Graphics`IneqGraphics`

This package provides functions for visualizing logical combinations of inequalities over the reals in 2D. It also provides functionality for visualizing regions over the complexes.

Visualization of inequalities.

This loads the package.

This plots the region x+y<1.

IneqPlot[x + y < 1, {x, -2, 2}, {y, -2, 2}]

Here is the simplest complex region.

ComplexIneqPlot[ 
      Abs[z]  ≤ 1,  {z, {-1, 1}, {-1, 1}} ]

This plots the region 1≤(((x+2y))^2)+4(y^2)≤4.

IneqPlot[ 
      1 ≤ (((x + 2  y))^2) + 4  (y^2) ≤ 
        4,  {x,  -3,  3},  {y,  -3,  3}, 
      PlotPoints -> 30 ]

In this case there is a logical combination of inequalities to describe the region 1<(x^2)-(y^2)<4∧xy<1∧x>0∧y>0.

IneqPlot[ 
      1 < (x^2) - (y^2) < 4 ∧ x y < 1 ∧ x > 0 ∧ y > 0,  {x,  
        0,  2.5},  {y,  -0.5,  1.5}, 
      PlotPoints -> 30]

This plots the symmetric difference between the two regions (((x+(1/2)))^2)+(y^2)≤1 and (((x-(1/2)))^2)+(y^2)≤1.

IneqPlot[ 
      Xor[ (((x + (1/2)))^2) +  (y^2) ≤ 
          1,  (((x - (1/2)))^2) +  (y^2) ≤ 1],  {x, -3, 
        3},  {y, -3, 3}, PlotPoints -> 60 ]

This plots the region ||x|-|y||≤1.

IneqPlot[ 
      Abs[ Abs[x]  -  Abs[y]]  ≤  1,  {x,  -2,  
        2},  {y,  -2,  2}]

This plots the region sin(x + y) > 1/2.

IneqPlot[Sin[x + y] > 1/2, {x, -8, 8}, {y, -8, 8}]

This plots the region sin((x^2) + (y^2)) > 1/2.

IneqPlot[Sin[x^2 + y^2] > 1/2, {x, -4, 4}, {y, -4, 4}, 
      PlotPoints -> 50]

This plots the region sin(2  x) + cos(3  y) ≤ 1.

IneqPlot[
      Sin[2*x] + Cos[3*y] ≤ 1, {x, -8, 8}, {y, -8, 8}, 
      PlotPoints -> 30]

This plots the region \(TraditionalForm\`\(\([\)\(x\)\(]\)\)\^2 + \(\([\)\(y\)\(]\)\)\^2 ≤ 1\), where [x] is Floor[x].

IneqPlot[
      Floor[x]^2 + Floor[y]^2 == 1, {x, -2, 2}, {y, -2, 2}, 
      PlotPoints -> 80]

This is an array of various norm-related regions. The center diagonal corresponds to the unit balls in (Subscript[ℓ, 1]), (Subscript[ℓ, 2]), (Subscript[ℓ, 3]) and (Subscript[ℓ, 4]) respectively or the regions |x(|^p)+|y(|^p)≤1 for p=1,2,3 and 4.

Show[ 
      GraphicsArray[ 
Table[ 
          IneqPlot[ 
            Abs[x]^p + Abs[y]^q ≤ 1,  {x,  -1,  
              1},  {y,  -1,  1},  Ticks ->  None,  
            DisplayFunction ->  Identity],  {p,  1,  4},  {q,  
            1,  4} ]
]]

This is the region |(z^3)+3(z^2)-(1 + 2  i)(|^2)≤|z+i(|^2).

ComplexIneqPlot[ 
      Abs[ (z^3) + 3  (z^2) - (1 + 2  I)]^2 ≤ 
        Abs[z + I]^2,  {z, {-4, 2}, {-2, 2}}, 
      PlotPoints -> 40 ]

This constructs an animation of a set of regions given by a bilinear or Möbius transformation.

Table[ 
      ComplexIneqPlot[
        1 < Abs[ ((z - a)/(a z - 1))] < 
          2 ,  {z, {-10, 2}, {-6, 6}}, 
        PlotRange -> {{-10, 2}, {-6, 6}}, 
        PlotPoints -> 30  ],  {a,  0,  4/10, 1/10}]

This is the region |sin((z^2) - i) + 
            1/2| ≤ 1.

ComplexIneqPlot[ 
      Abs[ Sin[(z^2) - I] + 1/2] ≤ 
        1,  {z, {-4, 4}, {-4, 4}}, PlotPoints -> 40]

Author: Robert Ipanaqué, robertchero@hotmail.com, http://www.unp.edu.pe/pers/ripanaque