Cartesian Coordinates vs Polar Coordinates
In Geometry, a coordinate system is a reference system, where numbers (or coordinates) are used to uniquely determine the position of a point or other geometric element in space. The coordinate systems allow the geometrical problems to be converted into a numerical problem, which provides the basis for Analytic Geometry.
Cartesian coordinate system and the Polar coordinate systems are two of the common coordinate systems used in mathematics.
Cartesian coordinate system uses the real number line as the reference. In one dimension, the number line extends from negative infinity to positive infinity. Considering the point 0 as the start, the length to each point can be measured. This provides a unique way of identifying a position on the line, with a single number.
The concept can be extended into two and three dimensions where number lines perpendicular to each other are used. They all share the same point 0 as the start. The number lines are termed as axes, and often called X axis, Y axis, and Z axis. The distance to a point along each axis starting from (0, 0, 0), which is also known as the origin, and given as a tuple is known as the coordinate of the point. A general point in this space can be represented by the coordinate (x,y,z). In a plane system where there are only two axes, coordinates are given as (x,y). A plane created by the axes are known as a Cartesian plane, and often referred to by the letters of the axes. E.g. XY plane.
This general point can be used to describe different geometrical elements by constraining the general point to behave in specific ways. For example, equation x^2+y^2=a^2 represents a circle. Rather than drawing a circle with radius a it is possible to denote the circle with more abstract way shown above.
Polar coordinates use a difference reference system to denote a point. Polar coordinates system uses the counter clockwise angle from the positive direction of x axis and the straight line distance to the point as the coordinates.
The polar coordinates can be represented as above in the two dimensional Cartesian coordinates system.
The transformation between polar and Cartesian systems is given by following relations:
r = √(x2 + y2) ↔ x = r cosθ, y = r sinθ
θ = tan-1 (x/y)
What is the difference between Cartesian and Polar Coordinates?
• Cartesian coordinates use number lines as the axes, and it can be used in one, two or three dimensions. Therefore has the ability to represent linear, planar, and solid geometries.
• Polar coordinates use an angle and a length as the coordinates, and it can represent only linear and planar geometries, though it can be developed into cylindrical coordinates system, to represent solid geometries.
• Both systems are used to represent imaginary numbers by defining the imaginary axis, and play a vital role in complex algebra. Though, in the plain form, Cartesian coordinates are real numbers (x,y,z) the coordinates in the polar system are not always real numbers; i.e. if the angle is given in degrees, coordinates are not real; if the angle is given in radians coordinates are real numbers.