A Mathematica Package for Graphing Equations and Inequalities in Non-Rectangular Coordinate Systems

Abstract

Solving and plotting inequalities are crucial mathematical skills that have a significant impact on problem-solving in a variety of disciplines. These tools provide an efective way to represent and analyze numerical and geometric relationships, facilitating decision-making, planning, and understanding complex situations in everyday life and professional fields. This paper presents and describes a new package coded in the Wolfram Language of the Mathematica symbolic calculation system, NRGraphics, which allows plotting equations and inequalities in non-rectangular plane coordinate systems. The authors have developed this package to provide a tool for the teaching process of graphical representation of equations and inequalities in non-rectangular plane coordinate systems.

Keywords

• eol>Non-rectangular coordinates
• Wolfram Language
• equations
• inequalities

1. Introduction

Solving inequalities constitutes an essential topic in mathematics with outstanding applications in many problems of theoretical and applied science [ 1, 2, 3, 4 ].

The symbolic calculation system Mathematica stands out on this topic among its competitors because it includes commands that solve inequalities that involve more than one variable [ 5 ]. In addition, it contains commands that allow obtaining the graphs of the regions of the plane in the Cartesian coordinate system, which are the solution set of inequalities of this type [ 6 ].

However, although there are many general-purpose or specific-purpose computer algebra systems [ 7 ], none have been found that allow obtaining the graphs of equations and inequalities in non-rectangular plane coordinate systems. 2. Domain theory and previous work The graphical representation of inequalities in the Cartesian coordinate system in the plane can be performed without significant complications with the RegionPlot command incorporated into Mathematica [ 6 ]. However, Mathematica does not include any command to graph inequalities in non-Cartesian coordinates in the plane.

There is a contribution from user Heike on the Mathematica Stack Exchange blog to graph equations in polar coordinates given implicitly [ 8 ]. From this contribution, the fundamental idea can be extracted to develop an algorithm to modify points on the plane graphed with

Mathematica via transformations.

On the other hand, Mathematica incorporates the PolarPlot command. However, this command is limited to polar coordinates and explicit expressions of = ( ). On the other hand, our package is not limited to polar coordinates and can operate, in particular, with implicit expressions of the form (, ) = 0. The following example shows us the existing limitation.

Mathematica Is impossible plot 2 − cos(2 ) = 0 with PolarPlot.

In[ 1 ]:= PolarPlot[r^2-Cos[2* ]==0,{r,0,1},{t,0,2*Pi}]

Out[ 1 ]= PolarPlot::nonopt

3. The package: NRGraphics

The NRGraphics package includes two commands: ContourNonCartesianPlot and

InequalityNonCartesianPlot. The syntaxes of both commands are:

ContourNonCartesianPlot[(, ), {, 1, 2}, {, 1, 2},options] and

InequalityNonCartesianPlot[(, ), {, 1, 2}, {, 1, 2},options] (, ) is an equation for and ; (, ) is an expression of inequalities and Boolean operators.

All graphical results are displayed in the rectangle:

1 ≤ ≤ 2 ∧ 1 ≤ ≤ 2 .

Both commands are similar to the ContourPlot and RegionPlot commands, respectively, that Mathematica incorporates.

This package is hosted at

And the way to install it is explained in https://support.wolfram.com/5648?src=mathematica

4. Results

First, we initialize the package:

Mathematica

Initialization of the NRGraphics.m package.

In[ 1 ]:= «NRGraphics.m

Below are examples of graphs obtained with the commands incorporated in the package and whose syntax was disclosed in the previous section.

Mathematica Plot the contours of 2 − cos(2) in polar coordinates.

In[ 2 ]:= ContourNonCartesianPlot[r^2-Cos[2*t],{r,0,1},{t,0,2*Pi}] Out[ 2 ]= See Fig. 1 (left).

Mathematica Plot an equation, 2 − cos(2) = 0, in polar coordinates.

In[ 3 ]:= ContourNonCartesianPlot[r^2-Cos[2*t]==0,{r,0,1},{t,0,2*Pi}] Out[ 3 ]= See Fig. 1 (right).

Mathematica Plot an equation, − 2(1 + cos()) = 0, in polar coordinates.

In[ 4 ]:= ContourNonCartesianPlot[r-2*(1+Cos[t])==0,{r,0,4},{t,0,2*Pi}] Out[ 4 ]= See Fig. 2 (left).

Mathematica Plot two equations, 2 − cos(2) = 0, − (1 + cos()) = 0, in polar coordinates.

In[ 6 ]:= ContourNonCartesianPlot[{r^2-Cos[2*t]==0,r-(1+Cos[t])==0}, {r,0,2},{t,0,2*Pi}] Out[ 6 ]= See Fig. 3.

Mathematica Plot an inequality, − 2(1 + cos()) ≤ 0, in polar coordinates.

In[ 7 ]:= InequalityNonCartesianPlot[r^2-Cos[2*t]<=0,{r,0,1},{t,0,2*Pi}] Out[ 7 ]= See Fig. 4 (left).

Mathematica Plot an inequality, 2 − cos(2) < 0 ∧ − (1 + cos()) < 0, in polar coordinates.

In[ 8 ]:= InequalityNonCartesianPlot[r^2-Cos[2*t]<0&&

r-(1+Cos[t])<0,{r,0,2},{t,0,2*Pi}]

Out[ 8 ]= See Fig. 4 (right).

Mathematica

Definition of the omega number.

Next, the following problem will be solved: Find a transformation : R2 → R2 from a plane to a plane that transforms a region of the plane in a region () of the plane, bounded by the graphs of 2 − 2 = 9, 2 − 2 = 1, = 2, = 4, for > 0 [ 9 ].

The proposed solution to the problem is provided below.

Mathematica Calculation of the coordinate functions = 1(, ), = 2(, ), of the transformation , from (, ) = (2 − 2, 2).

In[14]:= Reduce[{u,v}=={x^2-y^2,2xy}&& x>0&& u>0&& v>0,{x,y}] Out[14]= > 0&& > 0&& == Root[− 2 − 4#12 + 4#14&, 2]&& == /(2) Mathematica Plotting the region : 1 ≤ ≤ 9 ∧ 4 ≤ ≤ 8 in the plane.

In[15]:= InequalityNonCartesianPlot[1 ≤ ≤ 9 ∧ 4 ≤ ≤ 8,

{u,-1,11},{v,2,10},Transformation->Function[{u,v},{u,v}]] Out[15]= See Fig. 6 (left).

Mathematica

Defining the transformation and plotting the region () in the plane.

In[16]:= T=Function[{u,v},{x=Root[-v^2-4*u*#1^2+4*#1^4&,2],v/(2*x)}]; InequalityNonCartesianPlot[1 ≤ ≤ 9 ∧ 4 ≤ ≤ 8,

{u,-1,11},{v,2,10},Transformation->T]

Out[17]= See Fig. 6 (right).

5. Conclusions and future work

In this paper, the new NRGraphics.m package is presented and described. The NRGraphics.m package includes two commands to obtain graphs of equations and inequalities in plane nonrectangular coordinate systems. The coding of the package has been done in Mathematica v.11.2.0.0, but it can run in later versions. All the results shown in this paper have been obtained on an HP laptop with an Intel(R) Core(TM) i3-3110M CPU @ 2.40 GHz, with a RAM of 4.00 GB, 64-bit Operating System, and × 64 processor. The authors are committed to developing command packages linked to education, and a short-term objective is to improve the package so that it allows graphing equations in three-dimensional non-rectangular coordinate systems so that any subsequent results will be published by this means.

References

[1] E. F. Beckenbach , R. Bellman , An introduction to inequalities, number 3 in New mathematical library , 13 . print ed., Math. Assoc. of America , Washington, 1983 .

[2] G. H. Hardy , J. E. Littlewood , G. Pólya, Inequalities, Cambridge University Press, 1952 . Google-Books-ID: t1RCSP8YKt8C.

[3] D. S. Mitrinović , Analytic Inequalities, Springer, Berlin, Heidelberg, 1970 . URL: http://link. springer.com/10.1007/978-3- 642 -99970-3. doi: 10 .1007/978-3- 642 -99970-3.

[4] D. S. Mitrinović , J. Pečarić , V. Volenec , Recent Advances in Geometric Inequalities ( 1989 ). URL: https://typeset.io/papers/recent -advances-in-geometric-inequalities-4x9cek6agp.

[5] Manipulating Equations and Inequalities-Wolfram Language Documentation , 2016 . URL: https://reference.wolfram.com/language/tutorial/ManipulatingEquationsAndInequalities. html# 9389 .

[6] RegionPlot: Plot regions defined by inequalities- Wolfram Documentation , 2016 . URL: https://reference.wolfram.com/language/ref/RegionPlot.html?q=RegionPlot.

[7] List of computer algebra systems , 2023 . URL: https://en.wikipedia.org/w/index.php?title= List_of_computer_ algebra_systems&oldid=1178253292 , page Version ID : 1178253292 .

[8] one-more minute , Plotting an implicit polar equation , 2017 . URL: https://mathematica. stackexchange.com/q/547.

[9] J. A. Venero , Matemáticas III , Representaciones Gemar E.I.R.L. , Lima , 2020 .