Firstly, we take the definition of a circle to be the set up ALL points equidistant from a single point (panamajack already said this). If you try to introduce a “corner” into your circle, the point that forms the corner would no longer be equidistant with all the other points, thus violating the definition of a circle.
I guess the consensus is that i’m wrong. Thanks guys.
huh? why wouldn’t it?
Math is not my strong suit, but wouldn’t this be a sphere, and not a circle?
Since its all points on a single plane it’s a circle.
if it was all points on ALL planes then it’d be a sphere.
is the operative phrase, dtm.
Gotcha. I’m thinking 3-D when you guys are on 2-D.
Sorry, UncleBill. I don’t fly often enough.
… what?
Actually, assuming the room has a flat floor and ceiling, it’s cylindrical, not circular. But if it had a vaulted ceiling (or floor, for that matter), there would be corners where the wall meets the vaults.
Imagine a square and all the points around it. The vertecies will be a different distance from the center than the points along its sides. Imagine a shape with many sides. Its many verticies will be at a different distance from the center than the points along its sides.
I was about to argue… but then I thought about it some more. I get it now. thanks.
If it helps anyone ‘imagine’ a circle with ‘corners’ in it just open MS-Paint (or whatever graphics program you have available) and have it draw a circle. Look closely and you’ll see corners all around it. This is of course becuase your monitor is comprised of pixels so a curve can’t help but ‘stair-step’ across those pixels. Now make your pixels smaller and smaller. Your circle will appear smoother as you go but it never forms a ‘cornerless’ circle.
Of course as others have mentioned in a mathematical (or abstract) sense a circle by definition has no corners. Again as someone else mentioned you have the real world with no perfect circles and you have the abstract ideal.
ElJeffe good answer!
If I were feeling ornery, I’d point out that a circle isn’t, strictly speaking, a function; but there are ways around this that preserve the gist of your answer.
dantheman and Harmoix, we could just call it an n-sphere.
Apparently you’re not joking, although it was surely a joke when someone told you to go sit in the corner of a round room.
One common mathematical definition of a circle is in terms of a locus (set of points). A circle is defined to be the set of all points in a plane that are equidistant from a fixed point. The fixed point is of course the center of the circle, the distance from center to any point on the circle being the radius.
This definiton could also agree with a layman’s definition, since attaching a rope to a particular point on the ground, cutting the rope to some particular length, pulling it tout, and walking the end 'round all the way it will go, traces out a circle.
You seem to be attempting to define a “corner” in terms of three points: (1)Some point on the circle, (2) some other point, infinitely close to the previous point, and (3)some other point between the two.
The problem with that approach is what is commonly inferrred in mathematics from this particular usage of “infinitesimally close.” Indeed, if you are to actually have two distinct points, no matter HOW close together they are, then not only is there one but uncountably infinitely many other distinct points between the two. You seem to be incorrectly assuming that there is just ONE point between the two, therefore if this point is not in, say, the same vertical position as the other two, you therefore have a “corner” in the sense that the line segments connecting these three points form a sharp “angle.”
No so, for indeed if you consider any distinct points on the circle (again, no matter how close together they are, so long as thys are distinct then the curve connecting these two points (and indeed connecting all the other uncuntably infinitely many points between them, is not a striaght line but rather a circlular arc. it it wasn’t a circular arc, then you simply don’t have a circle!
In an attempt to get you to realize the flaw in your process, you will surely agree that this point (the one you allege is immediately adjacent to) some designated point, is either (1)the same point as the initial point, or (2) a different point from the initial one. If it’s the same point, then it’s the same point (obviously!) and there can be no other point “between”. If it’s a different point, which you apparently allege, then WHICH different point do you have in mind? IOW, you name me two distinct points on a circle, and I can always name you one that is in between the two, no matter HOW close together your two points are.
Therefore, you have not really described a second point, much less any third point between the two that supposedly “connects” them. Or, if you want to use a different terminology, “immediately adjacent to.” To put it another way, consider an ordinary real number line. Every real number corresponds to precisely one point on this number line, and conversely, every point on this number line corresponds to precisly one real number:
<--------------------------->
Now, let’s label a point. Say, the point corresponding to the number 0:
<-------------0------------->
There is NO real number, hence no “point” on this number line, that is THE SMALLEST number greater than 0. IOW, there is no single number that is “immediately adjacent to” 0.
Real numbers do, however, posses a property known as connectedness, but only in an abstract type of way. The set of real numbers is indeed “conncected” in the mathematical sense, but I don’t think this abstract mathematical property of real numbers is what you actually have in mind for this problem, though I could be wrong.
::snicker::
Did I really write that?
:smack:
from www.m-w.com (merriam-webster site)
Corner
Main Entry: 1cor·ner
Pronunciation: 'kor-n&r
Function: noun
1 a : the point where converging lines, edges, or sides meet :
Circle
Main Entry: 1cir·cle
Pronunciation: 's&r-k&l
Function: noun
1 a : RING, HALO b : a closed plane curve every point of which is equidistant from a fixed point within the curve c : the plane surface bounded by such a curve
You sure did. (Or perhaps it wasn’t you, but some Freudian sub-conscious level of you).
OK next question: do we really have to keep repeating it?
Well, if you want a mathematical proof:
we define cornerless as a function being “well behaved”, ie, it is differentiable at all points.
We define a generalised circle as the function fulfilling the criteria x^2 + y^2 = 1. All circles are merely a translation and enlargment.
Doing a polar differentiation (to avoid pesky infinities), its clear that the differentiation is 0 at all points thus, the circle is cornerless.
In this context, we are saying that a “corner” is a sharp turn or bend in the graph, as in f(x)=|x| at x=0. This is commonly called a “cusp.”
**
In polar coordinates, the unit circle x^2+y^2=1 is:
r = 1
Why do you think the derivative is 0 at all points on this circle? It’s quite clear, even from a polar plot, that this graph has horizontal tangets (derivative=0) at only two points, and equally clear that we have vertical tangents at two points.
Hint-- dy/dx in polar form is not simply dr/d(theta).