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?

 

[IRstuff]

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
faq731-376

 

[JohnRBaker]

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.

 

[hydroman247]

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.

 

[btrueblood]

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

What do you call a compass?

 

[JohnRBaker]

And I'll bet hydroman247 has never used a sliderule either ;-)

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

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.

 

[IDS]

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/

 

[JohnRBaker]

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.

 

[TheTick]

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.

 

[IDS]

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/

 

[dik]

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

 

[Cockroach]

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

 

[patprimmer]

Whatever

Regards
Pat
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