Comments on: What’s the difference between regular geometry and analytical geometry?

By: railbuff

railbuff — Wed, 07 Oct 2009 03:57:59 +0000

Regular or synthetic geometry is the geometry of points, lines, planes, triangles, etc. best represented as the geometry presented by Euclid in the first few books of his Elements.
It deals with the logical proofs of theorems about relationships, such as congruence of triangles, whether lines intersect or not, for example.
The logic is based on his axioms. e.g Through two distinct points, one and only one straight line may be drawn. This is said to be self-evident and is not proven. The usual line of attack deals with lines, then triangles, quadrilaterals, circles, tangents to circles each new theorem building on the ones proven before. One of the axioms states that through a point on on a line, one and only one line can be drawn parallel to the given line. This was an amazing accomplishment that withstood criticism for nearly 2000 years.
To see what happened in the 1700s-1800s, we would have to look at non-Euclidean geometry, but we haven’t time.
Analytic geometry was codified by Descartes, and it applies algebra to geometry - representing lines through equations. It is more amenable to stating real world problems - giving specific answers to specific questions. Synthetic geometry goals were more about the characteristics of all of a class - all triangles, while analytic does apply more readily to this triangle, not all of them.
On the other hand, analytic geometry also deals with proofs - by seeking general equations for certain geometric forms, like ellipses.

Hope this whets your appetite for more. Try Wikipedia
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Retired math teacher

By: James H

James H — Wed, 07 Oct 2009 03:10:59 +0000

I think the analytical geometry is the coordinate geometry where we use the distance formula, midpoint formula, and coordinate graphs.
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By: CogitoErgoCogitoSum

CogitoErgoCogitoSum — Wed, 07 Oct 2009 02:55:59 +0000

You say that you already got into analytical geometry? Then you should already have a taste of what it is and how it is distinguished from "regular geometry"

I am not sure what analytical geometry is. Except that, perhaps, it has numerical values applied to them… in much the same way that Euclidean geometry differs from Cartesian. Perhaps it involves computation more. Or continuously similar shapes embedded, like fractals or other such things?

"Regular geometry", that you learn in high school, is only two-dimensional Euclidean geometry. You can get into three dimensional, four dimensional, … or even a million dimensional geometry. You can get into hyperbolic, spherical or any other sort of curved-space, non-Euclidean geometry. There are a lot and a lot of different "geometries" out there.
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