2D Coordinates Systems

A 2D coordinate system is used to identify and locate points in the (two-dimensional) plane. The system is generally provided with a frame with an origin of coordinates (0,0).

There are several types of 2D coordinate systems:

— The Cartesian coordinate system, the most common, having a reference with 2 perpendicular axes noted x and y for abscissas and ordinates respectively.

— The polar coordinate system, identifying a point by its distance from the origin and by an angle

— Other less common systems such as the parabolic system, the barycentric or the elliptical

The base / referential change using cartesian coordinates $ (x, y) $ to another referential using polar coordinates $ (r, \theta) $ obeys the equations: $$ r = \sqrt{x^2 + y^2} \\ \theta = 2\arctan\left(\frac y{x+ \sqrt{x^2+y^2}} \right) $$ with $ \arctan $ the reciprocal of the function $ \tan $ (tangent).

The base / referential change from polar coordinates $ (r, \theta) $ to another referential using cartesian coordinates $ (x, y) $ follows the equations: $$ x = r \cos(\theta) \\ y = r \sin(\theta) $$

with $ r $ a positive real number and $ \theta $ an angle defined between $ ] -\pi, \pi ] $

The choice of the appropriate 2D coordinate system depends on the nature of the problem to be solved. Cartesian coordinate systems are often used to solve problems involving functions or polynomials, while polar coordinate systems are used to solve problems involving circles or complex numbers.