Analytical Geometry: Definition & Basics | Mahmoud

[SpanishOmelette]

Maybe a little bit of a fresh mathematical virgin thing to ask, but what exactly is the definition of analytical geometry?

I am asking for the reason I have recently acquired a book on the subject, but I found it by accident, and I would like to comprehend the most basic concepts of it!

I am interested in geometry.

Mahmoud

 

[HallsofIvy]

I would say "analytic geometry" rather than "analytical geometry" but either way, it means adding a coordinate system so that each point is assigned a pair of numbers, each line corresponds to an equation, etc.

 

[MarcusAgrippa]

Maybe a little bit of a fresh mathematical virgin thing to ask, but what exactly is the definition of analytical geometry?

I am asking for the reason I have recently acquired a book on the subject, but I found it by accident, and I would like to comprehend the most basic concepts of it!

I am interested in geometry.

Mahmoud

Analytical stands in contrast to synthetic. Geometry from the most ancient times till Descartes consisted of axioms and theorems about points planes and lines, the things that could be constructed from them such as triangles, rectangles, regular solids, etc, extended to include the measurement of length angle area and volume, and the objects that could be defined on the basis of measurement, such as circles, ellipses, parabolas and hyperbolae. Things constructed from the basic objects are called synthetic elements. In short, all of geometry until Descartes is synthetic geometry. Non-Euclidean geometries of Bolyai, Lobatchevski and Riemann were also originally developed as synthetic geometries.

When Descartes introduced the concept of coordinates, geometric problems could be reformulated as problems in algebra. The synthesis of geometry with algebra permitted by the introduction of coordinates is called analytic geometry. Analytic is a general work for anything that can be reduced to algebraic calculations and the use of equations and formulae. But beware: the word analytic is also used in mathematics with a very narrow specialised meaning when applied to functions. An analytic function is a function, real or complex, that admits derivatives to all orders - and thus allows the construction of a Taylor series for the function - and whose Taylor series converges in some open domain to the original function from which the series was constructed. This use of the term "analytic" must not be confused with the very much more general use of the word analytic which means able to be reduced to formal calculation in terms of formulae and equations.

There are other types of geometry also: algebraic geometry, differential geometry, integral geometry, and several others too.

 

[MarcusAgrippa]

The term "analytical" is sometimes used in place of "analytic" to avoid confusion with the term "analytic" used in its narrow sense of an infinitely differentiable function whose Taylor series converges to the original function in some open domain.