topic, one which would be expected to be discussed in the mathematical forum, not an engineering one. But you can see the process in the video display of CAD packaging as an end user increases the resolution of the display. SolidWorks shows this quite well, for example.

I've shown the convergence to Pi asymptotically from the left using inscribing of polygons of increasing sides. You could do the same thing by circumscribing that polygon and using the midpoint of the side as the tangent point to the circle. The arithmetic changes slightly, but it is the same process. In that case, you would converge on Pi from the right.

So we're showing the sum of an infinite series converging to a limit. The fact that circumference is Pi times diameter as a circle property falls out from the perimeter of the polygon. But the process, obvious to some in this forum, is the point I wanted to make.

I just wasn't expecting such a parochial mentality of some readers, which on occasion is refreshing, in an intuitively obvious demonstration. But I'm not in the faith conversion business, but I digress.

Regards,
Cockroach

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

Anyone who was even only mildly lucid in Calculus I understands that method of calculating pi (which is not "proof" that you have made a circle from segments).  Anyone who was merely present in high school geometry understands that it is still not a circle.

As for lack of imagination, I suppose I would defer to those that do not have the ability to imagine curvature and are still stuck pasting straight segments together.  Then again, there are no curves, but only straight lines travelling through curved space.

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

The line describing a circle is infinitesimally close to being a straight one.

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

This pretty much summarizes the method.

Asymtotic convergence from the left uses verticies of a polygon of increasing sides inscribed from a common point or centre of curvature. That from the right are circumscribed polygons of increasing sides that use the midpoint of a side tangent to a curve with common centre. Because of symmetry, such points, whether they are polygon verticies or midpoints to the sides of said polygon lying tangent to the curve, are equi-distant from the centre of curvature. Hence the notion of a "circle".

So a circle can be regarded as a polygon of infinite sides, no matter which way you choose to describe it. Points on a curve or tangent lines that are subtended at the midpoint of each side, convergence to a constant defined by perimeter divided by twice the radius of curvature, is a unique circular poperty that falls from the logic of the method. Regardless, the concept is a polygon of increasing sides thus geometrically forming a circle with points lying equi-distant from a common centre of curvature.

End of story.

Regards,
Cockroach

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

Cockroach,
"I get a number approaching but never reaching Pi."
Most of us don't have the patience to wait for an infinite series to reach its asymptote.  
We'd rather just reach for the pi.




STF

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

Just because calculus uses a very close simulation to a circle, that does not make it a real circle, just an infinitely close approximation.

Just because a CAD program cannot actually draw a circle, that does not mean it's best attempt at an approximation is a circle.

Regards
Pat
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RE: infinite or none? does a circle has infinite corners or none?

I haven't read every reply in the whole thread. I don't think anyone here is confused about any of the concepts.

I think both "sides" (no pun intended) would agree with a statement worded as follows:   In the limit as N approaches infinity, an N-sided polygon approaches a circular shape.

Personally I think the notion of a limit is a necessary part of that statement. The phrase "polygon with an infinite number of sides" does not seem mathematically precise and perhaps creates ambiguity which results in disagreement when people interpret the ambiguous phrase differently.

Sorry if someone else already said the exact same thing.
Also sorry for prolonging this. Barring any negative reply to my comments, I'm done.


=====================================
(2B)+(2B)'  ?

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

Correction in bold. Should have been:
"In the limit as N approaches infinity, an N-sided regular polygon approaches a circular shape."

=====================================
(2B)+(2B)' ?

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

Pretty sure the circle only has two sides, inside and outside. And with that, I'm steppin' outside.

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

You mean it's a 'half-space'?

Now that should really confuse people hairpull

John R. Baker, P.E.
Product 'Evangelist'
Product Engineering Software
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To an Engineer, the glass is twice as big as it needs to be.

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

Cockroach, re. your definition/proof of circle as polygon with infinite sides (and I say this half serious)- it's all well but the first polygon (triangle) you mention was already inscribed in something called circle... So, by that definition a circle is already well established fact when we start inscribing polygons into it. On a more serious note, as others have noted before me- when a push comes to shove, a circle (which is defined as a set of points in a plane equidistant from a given point that is its centre) can be approximated by a polygon whose number of sides converges towards infinity (the greater the number of sides, the closer approximation to the actual circle).

Re. CAD analogy someone drew to Solidworks is not as to the point as someone did to AutoCAD (with answer being 42), because Solidworks and other modern CAD packages use 'proper' circles, and the limiting factor is resolution of the display, whereas AutoCAD actually (prolly for purposes of speed of display, &c) actually approximates circles with polygons (number of sides is set in options) to display, but calculates as it should with circles...

As for the intention of the OP, I must say the (intended) irony escaped me at the first, but I do see it. Admittedly, it's indeed not an engineering question per se, but interesting none the less. Increasing the number of sides of a polygon until there are none.

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

Actually WolfHR, I pointed out that the vertices in the case of a polygon, or the midpoints of the sides of the polygon, lie equi-distant from the centroid of said polygon. This is important, I drew the circle in dashed lines in Oder to accent this feature but more importantly to show that as the number of polygon sides increase, the figure starts to move closer and closer to a circle. I also pointed out that if possible to reach any large number, the true circle will never be obtained, for I can't count to infinity. What I am doing is to populate points equi-distant from a centre of area of a n sided polygon, the path of which is a circle. I never presupposed a circle and could of started with a polygon of any size, then determine the distance of these points from a theoretical centre. Don't let that detract from the mathematical argument.

ElectricPete brings to mind the notion of limit. I am not summing and allowing infinitesimal accumulations like an integration process. Rather, I an incrementally increasing the number of sides in a polygon from three to a very large number. The concept of circularity evolves out of the process as the circumference of a circle equals Pi times diameter. These fall out of the argument.

But two well thought out points, well presented, so I thank you. Try it for yourself, start with an equilateral triangle of any size and locate the centre. Then increase the sides of the polygon by one and recompute. You'll get the very same result whether you use the vertex of said polygons or midpoint of the sides. It is a beautiful proof! Try it for yourself over a few beers.

Regards,
Cockroach

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

I have no doubt, Cockroach, and I entirely see your point and concur*. Although... I'm a lazy git, so I'd start with triangle and just because of convenience increase the number of sides twofold. Admittedly, if we were going all the way to infinity, my method would not be any faster than the one you suggested (but both of us know we'd never get there), but it seems to be faster to reach large numbers, and halving the angles also seems more convenient.

* the point in previous post was an attempt at a bit of joke- we show that polygons tend to approximate circle more closely as number of sides increases, and start doing it by inscribing a triangle in a circle... so the intended joke was why would we approximate the circle if we took a compass and have already drawn it?

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

I'd look at it this way:

If my mom put me in a round room, and told me to go stand in the corner for 10 minutes..... I would be in there for the rest of my life.

Charlie
www.facsco.com

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

Well Cockroach I almost gave you a star and would have if your attempt weren't flatly wrong. You just may be more into math and proofs than actuality.

The more points you make the closer you will be getting to the correct answer or will you?
A Circle is a Circle now a computer may come awfully close to representing a circle, but if you get a pushpin/thumbtack a piece of string and a pen and make two tiny loops the size of the pushpin diameter and the other the size of the pen tip. Push the pin through the pin hole into a piece of paper. stretch the string to it's max extent and put pen to page and go 360º around so you end where you start and you'll have a circle. Of course no one can draw a perfect circle although it may be possible it's highly not probable.

Everyone who thinks of a circle as an infinite sided polygon, let's have a race. you can start by drawing your polygon and I can start walking halfway to the nearest wall.

No one will win because you'll never have infinite number of sides because that number does not exist and I'll never reach the wall.

Cheers everyone and thank you for the mindlessly inane and quite stupid imho discussion.
atom...............infinity
The atom smiley is a Gif which is the closest that file type can get to a circle unless you make it ∞x∞ pixels and you may come close but never achieve a perfect circle.
Even with the best cad system and graphics card, if you zoom in far enough to a circle you'll see the linear approximation even with a Cray supercomputer. Pretty sure SolidWorks has a limitation to how many sides it can have in a polygon. If I find that out I'll post it on the SolidWorks forum which I recognize a few forum posters from.

One more thought before I leave the discussion Can you get a perfect circle from an etch a sketch with precisely programmed stepper motors at the knobs?

P.S. patprimmer If it's curved it's not a line at least not any longer. smile

Quote (patprimmer)

A circle is one continuous curved line where all points are equidistant from its centre.
If it were a collection of infinitely short straight lines, it would also have infinitely small variations in distance from the central point and would therefore fail the definition of a circle.
Note: quote has been spell corrected were was "where"
But I completely agree with your point.

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

Well if you want to be pedantic, if it's been changed it is no longer a quote.

http://www.merriam-webster.com/dictionary/line

It seems Merriam Webster disagrees with your definition of a line around points 7, 8 and 9

Just to be pedantic about it.

Regards
Pat
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RE: infinite or none? does a circle has infinite corners or none?

What's all this crap about computers NOT being able to represent a circle except as some sort of approximation. It's just plan BS! A circle, besides being a periodic curve (as I've already pointed-out), can be represented using a conical form, that is, a mathematically determinant object. Meaning that NO approximation is required when performing mathematical operations involving circles or for that matter, any other shape that can be represented as a conical form, such as a line, ellipse, sphere, cylinder, cone, tori, etc.

I suspect that where this perception that somehow computers need to approximate a circle is when it needs to display one a computer screen, but this has absolutely NOTHING to do with how the computer software represents a circle mathematically. It's purely a 'mechanical' limitation of the display technology, but even that is not always what you'd think. There was a time before raster displays (like your TV) when we were using direct view displays, that is the displayes were not scan line based but with objects being directly drawn on the screen. However these were limited to monochromatic displays, the most common being White on Black with Display Refreash Devices...



...or Green on Black with Direct View Storage Terminals...



...but circles were circles, period.

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.

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

Where can I get one of these computers that allow methematical operations to be carried out with no approximation, rather than using floating point numbers with a finite number of digits?

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

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

Anyways, back to to the question originally posed. Either approach results in the same answer, which is there are none. As defined as set of point equidistant from the center, there are no corners. As a regular polygon at the limit of the number of vertices at infinity, the answer is likewise there are no corners, since in the limit, the included angle becomes a straight angle, and cannot be defined as any sort of "corner."

TTFN
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RE: infinite or none? does a circle has infinite corners or none?

While most versions of engineering software (specifically CAD/CAE/CAM) performs their computations using floating point numbers, that does not mean that there are not closed-formed solutions for calculations involving mathematically determimant objects. For example, the intersection of two Spheres will result in a Circle and NO approximation is needed since this is a closed solution, there is only ONE answer and it doesn't take calculus to compute the results. Now if those 'Spheres' were represented as NURB's or B-surfaces, that would be a different story and while some systems which only represent geometry as NURB's, that's not true of all software (and certainly not our software). The same is true is you were to intersect a Plane with a Cone or Cylinder, or for that matter, another Plane. Why waste compute cycles running a tolerance sensitive, convergent computation when a couple of well understood geometric equations gives you the exact answer in a single pass.

And then there's that small number of systems which don't depend on floating point computations at all, but rather uses integer math, which while it does limit the sorts of computations and geometric representations which can be handled, it's often sufficient for things like AEC and GIS applications. The advantage being that computations are generally fast resulting in efficient and compact data representations.

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.

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

A circle is a circle that has no corners.

It is not possible to draw or simulate one with current technology. What we call a circle in drawings are not but they are a close enough approximation that it is acceptable.

That is all there is to it.

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

"It is not possible to draw or simulate one with current technology."

What do you call a compass?

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

And I'll bet hydroman247 has never used a sliderule either winky smile

I suspect that they only way this argument will be settled is when the primary protagonists agree to meet on the field of honor with Triangles and T-Squares swords

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.

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

JohnRBaker - iterative solutions vs analytical solutions is not the point. Whilst it is true that there is only one answer, in general that answer can only be represented by an approximation. It makes no difference whether it is a very rough approximation such as a line drawn on paper by a pair of compasses, a better approximation such as a number stored with 80 binary digits, or a very precise approximation such as a number stored with some arbitrarily large, but finite, precision, they are all approximations. The things we deal with on a daily basis and call circles are all approximations to the mathematical abstraction of a perfect circle. On that basis referring to a regular polygon with a large number of sides as a circle is perfectly reasonable and correct.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

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

I wasn't referring to how it is 'represented' graphically but rather that mathematically there is no need for an approximation when computing, storing and using a circle within an engineering software package.

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.

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

The food is not the lunch.

It is common engineering folly to confuse the representation or model of a phenomenon as the actual phenomenon.

A circle is an abstract, one with a clear definition. A drawing is only a representation. An n-sided polygon is only an approximation. Whatever intellectual gymnastics one performs to understand a circle is only a mental model.

Strangely, we can only tell how close something comes to being circular within an uncertainty, but we can never verify that something is perfectly circular.

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

Quote:

I wasn't referring to how it is 'represented' graphically but rather that mathematically there is no need for an approximation when computing, storing and using a circle within an engineering software package.

I wasn't referring to how it is represented graphically either, but I'm going to leave it at that.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

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

A corner is located at the intersection of two planes... a circle has no planes... it is the loci of points an equidistant from a single point...

No corners... not an infinite number of them!

Dik

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

You guys are so simple!

Despite the second degree polynomial equation, I can produce a tangent at any point to the curve. Correct? The path can be said to contain infinitely many points, to which I can draw a tangent line.

So the circle, whose circumference is composed of many points approaching infinity in the limit, must have a tangent line associated with each of these infinitely many points. Hence the notion that a circle has infinite tangents, I.e. sides of a polygon circumscribed to such a circumference.

Pay attention to the geometry! Simple Archimedes with a little Euclid. Do the differential calculus, it is intuitively obvious! OMG!

Regards,
Cockroach

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