Definition of Analytical Geometry

Thegeometryis the area within mathresponsible for the analysis of the properties and the measures that the figures, either in space or in the plane, meanwhile, within geometry we find different classes: Descriptive Geometry, Plane Geometry, Space Geometry, Projective Geometry, and Analytic Geometry.

Branch of geometry that analyzes geometric figures through a coordinate system

For its part, analytic geometry is a branch of geometry that focuses on the analysis of geometric figures starting from a coordinate system and using the methods of algebra and mathematical analysis.

We must say that this branch is also known as Cartesian geometry and that it is a part of geometry that is widely used in various fields such as physics and science. engineering.

The main claims of analytic geometry consist in obtaining the equation of the coordinate systems from the geographic location they have and once the equation is given in the coordinate system, decide the locus of the points that allow to verify the given equation.

It should be noted that a point on the plane that belongs to a coordinate system will be determined by two numbers, which are formally known as abscissa and coordinate of the point. In this way, two ordered real numbers will correspond to every point in the plane and vice versa, that is, to every ordered pair of numbers a point in the plane will correspond.

Thanks to these two questions, the coordinate system will be able to obtain a correspondence between the geometric concept of the points of the plane and the algebraic concept of the ordered pairs of numbers, thus applying the bases of analytical geometry.

Likewise, the aforementioned relationship will allow us to determine plane geometric figures, by means of equations with two unknowns.

Pierre de Fermat and René Descartes, its pioneers

Let's do a bit of history, because as we know mathematics and of course geometry have also been subjects that were approached from there far back in time by various men of science and intellectuals, who with few tools but much enthusiasm and lucidity managed to contribute an enormous baggage of conclusions and topics about them, which would later become principles and theories that continue to be taught to the day of today.

The French mathematicians Pierre de Fermat and René Descartes are the two names behind and closely linked to this branch of geometry.
Precisely the name of Cartesian geometry has had to do with one of his pioneers, and as a tribute it was decided to name it that way.

In the case of Descartes, he made important contributions that would later be immortalized in his work, Geometry, which would be released in the seventeenth century; on the side of Fermat and almost on a par with his colleague, he also contributed his own through the work Ad locos blueprints et solidos isagoge

Today both are recognized as the great developers of this branch, however, in their time, Fermat's works and proposals were better received than those of Descartes.

The great contribution made by these is that they appreciated that algebraic equations correspond to geometric figures and that implies that lines and certain geometric figures can also be expressed as equations, and at the same time the equations can be represented as lines or figures geometric.

Thus the lines can be expressed as polynomial equations of the first degree and the circles and the other conic figures as polynomial equations of the second degree.