point, line and circle

[tgardzie22]

A point,a line and a circle are different but can be same.If the length of a line =0 it becomes a point.If the diameter of a circle=0 it also becomes a point.One can say:if the"length" of a point= infinity it becomes a line and if the "diameter" of a point=infinity it becomes a circle.What if any value for poin,line and circle is

negative(less than zero)?

[Sato]

A line with length 0 is only a point; a "line" is defined as something with length, and so if it does not have length, it isn't a line. Similarly, a necessary condition for some object to be a circle (by its definition) is that it has a circumference, and if the diameter is 0, then so is the radius, in turn the circumference is nil, and so it is not a circle, just a point. A point is defined as an individual with no magnitude (besides maybe relative position), and so it cannot be stretched to infinity. You can, however, consider the distance between two points, or all the points equidistant from some particular point, and you will have a line or a circle.

[tgardzie22]

Why thoreticly a point cannot streach and line and circle cannot shrink? May be I should move this topic to phisics?

[Bignose]

Why thoreticly a point cannot streach and line and circle cannot shrink? May be I should move this topic to phisics?

It's not that 'theoretically' a point cannot stretch and so on. It's just using definitions properly. Once you 'stretch' a point, it isn't really a point anymore. By definition.

 

Really you're asking "if you chance the definitions of mathematics, what does mathematics say?" and the answer is "well, since you changed the definitions, mathematics really doesn't say anything..."

 

So, yeah, It isn't really a questions for mathematics, physics, or anything else. If you change definitions this radically, why stop there? You can say anything you want. Doesn't mean it is actually scientific or mathematics.

Edited January 25, 2015 by Bignose

[John]

These questions aren't entirely invalid.

"Line" and "point" are usually taken as primitive objects in geometry, and the properties of a line or point in any given geometry depend on the axioms used to describe them. Usually we define a line as containing at least two distinct points, so in that sense shrinking a line to length zero doesn't really work. You might enjoy reading about the concept of duality in projective geometry, in which we interchange the roles of points and lines in a projective plane while preserving incidence. However, this isn't directly related to what you seem to be describing.

Shrinking a circle such that r = 0 produces an example of a degenerate conic.

 

As for negative lengths (or negative sizes in general), in measure theory there is the concept of signed measure (measure is, in a sense, a generalization of the concept of size).

 

The study and application of these concepts involve some fairly high-level mathematics (namely, it seems: geometry, algebraic geometry, and measure theory, of which I've studied only the first and a little of the third, so I can't provide too much information or insight), but if they're more than just passing points of curiosity for you, they're there for the learning.

Edited January 25, 2015 by John

[ajb]

Why thoreticly a point cannot streach and line and circle cannot shrink? May be I should move this topic to phisics?

This reminds me of blowups or maybe homotopy. (This is all mathematics by the way, of course these ideas can be very important in physics)

[studiot]

The ancient Greeks thought long and hard about this question and they came up with a notion that, when put in modern terms boils down to the idea of dimension.

 

Euclid definition 1

A point is that which has no part

 

Euclid definition 2

A line is a breadthless length

 

Euclid definition 3

The extremities of a line are points

 

(John made this point)

 

Euclid definition 15

A circle is a plane figure contained by one line such that all the lines falling upon it from one point are equal.

 

A modern translation of the above is that you require zero dimensions for a point, one dimension for a line and two dimensions for a circle.

Extending that you require three dimensions for a sphere. The corresponding figure for dimensions greater than three is called a ball or an n-ball, and sometimes a ball is used in three dimensions.

Edited January 25, 2015 by studiot

[ajb]

A modern translation of the above is that you require zero dimensions for a point, one dimesnion for a line and two dimensions for a circle.

Just a warning to avoid potential confusion. By a 'circle' one could mean the circle 'O' or one could mean the disk, that is the interior of the circle. It depends to some extent if you like geometry or topology as the two meanings get mixed a little. So, for me a circle is a one dimensional manifold while the disk is a two dimensional manifold with a boundary.

[michel123456]

The OP is about negative.

[studiot]

 

The OP is about negative.

 

 

Yes the OP starts from a misconception and several have tried to explain this in different ways.

 

A point has no length, of any value.

[Strange]

The OP is about negative.

 

That doesn't really mean anything. Unless you treat the length of a line as a vector, in which case negative length simply means a line pointing in the opposite direction.

[michel123456]

 

That doesn't really mean anything. Unless you treat the length of a line as a vector, in which case negative length simply means a line pointing in the opposite direction.

Yes. Negative length would mean negative space. And there is no such a thing.

[Sensei]

[math]Length =\sqrt{x^2+y^2}[/math] for 2 dimensions

[math]Length =\sqrt{x^2+y^2+z^2}[/math] for 3 dimensions

[math]Length =\sqrt{x^2+y^2+z^2+...+zz^2}[/math] for n-dimensions

 

Where x,y,z....zz are relative. i.e. x=x0-x1,y=y0-y1 etc.

 

Powering negative number causes it to be always positive, then getting square from only positive will return only positive results.

 

Line must be closed line segment (have two ending it points)

http://en.wikipedia.org/wiki/Line_segment

if line is half-open line segment, or open line, it's length is infinite and above equation for length of line cannot be used.

 

http://en.wikipedia.org/wiki/Euclidean_distance

Edited January 25, 2015 by Sensei

[michel123456]

[math]Length =\sqrt{x^2+y^2}[/math] for 2 dimensions

[math]Length =\sqrt{x^2+y^2+z^2}[/math] for 3 dimensions

[math]Length =\sqrt{x^2+y^2+z^2+...+zz^2}[/math] for n-dimensions

 

Where x,y,z....zz are relative. i.e. x=x0-x1,y=y0-y1 etc.

 

Powering negative number causes it to be always positive, then getting square from only positive will return only positive results.

 

Line must be closed line segment (have two ending it points)

http://en.wikipedia.org/wiki/Line_segment

if line is half-open line segment, or open line, it's length is infinite and above equation for length of line cannot be used.

 

That is not accurate.

Mathematically, a square root has two results, one positive, one negative.

Simply one has to discard the negative result because it does not seem to have any physical correspondence with reality.

[Sensei]

 

Simply one has to discard the negative result because it does not seem to have any physical correspondence with reality.

 

I am going from home to shop, straight line with length.

Then from shop to home, it's the same equal length.

One can calculate length using above equation looking from home to shop, or from shop to home, and it'll be the same scalar length.

Powering negative number causes it to be always positive, then getting square from only positive will return only positive results.

 

That is not accurate.

Mathematically, a square root has two results, one positive, one negative.

 

Rethink once again bold by you sentence. You have positive number and taking square from positive. What will be result?

[michel123456]

 

Rethink once again bold by you sentence. You have positive number and taking square from positive. What will be result?

I will not change mathematical rules.

 

[math]-2^2=4[/math]

 

I know we have an obsession with positive results, but negative results still exist. Mathematically speaking.

Edited January 25, 2015 by michel123456

[Sensei]

I will not change mathematical rules.

 

[math]-2^2=4[/math]

 

I know we have an obsession with positive results, but negative results still exist. Mathematically speaking.

 

Powering number to 2 is equal to multiplication of number by itself.

Multiplication of negative real number by negative real number, will give positive real number.

[math](-2)*(-2)=+4[/math]

[michel123456]

Yes, that's what I said.

[latex]\surd4=\pm2[/latex]

[studiot]

 

Yes. Negative length would mean negative space. And there is no such a thing.

 

 

This discussion brings a chuckle.

 

Negative length is a perfectly valid mathematical concept, much used by those concerned with geometrical optics.

 

 

 

 

I think ajb meant his topologies to be Hausdorff.

I think that means they are also metric spaces, which means they have a metric.

(I'm sure ajb will correct me if I am wrong here)

 

The first metric space axiom is that it possesses a non negative distance function.

 

Using the square root as a distance function therefore precludes the negative root by the first metric space axiom.

 

However this colloquial definition of a Hausdorff space exemplifies the idea that a (finite) line require two points:

 

"A space is Hausdorff if any two points can be housed off by separation in disjoint sets."

[ajb]

I think ajb meant his topologies to be Hausdorff.

It is usual to look for Hausdorff topologies, but sometime they are not enough.

 

I think that means they are also metric spaces, which means they have a metric.

(I'm sure ajb will correct me if I am wrong here)

All metric spaces are Hausdorff. In converse, every Hausdorff second-countable regular space is metrizable; that is homomorphic to a metric space.

 

The first metric space axiom is that it possesses a non negative distance function.

The 'next best' thing is to equip you space with a quadratic form. Using that you can define most of what you would like from a metric or a norm, depending on the properties of that form. Such things are important in physics, for example the Minkowski metric is an indefinite (i.e. nether positive of negative) quadratic form.

Edited January 26, 2015 by ajb

[michel123456]

Yes. Negative length would mean negative space. And there is no such a thing.

Maybe I was wrong.

 

I was pondering: if you take a simple orthogonal triangle, mathematically one can get the value for the hypotenuse through the Pythagorean theorem

[math]a^2+b^2=c^2[/math]

And thus obtain mathematically a set of 2 hypotenuses, one positive, one negative. And I was wondering, were is the negative one?

Then I thought that in fact there are 2 triangles that correspond to the original definition. See below

 

 

The 2 triangles are mirrored like the left & right hand

 

So there are 2 triangles and subsequently 2 hypothenuses. If one is considered positive, then the other will be negative. And there is no need for negative space.

[ajb]

You are trying to define some notion of parity or orientation here. This is not the same as 'negative lengths'.

 

We kind of have these zero and negative lengths when we look at pseudo-Riemannian metrics as found in relativity.

[studiot]

Michael, your mirror image is almost there but the image is still real.

I have already pointed out that virtual images in optics lead to negative lengths.