Different Types of Coordinate Systems

The coordinate system’s primary role is to determine points on the plane in a unique manner. Their study of their system led to coordinate geometry advancements. The study was a development effort by Rene Descartes, a French Philosopher, and Mathematician; as it is widely known, a number line is a line representing positive and negative directional integers initiating from the origin. The idea of placing two lines perpendicular to a plane and point localisation on them by reference was an invention by Rene Descartes. The article will feature different types of coordinate systems. 

Number line coordinate system

One of the simplest types of the coordinate system is primarily used for point identification on a line by using real numbers using a number line.  

At this point of origin, O is chosen on the line. The point P coordinate on the line is defined as the signed distance from point O to P, where this distance is taken as positive or negative depending on the side point P is noted. Each point o the line is given a unique coordinate, and a whole number signifies each. 

Cartesian coordinate system

As stated in the introduction, the concept of perpendicular lines in mutual form is for the detonation of a point coordinate. For localisation of the point position in a plane made up of two perpendicular lines, the Cartesian coordinate system is used. It is a type of coordinate system in which the points are denoted or represented in (x,y) coordinates in two-dimensional form and the x and y-axis, respectively.  

The coordinate about the x-axis is the perpendicular distance from the axis of y. It is measured along the x-axis, and the whole concept is called Abscissa. The Y coordinate point is the perpendicular distance from the x-axis, calculated or measured along the y-axis, basically known as the Ordinate.

Polar Coordinate System

In this type of coordinate system, a point is chosen firstly as a pole, and the ray from a particular point is taken as the polar axis. Two parameters are the possible angle while the other is the radius. The angle with a polar axis named Ɵ has a single line through the pole and is measured from axis to line in an anticlockwise direction. The points will have several different distances from the origin point ®. Therefore, the polar coordinate system point is depicted as coordinates pair (r, Ɵ). The pole representation by (0, Ɵ) for any Ɵ value, where r = 0.

Polar coordinates of the same point are as follows: (r, Ɵ), (r, Ɵ + 2π), and (-r, Ɵ + π). The distance from the pole is called the radial distance, radial coordinate,or simply radius, and the polar angle, angular coordinate, or azimuth.

X = r cos Ɵ and y = r sin Ɵ

r = (x2 + y2) ½ and tan Ɵ = (yx)

The curve polar equation consists of the form (r, Ɵ).

In the case of the circle equation with (R, β) centre and radius, a is r2 – 2rR cos(Ɵ – β) + R2 = a2.

The lines running through the pole, also called radial lines, are depicted by the equation: Ɵ = β.

Some Important Terms

Cartesian Plane:

Cartesian planes constitute quadrants and axes. The plane is named Coordinate or  Cartesian Plane, or the x-y plane. The axes are named coordinate axes. 

Quadrants:

The plane is divided into four parts, and these parts are named quadrants, specifically a one-fourth part. Further, these quadrants are numbered in an anticlockwise manner from OX as I, II, III, and IV.

One dimensional Cartesian coordinate system:

For one dimension, the cartesian coordinate system is only a line. A point O is chosen, line origin, length unit, and line orientation. The orientation determines which of the half’s lines is positive and which is negative. Point P on the line can be dignified by the distance from O taken with either of the signs. 

Number line:

The line with a Cartesian coordinate system chosen is called a number line. Every number has a unique location on the line. Every point on this line can be denoted as a number.

Plane Cartesian Formulae 

Two points distance

Two point distance of plane (x1, y1) and (x2, y2) is as follows:

d = [(x2 – x1)2 + (y2 – y1)2 ]1/2

When it comes to a 3D system, the distance formula is as follows between (x1, y1, z1) and (x2, y2, z2):

d = [(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2 ]1/2

Vector Representation

In 2D, the origin vector to the point of coordinates of Cartesian (x, y) can be stated as r = xi + yj where i is (1,0), and j is (0,1), denoting them as unit vectors in the x-axis and y-axis direction respectively.  

In 3D, r = xi + yj + zk, where k is (0,0,1) as unit vector in z-axis direction.

Conclusion

Mathematics in older times was divided into geometry and Algebra. Rene Descartes combined these two branches as a French Mathematician of the primary time. He developed the concept of the frame of reference to incorporate both algebra and geometry. Most different types of coordinate systems use a coordinate, two numbers, and some point of reference fixed, particularly named origin point. The x value is the primary indicator of how and which direction from the origin, such as left or right purpose. The second number signifies how far or below the purpose is from the point of origin, referred to as the Y value. The point of origin has a coordinate of 0, 0.