geometry

[noxid]

what is the difference between euclidean and non euclidean geometry-thankyou

[Unity+]

what is the difference between euclidean and non euclidean geometry-thankyou

The main difference between the two is their use of different axioms, or assumed rules. For example, Euclidian geometry has the use of rules of parallel lines(Euclid's fifth postulate). Non Euclidian geometry consists of lines that curve towards or away from each other.

 

http://en.m.wikipedia.org/wiki/Non-Euclidean_geometry

[studiot]

Well geometry is a very wide subject that has developed considerably in the nearly two and half thousand years since Euclid.

 

It is this development that has muddied the waters somewhat in the distinction, since in Euclid's day there was only one sort of geometry, some of which we no longer include in modern definitions of 'Euclidian Geometry'.

 

To distinguish I offer a description of modern Euclidian Geometry.

 

Any other sort of Geometry is non-Euclidian.

 

The main characteristic feature of modern Euclidian Geometry is the idea that the distance between two points is given by the square root of the sum of the squares of the coordinate distances between them.

 

In other words the scale factor is the same in all directions at all points in the region of interest.

 

This is not true for instance in Projective Geometry used by artists for perspective and cartographers for mapping, and draftsmen for some aspects of engineering drawing.

 

It is also not true in the Geometry of most surfaces, particularly spheres.

 

One consequence of this property of Euclidian Geometry is that it aligns with what we now call Vector Geometry. That is we can prove theorems in Vector Geometry by Euclid or vice versa. Vector Geometry is itself a development of Coordinate Geometry. So Coordinate Geometry is Euclidian.

We now identify Euclidian Geometry with what we call linear algebra and vector spaces.

 

Another form of Geometry involves the application of the Calculus to Geometry. Euclid did not know about the calculus and this sort, called Differential Geometry, has some Euclidian and some non Euclidian aspects.

Edited April 29, 2014 by studiot

[John]

Simply put, Euclidean geometry satisfies Euclid's five postulates. Notably, in Euclidean geometry, the Euclidean parallel postulate holds. While stated by Euclid in terms of transversals and interior angles, this postulate equivalently says that given a line m and a point P not on m, there is at most one line through P parallel to m.

"Non-Euclidean geometry" is generally taken to mean any geometry in which the Euclidean parallel postulate (or anything logically equivalent to it) is not assumed. Perhaps the best-known examples of this are hyperbolic geometry (in which, given m and P as before, there are at least two lines through P parallel to m) and elliptic geometry (in which parallel lines do not exist).

To avoid possible confusion when reading about or discussing this, note that Euclid's first four postulates are enough to prove that parallel lines exist (and thus you may note that elliptic geometry requires throwing out more than just the fifth postulate). However, the statement that "there is at most one parallel to a given line through any point not on the line" cannot be proven from the other four axioms.

Edited April 30, 2014 by John