Cartesian Plane ‚ÄĒ Coordinate System

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.

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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 hor¬≠i¬≠zon¬≠tal axis is known as the abscis¬≠sa axis and is usu¬≠al¬≠ly the x‚ÄĎaxis. The ver¬≠ti¬≠cal axis is known as the ordi¬≠nate axis and is usu¬≠al¬≠ly the y‚ÄĎaxis.

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

Quad¬≠rants are char¬≠ac¬≠ter¬≠is¬≠tic regions of the plane defined by changes in signs on coor¬≠di¬≠nate 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: Iden¬≠ti¬≠fi¬≠ca¬≠tion of the quad¬≠rants of the Carte¬≠sian plane.

As shown in fig¬≠ure 2, quad¬≠rants are defined in the fol¬≠low¬≠ing regions. 

First quad¬≠rant: { x > 0 , y > 0 }

Sec¬≠ond quad¬≠rant: { x < 0 , y > 0 } 

Third quad¬≠rant: { x < 0 , y < 0 } 

Fourth quad¬≠rant: { x > 0 , y < 0 } 

Representation of functions in the Cartesian plane

The Carte¬≠sian plane also serves to rep¬≠re¬≠sent func¬≠tions, i.e. rules that assign each point of the x‚ÄĎaxis to a point of the y‚ÄĎaxis. Fig¬≠ures 3 to 6 show some exam¬≠ples of func¬≠tion graph¬≠ics in the Carte¬≠sian 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 func¬≠tion of x.

Cartesian system in Unity

Carte¬≠sian coor¬≠di¬≠nates are used in var¬≠i¬≠ous places in Uni¬≠ty, for exam¬≠ple to define the posi¬≠tion of a GameOb¬≠ject in the plane or space. But for the pur¬≠pose I want¬≠ed to make a sim¬≠ple func¬≠tion plot¬≠ter using spheres that move on the x‚ÄĎaxis and their posi¬≠tion on the y‚ÄĎaxis is cal¬≠cu¬≠lat¬≠ed using the expres¬≠sion 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

Rec¬≠tan¬≠gu¬≠lar coor¬≠di¬≠nates are often used in Unity.

Under¬≠stand¬≠ing how the Carte¬≠sian plane is used to rep¬≠re¬≠sent points and func¬≠tions in the plane, along with knowl¬≠edge of ana¬≠lyt¬≠i¬≠cal geom¬≠e¬≠try can help us cre¬≠ate new and inter¬≠est¬≠ing solutions. 

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