I'm trying to make this plot for the Łukasiewicz t-norm
and I don't know how to make the z axis go only from 0 to 1,
mine goes from -1 to 1 and I need that flat area to be visible.
Now I have
def luka_tnorm(a, b):
c = a + b - 1
return max(c, 0)
plot3d(luka_tnorm(a, b), (a, 0, 1), (b, 0, 1))
My plot now looks like this
Try this:
sage: plot3d(luka_tnorm, (0, 1), (0, 1))
Launched html viewer for Graphics3d Object
sage: plot3d(luka_tnorm, (0, 1), (0, 1), mesh=True)
Launched html viewer for Graphics3d Object
Related
I'm new to Mathematica. I'm trying to produce whole lotta plots, but I don't know how to make Mathematica name them after variables. Here for example I have code producing sine plots for different coefficients infront of x variable:
d = 2;
For[n = 1, n <= 3, n = n + 1, {Do[Print[Plot[Sin[n*x] + d, {x, 0, 6 Pi}]]], Export["E:\\plots\\a.pdf", Plot[Sin[n*x] + d, {x, 0, 6 Pi}]]}]
And in program it produces 3 plots, and each time each plot is exported to a.pdf. Sadly every next time it is overwritten, so I end up with single a.pdf plot when n = 3.
Here is what I would like to achieve. After running program it would produce me 3 plots of names:
S, n=1, d=2.pdf
S, n=2, d=2.pdf
S, n=3, d=2.pdf
or
S n1 d2.pdf
S n2 d2.pdf
S n3 d2.pdf
Something like this perhaps ?
Do[Export["Plot" <> ToString[n] <> "_" <> ToString[d] <> ".pdf",
Plot[Sin[n*x] + d, {x, 0, 6 Pi}]], {n, 1, 3}]
You'll probably want to adjust the file names.
Assuming I have a vector of say four dimensions in which every variable lays in a special interval. Thus we got:
Vector k = (x1,x2,x3,x4) with x1 = (-2,2), x2 = (0,2), x3 = (-4,1), x4 = (-1,1)
I am only interested in the points constraint by the intervals.
So to say v1 = (0,1,2,0) is important where v2 = (-5,-5,5,5) is not.
In additon to that the point i+1 should be relatively close to point i among my journey. Therefore I dont want to jump around in space.
Is there a proper way of walking through those interesting points?
For example in 2D space with x1,x2 = (-2,2) like so:
Note: The frequenz of the red line could be higher
There are many ways to create a space-filling curve while preserving closeness. See the Wikipedia article for a few examples (some have associated algorithms for generating them): https://en.wikipedia.org/wiki/Space-filling_curve
Regardless, let's work with your zig-zag pattern for 2D and work on extending it to 3D and 4D. To extend it into 3D, we just add another zig to the zig-zag. Take a look at the (rough) diagram below:
Essentially, we repeat the pattern that we had in 2D but we now have multiple layers that represent the third dimension. The extra zig that we need to add is the switch between bottom-to-top and top-to-bottom every layer. This is pretty simple to abstract:
In 2D, we have x and y axes.
We move across the x domain switching between positive and negative
directions most frequently.
We move across the y domain once.
In 3D, we have x, y, and z axes.
We move across the x domain switching between positive and negative directions most frequently.
We move across the y domain switching between positive and negative directions second most frequently.
We move across the z domain once.
It should be clear how this generalizes to higher dimensions. Now, I'll present some (Python 3) code that implements the zig-zag pattern for 4D. Let's represent the position in 4D space as (x, y, z, w) and the ranges in each dimension as (x0, x1), (y0, y1), (z0, z1), (w0, w1). These are our inputs. Then, we also define xdir, ydir, and zdir to keep track of the direction of the zig-zag.
x, y, z, w = x0, y0, z0, w0
xdir, ydir, zdir = +1, +1, +1
for iw in range(w1 - w0):
for iz in range(z1 - z0):
for iy in range(y1 - y0):
for ix in range(x1 - x0):
print(x, y, z, w)
x = x + xdir
xdir = -xdir
print(x, y, z, w)
y = y + ydir
ydir = -ydir
print(x, y, z, w)
z = z + zdir
zdir = -zdir
print(x, y, z, w)
w = w + 1
This algorithm has the guarantee that no two points printed out after each other have a distance greater than 1.
Using recursion, you can clean this up to make a very nice generalizable method. I hope this helps; let me know if you have any questions.
With the work of #Matthew Miller I implemented this generalization for any given multidimenisonal space:
'''assuming that we take three points out of our intervals [0,2] for a,b,c
which every one of them is corresponding to one dimension i.e. a 3D-space'''
a = [0,1,2]
b = [0,1,2]
c = [0,1,2]
vec_in = []
vec_in.append(a)
vec_in.append(b)
vec_in.append(c)
result = []
hold = []
dir = [False] * len(vec_in)
def create_points(vec , index, temp, desc):
if (desc):
loop_x = len(vec[index])-1
loop_y = -1
loop_z = -1
else:
loop_x = 0
loop_y = len(vec[index])
loop_z = 1
for i in range(loop_x,loop_y,loop_z):
temp.append(vec[index][i])
if (index < (len(vec) - 1)):
create_points(vec, index + 1, temp, dir[index])
else:
u = []
for k in temp:
u.append(k)
result.append(u)
temp.pop()
if (dir[index] == False):
dir[index] = True
else:
dir[index] = False
if len(temp) != 0:
temp.pop()
#render
create_points(vec_in, 0, hold, dir[0])
for x in (result):
print(x)
The result is a journey which covers every possible postion in a continous way:
[0, 0, 0]
[0, 0, 1]
[0, 0, 2]
[0, 1, 2]
[0, 1, 1]
[0, 1, 0]
[0, 2, 0]
[0, 2, 1]
[0, 2, 2]
[1, 2, 2]
[1, 2, 1]
[1, 2, 0]
[1, 1, 0]
[1, 1, 1]
[1, 1, 2]
[1, 0, 2]
[1, 0, 1]
[1, 0, 0]
[2, 0, 0]
[2, 0, 1]
[2, 0, 2]
[2, 1, 2]
[2, 1, 1]
[2, 1, 0]
[2, 2, 0]
[2, 2, 1]
[2, 2, 2]
I'm solving and plotting the equations of motion for the double pendulum using Mathematica's NDSolve.
I've successfully plotted the Angular position using a standard plot. But when I come to use the parametric plot for the position of each mass. I get no errors but simply no plot.
eqn1 = 2 th''[t] + Sin[th[t] - ph[t]] (ph'[t])^2 + Cos[th[t] - ph[t]] (ph''[t]) + (2 g/l) Sin[th[t]]
eqn2 = ph''[t] + Sin[th[t] - ph[t]] (th'[t])^2 + Cos[th[t] - ph[t]] (th''[t]) + (g/l) Sin[th[t]]
eqnA = eqn1 /. {g -> 10, l -> 1}
eqnB = eqn2 /. {g -> 10, l -> 1}
sol = NDSolve[{eqnA == 0, eqnB == 0, th[0] == 0.859, th'[0] == 0, ph[0] == 0.437, ph'[0] == 0}, {th, ph}, {t, 0, 10}]
Plot[{th[t], ph[t]} /. sol, {t, 0, 10}]
r1 = {lSin[th[t]] + lSin[ph[t]], -lCos[th[t]] - lCos[ph[t]]} /. {l -> 1, g -> 10}
ParametricPlot[r1 /. sol, {t, 0, 10}]
Replace
r1 = {lSin[th[t]] + lSin[ph[t]], -lCos[th[t]] - lCos[ph[t]]} /. {l->1, g->10}
with
r1 = {l*Sin[th[t]] + l*Sin[ph[t]], -l*Cos[th[t]] - l*Cos[ph[t]]} /. {l->1, g->10}
and your ParametricPlot should appear.
One useful trick you might remember, when any plot doesn't appear you can try replacing the plot with Table and see what it shows. Often the table of data provides the needed hint about why the plot isn't appearing.
I have tried many different things but I have not found a way to infer to the point selected by red circle.
You're looking for BoundingBox.corner(n) in which n:
0 = [0, 0, 0] (left front bottom)
1 = [1, 0, 0] (right front bottom)
2 = [0, 1, 0] (left back bottom)
3 = [1, 1, 0] (right back bottom)
4 = [0, 0, 1] (left front top)
5 = [1, 0, 1] (right front top)
6 = [0, 1, 1] (left back top)
7 = [1, 1, 1] (right back top))
If you group is your Group and you want the left front bottom corner::
group.local_bounds.corner(0)
What is the best way of plotting several functions with different domains into the same plot? Is there a way to do this with plot2d, or do I have to use draw2d instead?
I especially like the possibility in plot2d to give several functions in a list, whereas I would have to add the different functions in draw2d as separate parameters, if I understand the documentation correctly.
An example of what I mean:
f(x, a) := sqrt(a) * exp(-(x-a)^2);
fmax(x) := sqrt(x);
In this example I would like to plot f(x, a) for several a (e.g. using makelist(f(x, a), a, [0, 0.5, 1, 2, 5])) from -1 to 10 and fmax from 0 to 5 (to show where the maxima of the f(x, a) family of curves are located).
You can try draw2d
f(x, a) := sqrt(a) * exp(-(x-a)^2);
fmax(x) := sqrt(x);
flist: makelist(f(x, a), a, [0, 0.5, 1, 2, 5]);
par: map(lambda([f], explicit(f, x, -1, 10)), flist);
par: append([explicit(fmax, x, 0, 5), color=red], par);
load(draw);
apply(draw2d, par);
One approach I am not particularly happy with is to declare the functions with smaller domains as parametric curves, with the x axis parameter being simply x:
f(x, a) := sqrt(a) * exp(-(x-a)^2);
fmax(x) := sqrt(x);
plot2d(endcons([parametric, x, fmax(x), [x, 0, 5], [nticks, 80]],
makelist(f(x, a), a, [0, 1/2, 1, 2, 5])),
[x, -1, 10]);
This was frustrating me for hours but I found a way to have multiple differently domained functions on the same graph.
wxplot2d([if x < 0 then -x else sin(x), if x > -1 then x^2],[x,-%pi,%pi],[y,-2,2]);