Cartesian Plane – Coordinate System

By GameDevTraum

Introduction

The Cartesian coordinate system is named after the philosopher and mathematician René Descartes, considered the creator of analytical geometry.

This system allows us to represent points on a line, in the plane and in space using arrays of numbers. For example: (1,5), (-3,0), (4,1,-1), etc.

What is a Cartesian plane?

The Cartesian plane is defined by two straight lines perpendicular to each other, one horizontal and one vertical, these straight lines are known as Cartesian axes. Using these axes we can identify any point on the plane with a single ordered pair of numbers.

The horizontal axis is known as the abscissa axis and is usually the x-axis. The vertical axis is known as the ordinate axis and is usually the y-axis.

The point where the abscissa and ordinate axes are cut is known as coordinate origin or simply origin and is the point (0.0).

The following figure shows a Cartesian plane with the point (2,1) plotted.

representation in Cartesian plane with point 2 1 represented
Fig. 1: Plano cartesiano xy con el punto (2,1) representado.

Quadrants of the Cartesian plane

Quadrants are characteristic regions of the plane defined by changes in signs on coordinate axes.

There are four quadrants, the first is the upper right region and they are listed counterclockwise.

four quadrants of the Cartesian plane
Fig. 2: Identification of the quadrants of the Cartesian plane.

As shown in figure 2, quadrants are defined in the following regions.

First quadrant: { x > 0 , y > 0 }

Second quadrant: { x < 0 , y > 0 }

Third quadrant: { x < 0 , y < 0 }

Fourth quadrant: { x > 0 , y < 0 }

Representation of functions in the Cartesian plane

The Cartesian plane also serves to represent functions, i.e. rules that assign each point of the x-axis to a point of the y-axis. Figures 3 to 6 show some examples of function graphics in the Cartesian plane.

graph of a linear function in the Cartesian plane.
Fig. 3: Graph of a linear function.
graph of a cubic function in the Cartesian plane.
Fig. 5: Graph of a cubic function.
graph of a quadratic function or parabola in the Cartesian plane.
Fig. 4: Graph of a quadratic or parabola function.
graph of a sine function of x in the Cartesian plane.
Fig. 6: Graph of a sine function of x.

Cartesian system in Unity

Cartesian coordinates are used in various places in Unity, for example to define the position of a GameObject in the plane or space. But for the purpose I wanted to make a simple function plotter using spheres that move on the x-axis and their position on the y-axis is calculated using the expression of the function.

The buttons allow us to change the expression of the function

scene in unity in which spheres are observed forming a linear function
Fig. 7: The spheres move according to a linear function.
scene in unity in which spheres are observed forming a quadratic function
Fig. 9: The spheres move according to a quadratic function.

scene in unity in which spheres are observed forming a sine wave function
Fig. 8: The spheres move according to a sinusoidal function.
scene in unity in which spheres are observed forming a sine wave function in one section and a linear function in another.
Fig. 10: The spheres move according to a sinusoidal function in one section and a linear function in another.

Conclusion

Rectangular coordinates are often used in Unity.

Understanding how the Cartesian plane is used to represent points and functions in the plane, along with knowledge of analytical geometry can help us create new and interesting solutions.