MuPAD: Drawing asymptotically flat spacetime

Today, I will show how to draw an asymptotically flat spacetime with two ends using MuPad. To draw ends, we use hyperbolic functions.

First, we talk about the end. The function that I am going to use is . It is clear that blow up at and approaches at infinity. To use , we use the following

h := proc(t)
begin
  (1-4*t)/(t-2)+12
end_proc

I have added to the function so that approaches at infinity. Then to draw the (upper) end, I use the following function

f := proc(x, y)
begin
  if 
    x^2 + y^2 > 3.4 
  then
    h(x^2+y^2)
  else
  end_if
end_proc

I have used the number because I do not want my end is too tall, keep in mind that is the blow-up number. We can calculate

h(3.4)

to see that this number is nothing but which is close to as I need. We are now able to draw the first end using the following

plot(
     plot::Function3d(f, x = -6 .. 6, y = -6 .. 6, Submesh = [3, 3]),
     ViewingBox = [Automatic, Automatic, 3 .. 8], 
     Scaling = Constrained)

Say HI in MuPAD

I found this interesting note regarding to Matlab. In that note, they plotted the word HI using Matlab. Here I try to use MuPAD in order to get a slightly better picture.

As mentioned in the note, the full function we need to use is

If you plot that full function, what you are going to have is the following picture

MuPad: Heart in 3D

This is not mathematics. I just found an equation so that we can draw a heart in 3D. Indeed, the following equation

will generate a heart. I have tried and the following pictures show that fact.

MuPAD: Drawing a surface with a line on

I took me years to figure out how did we plot such a picture in this entry. Thanks to MuPAD, we can do it quite easily. What I got is the following

Firstly, we need to choose a function which has a mountain-pass shape. Thank to a special solution to the Toda system considered in this entry, we can choose

.

MuPAD

MuPAD was a Computer algebra system (CAS). Originally developed by the MuPAD research group at the University of Paderborn, Germany, it was developed by the company SciFace Software GmbH & Co. KG in cooperation with the MuPAD research group and partners from some other universities since 1997.

I found MuPAD a very cool tool for plotting graph of functions. I will show you in details some examples.

1. We start with the following function

.

To plot its graph, I use the following

f := plot::Function3d(exp(-x^2-y^2)*(cos(3*y)+sin(7*x)), x=-2..2, y=-2..2, Submesh=[1,1], FillColorFunction=((x, y, z) -> [(z+2)/4, 0.5, (2-z)/4])): plot(f)

and this is what we get