Can one Calculate Hypotenuse in Following Case?

[Edwin]

I understand that to calculate the length in one dimension requires two points. And to calculate the lengths of the sides of a triangle in two dimensions requires 3 points.

The question I have is for the points A, B, and C at the vertices of a triangle: If one of the points is missing, that is, not on the graph, is it possible to calculate the lengths of all three sides of the triangle?

In other words...

If the length between A and B is the double closed interval [a, b], and the length between A and C is the double open interval (a, c) and the distance between B and C is the double open interval (b, c), can the lengths of the three sides of the triangle be calculated exactly? That is, with the existence of only two (A and B) out of the three points (A, B, and C) on the graph of the triangle, is it possible to have a triangle?

Inquisitively,

Edwin G. Schasteen
egschasteen@yahoo.com

The second question I have, which arrises from the question above...

Can the "limit as x approaches a" (lim x->a) be formally defined (three bar equal sign instead of two) as "the half open interval from x to a where a is the missing point in the interval" [x, a)?


That is:

Does lim x-> def= [x, a)?


Inquisitively,

Edwin G. Schasteen
egschasteen@yahoo.com

 

[dextercioby]

A triangle is completely specified by 3 of its 6 elements,but necesarily one side.So,if u give two sides and the angle between them,then the triangle is uniquetly determined.

 

[HallsofIvy]

First question: what ARE you given? Certainly, if you are given only two points, then there exist an infinite number of other points that WILL make a triangle! If you are given two points, you can find the distance between them. You would still need to know two angles, or one angle and another length, or the other two lengths. Even then, there may be more than one point that would fit.

By the way you seem to have some confusion of terminology:
"the length between A and B is the double closed interval [a, b], "
"length" is a number, not an interval so it is not clear what you are given.

Also, the second question:
"Can the "limit as x approaches a" (lim x->a) be formally defined (three bar equal sign instead of two) as "the half open interval from x to a where a is the missing point in the interval" [x, a)? "

No, it can't. For one thing you don't say the limit of WHAT which is certainly important. For another, a limit is a NUMBER, not an interval!

 

[jdavel]

A triangle is completely specified by 3 of its 6 elements,but necesarily one side...

Not quite! Side-side-angle doesn't work.

 

[dextercioby]

Not quite! Side-side-angle doesn't work.

The case side-angle-angle determines uniquely the triangle.The case angle-side-angle obviously does so.As u pointed out,the case side-side-angle does not uniquely determine the triangle.
Yes,u were right,the way i spelled it,i didn't include the possible cases with their all subcases.I had only thought of the 4 big cases.Didn't go through every possibility.How stupid of me...
Thenx again...I hate when I'm sloppy.I could blame it on the fatigue...

 

[Edwin]

Thanks!

Thanks for the information!

I heard that distance is used to describe the difference between two points in geometry, and that intervals are used to describe the difference between points in differential geometry. Is this true?
I, also, heard that Minkowsky space is used to describe space-time used in General Relativity. I heard that Minkowski Space uses intervals to describe the difference between points. Are these statements true?


Inquisitively,

Edwin G. Schasteen

 

[dextercioby]

Thanks for the information!

I heard that distance is used to describe the difference between two points in geometry, and that intervals are used to describe the difference between points in differential geometry. Is this true?
I, also, heard that Minkowsky space is used to describe space-time used in General Relativity. I heard that Minkowski Space uses intervals to describe the difference between points. Are these statements true?


Inquisitively,

Edwin G. Schasteen

An algebraic view if the notion of distance in simple geometry would tell us that the distance between 2 points in (ordinary) space is the number associated with the length of the closed interval between the 2 points.

Since differential geometry is nothing but topology+algebra+calculus,i guess the definition is basically the same,but the means to calculate it would be different,not applying ordinary geometry's theorems,but (sometimes) complicated formulas using integrals,derivatives and so on...

The notion of "Minkowski (sic) space" is used for describing a 4 dimensional flat manifold called "spacetime (countinuum)" and may be encountered in SR and QFT.GR uses curved 4 dimensional manifolds and normally uses the terminology "curved spacetime".It's just a matter of name conventions.Minkowski was not aware of GR (because at 1909 it didn't exist),so we cannot call the manifold encountered in GR "Minkowski space".Mathematically speaking,we're dealing with 2 differents objects.

Yes,the interval is the fundamental conserved quantity in Special Relativity.The distance between 2 different points in Minkowski space (called "events") is the the relativistic interval between the events.

 

[Edwin]

4-Manifolds

Thanks for the information. So four dimensional manifolds used in General Relativity are a curved version of the flat manifolds known as Minkowski Space. Just out of curiousity, Does General Relativity hold that the curvature of the surface of this manifold is everywhere negative? If the answer is no, does it hold that at anypoint the curvature of space is positive?

Inquisitively,

Edwin G. Schasteen

 

[dextercioby]

Thanks for the information. So four dimensional manifolds used in General Relativity are a curved version of the flat manifolds known as Minkowski Space. Just out of curiousity, Does General Relativity hold that the curvature of the surface of this manifold is everywhere negative? If the answer is no, does it hold that at anypoint the curvature of space is positive?

Inquisitively,

Edwin G. Schasteen

I'm glad u got it right.For the other question,it's a bit more complicated.GR uses Einstein's equations for finding the gravitational field described by the 10 independent components of the metric tensor.Einstein equation link the "fabric/geometry" of spacetime continuum with the matter inside it.
Every geometrical property of spacetime is dictated by the matter inside it,via its energy-momentum 4tensor.

By curvature,which quantitiy u mean??The Ricci scalar??If so,in the absence of matter it is zero.In the presence of matter it could be different.Either positive or negative.As u can see,in GRelativity,everything is relative...

Daniel.