Ngày 01 tháng 04 năm 2012

A some examples in MUPAD


Re(cos(x+I*y)





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
f(x,y)=-(x^2+y^2)(\cos (3y) + \sin (7y)).
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
exp(-x^2-y^2)*(cos(3*y)+sin(7*x))
To obtain the following picture
you need
f := plot::Function3d(exp(-x^2-y^2)*(cos(3*y)+sin(7*x)),
x=-2..2, y=-2..2,
Submesh=[1,1]):
graphLines:=plot::modify(f, Filled = FALSE, XLinesVisible = FALSE, YLinesVisible = FALSE,
ZContours = [Automatic, 15], LineColorFunction=((x, y, z) -> [(z+2)/4, 0.5, (2-z)/4])):
linesProjection := plot::Transform3d(
[0, 0, -1.5], // shift vector
[1, 0, 0, // transformation matrix
0, 1, 0,
0, 0, 0],
graphLines):
plot(f,linesProjection)
2. With the following function
{\rm Re}(\cos (x+iy).
To plot its graph, I use the following
f := plot::Function3d(Re(cos(x+I*y)), 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 getRe(cos(x+I*y)3. If our function has singularities, for example, Gamma function f(x,y)=|\Gamma (x+iy)|, we have
Re(cos(x+I*y)f := plot::Function3d(abs(gamma(x+I*y)), x=-4.5..4.5, y=-3..3, Submesh=[3,3], XSubmesh = 1, YSubmesh = 2): plot(f)
4. If you intend to plot sereval graphs, the following is a good example.
sin(x^2 + y^2), cos(x^2 - y^2)
Here we used functions \sin(x^2 + y^2)\cos(x^2 - y^2). The code I used is
plotfunc3d(sin(x^2 + y^2), cos(x^2 - y^2), x = 0..1, y = 0..2)
5. The Mobius band can be drawn by using the following
x := cos(alpha) * (1 + r * cos(alpha/2)):
y := sin(alpha) * (1 + r * cos(alpha/2)):
z := r * sin(alpha/2):
plot(plot::Surface([x,y,z], r = -0.5 .. 0.5, alpha = -PI .. PI,
Mesh = [35, 31], LineColor = RGB::Black.[0.2]),
Axes = None, Scaling = Constrained)
6. If you need a transparent picture, probably you need to add a factor like abs(sin(z)) in the FillColorFunction. Following is an example
K := (x,y,z)->[abs(sin(x)), abs(sin(y)), abs(sin(z)), abs(sin(z))]:
F1 := plot::Function3d(
1.7*x*exp(-1/2*(x^2+y^2)),
x=-4..4, y=-4..4,
FillColorFunction=K):
F2 := plot::Function3d(-1, x=-4..4, y=-4..4):
plot(F1, F2)
The original one is
7. I will end this topic by showing some more pictures obtained by zooming a little bit closed.
sin(x
sin(x
sin(x
sin(x
plotfunc3d(sin(x - -PI)*sin(y - -PI), Submesh = [4, 4], x = -PI .. PI, y = -PI .. PI)
sin(x
sin(x
In conclusion, if you want to plot a general surface in 3D, you need its parametrization in order to draw. For details, I prefer the reader to this paper.