- What is analytical geometry?
- History of analytical geometry
- Applications of analytical geometry
- Analytical Geometry Formulas
We explain what analytical geometry is, its history, characteristics and most important formulas. Also, its various applications.
What is analytical geometry?
Analytical geometry is a branch of mathematics dedicated to the in-depth study of geometric figures and their respective data, such as areas, distances, volumes, intersection points, tilt angles, and so on. To do this, he uses basic techniques of mathematical analysis and algebra.
It uses a coordinate system known as the Cartesian plane, which is two-dimensional and is composed of two axes: one of abscissa (x-axis) and another of ordered (Axis y). There you can study all the geometric figures let them be from our interest, assigning to each point of the same a specific place of coordinates (x, y).
Thus, analytical geometry analyzes usually comprise the mathematical interpretation of a geometric figure, that is, the formulation of equations. Or it could be the opposite: the graphical representation of a mathematical equation. This equivalence is found in the formula y = f (x), where f is a function of some kind.
Analytical geometry is a fundamental field of math which is usually part of the high school curriculum.
History of analytical geometry
The founder of this field of study is considered the French philosopher René Descartes (1596-1650), with the appendix entitled “The Geometrie”In his famous work Discourse of the method.
However, in the 11th century, the Persian mathematician Omar Khayyam (c.1048-c.1131) used similar ideas, which Descartes could hardly know. In other words, they both probably invented them on their own.
Given the secrecy of Descartes' ideas, the Dutch mathematician Franz van Schooten (1615-1660) and his collaborators expanded, developed, and disseminated analytic geometry in the West. It used to be called "Cartesian Geometry", to pay homage to its creator, but that term today prefers to be used to refer only to the appendix written by Descartes.
Applications of analytical geometry
Analytical geometry is one of the most useful conceptual tools in science. humanity, and today its applications can be seen in, to name a few examples:
- The hanging bridges. From the old wooden suspension bridges, to their modern versions with steel cables, the geometric principle of the parabola is applied in each of them.
- Satellite dishes. Satellite dishes to capture information satellite have the shape of a paraboloid, generated by its reflector that rotates on the axis, chasing the signal. Thanks to the reflection property of the parabola, the dish of the antenna can reflect the satellite signal towards the feeding device.
- Astronomical observation. The celestial bodies they orbit on a path that describes an ellipse, as deduced by Johannes Kepler (1571-1630), and not a circumference, as Copernicus (1473-1543) believed. These calculations were possible only using Analytical Geometry.
Analytical Geometry Formulas
Analytical geometry offers formulas for geometric figures.
Geometry studies geometric figures and obtains their basic equations, such as:
- The lines are described by the formula ax + by = c.
- Circles are described by the formula x2 + y2 = 4.
- Hyperbolas are described by the formula xy = 1.
- The parables are described by the formula y = ax2 + bx + c.
- Ellipses are described by the formula (x2 / a2) + (y2 / b2) = 1.