ANALYTICAL GEOMETRY: WHAT IT STUDIES, HISTORY, APPLICATIONS - MATHS - 2025

The analytic geometry studies lines and geometric shapes by applying basic algebra techniques and mathematical analysis in a given coordinate system.

Consequently, analytical geometry is a branch of mathematics that analyzes in detail all the data of geometric figures, that is, the volume, the angles, the area, the points of intersection, their distances, among others.

The fundamental characteristic of analytical geometry is that it allows the representation of geometric figures through formulas.

For example, the circumferences are represented by polynomial equations of the second degree while the lines are expressed by polynomial equations of the first degree.

Analytical geometry arises in the seventeenth century due to the need to provide answers to problems that until now had no solution. Its top representatives were René Descartes and Pierre de Fermat.

Today many authors point to it as a revolutionary creation in the history of mathematics, since it represents the beginning of modern mathematics.

History of analytical geometry

The term analytical geometry arose in France in the seventeenth century due to the need to give answers to problems that could not be solved using algebra and geometry in isolation, but the solution lay in the combined use of both.

Main representatives of analytical geometry

During the seventeenth century two French by chance in life carried out research that in one way or another ended in the creation of analytical geometry. These people were Pierre de Fermat and René Descartes.

At present it is considered that the creator of analytic geometry was René Descartes. This is due to the fact that he published his book before Fermat's and also in depth with Descartes on the subject of analytical geometry.

However, both Fermat and Descartes discovered that lines and geometric figures could be expressed by equations and equations could be expressed as lines or geometric figures.

According to the discoveries made by the two, it can be said that both are the creators of analytical geometry.

Pierre de Fermat

Pierre de Fermat was a French mathematician who was born in 1601 and died in 1665. During his life he studied the geometry of Euclid, Apollonius and Pappus, in order to solve the measurement problems that existed at that time.

Later these studies triggered the creation of geometry. They ended up being expressed in his book "Introduction to flat and solid places" (Ad Locos Planos et Solidos Isagoge), which was published 14 years after his death in 1679.

Pierre de Fermat applied analytic geometry to Apollonius's theorems on geometric places in 1623. He was also the first to apply analytical geometry to three-dimensional space.

Rene Descartes

Also known as Cartesius, he was a mathematician, physicist, and philosopher who was born on March 31, 1596 in France and died in 1650.

René Descartes published in 1637 his book "Discourse on the method of conducting reason correctly and seeking the truth in science" better known as "The Method" and from there the term analytical geometry was introduced to the world. One of its appendices was "Geometry."

Fundamental elements of analytical geometry

Analytical geometry is made up of the following elements:

The Cartesian coordinate system

This system is named after René Descartes.

He was not the one who named it, nor the one who completed the Cartesian coordinate system, but he was the one who spoke of coordinates with positive numbers allowing future scholars to complete it.

This system is composed of the rectangular coordinate system and the polar coordinate system.

Rectangular coordinate systems

Rectangular coordinate systems are called the plane formed by the tracing of two number lines perpendicular to each other, where the cut-off point coincides with the common zero.

Then this system would be made up of a horizontal line and a vertical one.

The horizontal line is the X axis or the abscissa axis. The vertical line would be the Y axis or the ordinate axis.

Polar coordinate system

This system is in charge of verifying the relative position of a point in relation to a fixed line and to a fixed point on the line.

Cartesian equation of the line

This equation is obtained from a line when two points are known through which it passes.

Straight line

It is one that does not deviate and therefore has neither curves nor angles.

Conics

They are the curves defined by the lines that pass through a fixed point and by the points of a curve.

The ellipse, circumference, parabola, and hyperbola are conic curves. Each of them is described below.

Circumference

Circumference is called the closed plane curve that is formed by all the points of the plane that are equidistant from an interior point, that is, from the center of the circumference.

Parable

It is the locus of the points in the plane that are equidistant from a fixed point (focus) and a fixed line (directrix). So the directrix and the focus are what define the parabola.

The parabola can be obtained as a section of a conical surface of revolution through a plane parallel to a generatrix.

Ellipse

The closed curve that describes a point when moving in a plane is called ellipse in such a way that the sum of its distances to two (2) fixed points (called foci) is constant.

Hyperbola

Hyperbola is called the curve defined as the locus of the points in the plane, for which the difference between the distances of two fixed points (foci) is constant.

The hyperbola has an axis of symmetry that passes through the foci, called the focal axis. It also has another, which is the bisector of the segment that has the fixed points at its ends.

Applications

There are many applications of analytical geometry in different areas of daily life. For example, we can find the parabola, one of the fundamental elements of analytical geometry, in many of the tools that are used daily today. Some of these tools are as follows:

Satellite dish

Parabolic antennas have a reflector generated as a result of a parabola that rotates on the axis of said antenna. The surface that is generated as a result of this action is called a paraboloid.

This ability of the paraboloid is called the optical property or reflection property of a parabola, and thanks to this it is possible for the paraboloid to reflect the electromagnetic waves it receives from the feeding mechanism that makes up the antenna.

Hanging bridges

When a rope supports a weight that is homogeneous but, at the same time, is considerably greater than the weight of the rope itself, the result will be a parabola.

This principle is fundamental for the construction of suspension bridges, which are usually supported by wide steel cable structures.

The principle of the parable in suspension bridges has been used in structures such as the Golden Gate Bridge, located in the city of San Francisco, in the United States, or the Great Bridge of the Akashi Strait, which is located in Japan and connects the Island of Awaji with Honshū, the main island of that country.

Astronomical analysis

Analytical geometry has also had very specific and decisive uses in the field of astronomy. In this case, the element of analytic geometry that takes center stage is the ellipse; Johannes Kepler's law of motion of the planets reflects this.

Kepler, a German mathematician and astronomer, determined that the ellipse was the curve that best fit the motion of Mars; He had previously tested the circular model proposed by Copernicus, but in the midst of his experiments, he deduced that the ellipse served to draw an orbit perfectly similar to that of the planet he was studying.

Thanks to the ellipse, Kepler was able to affirm that the planets moved in elliptical orbits; this consideration was the statement of the so-called second law of Kepler.

From this discovery, later enriched by the English physicist and mathematician Isaac Newton, it was possible to study the orbitational movements of the planets, and increase the knowledge that was had about the universe of which we are part.

Cassegrain telescope

The Cassegrain telescope is named after its inventor, the French-born physicist Laurent Cassegrain. In this telescope the principles of analytical geometry are used because it is mainly composed of two mirrors: the first is concave and parabolic, and the second is characterized by being convex and hyperbolic.

The location and nature of these mirrors allow the defect known as spherical aberration to not take place; This defect prevents light rays from being reflected in the focus of a given lens.

The Cassegrain telescope is very useful for planetary observation, as well as being quite versatile and easy to use.

References

1. Analytic Geometry. Retrieved on October 20, 2017, from britannica.com
2. Analytic Geometry. Retrieved on October 20, 2017, from encyclopediafmath.org
3. Analytic Geometry. Retrieved on October 20, 2017, from khancademy.org
4. Analytic Geometry. Retrieved on October 20, 2017, from wikipedia.org
5. Analytic Geometry. Retrieved on October 20, 2017, from whitman.edu
6. Analytic Geometry. Retrieved on October 20, 2017, from stewartcalculus.com
7. Plane analytical geometry Retrieved on October 20, 2017