# How to generate a random pulsar graph between two points?

I am going crazy trying to create this. I am trying to create a pulsar graph that starts and ends on two points that have the same y smoothly along the x axis. The pulsar should have random valleys and peaks. The height of the graph should also be variable.

Here the first graph is what I want, while the second one is the best that I think I can produce. I have tried lots of different things from generating some random points and trying to interpolate them(the result is not smooth along the x axis) to adding random functions using Exp[](but failing on making it fall between two points).

This is to create something similar to

or but still have more freedom on where to put the waves. I hope I managed to explain myself correctly.

P.D. I managed to find this similar answer in the past, but I cant make the graph between two random points and hence creating this question.

Edit: After lots of trial an error I found a way to do this.

I created 3 functions.

Jaggify[list_, height_] :=
Union[Flatten[
Table[{list[[x]], {(list[[x, 1]] + list[[x + 1, 1]])/
2., (list[[x, 2]] + list[[x + 1, 2]])/2. +
height RandomReal[]^5}, list[[x + 1]]}, {x, 1,
Length[list] - 1}], 1]]


This function receives a list of points. Then it iterates over pair of points and adds a new one in between. The height of the new point is in the middle of the two adjacent points plus a random value.

RandomPulsarPoints[start_, end_, y_, peaks_, height_] :=
Module[{list, seedlist},
list = Table[{start + x, y}, {x, 0, end - start, (end - start)/
peaks}];
seedlist = Table[
{list[[linum, 1]],
list[[linum, 2]] +
If[linum == 1 || linum == 2 || linum == Length[list] - 1 ||
linum == Length[list]
, 0
, RandomChoice[{1/linum^3,
1 - 1/linum^3} -&gt; {height^3 RandomReal[],
height (RandomReal[])^3}]]}, {linum, 1, Length[list]}];
Join[{First[seedlist]},
Jaggify[Jaggify[Most[Rest[seedlist]], height],
height/2], {Last[seedlist]}]
]


This function generates a list of random points between start and end. The first two points and the last two have the same y value to force the graph to start and end smoothly. You can define how tall and how many peaks are there going to be in that interval.

pulsar[start_, end_, y_, height_, peaks_] :=
Module[{length = end - start},
If[length == 0 || peaks == 0, BSplineCurve[{{start, y}, {end, y}}],
BSplineCurve[RandomPulsarPoints[start, end, y, peaks, height]]]]


This final function just uses the previous functions to create a BSplineCurve.

Graphics[pulsar[0, 10, 0, 3, 10]] I hope this is useful to someone else. :)

I think this question can be closed now.

• This and this. – march Dec 17 '15 at 19:48
• Plotting the graphs one on top of each other is not the issue. I can do that just fine. The second link I already found it before but I am unable to make it so that I can give two points and make the pulsar graph start and end between the two points. – listix Dec 17 '15 at 19:53
• I believe it's almost a dup of this. Won't vote to close, my vote is bonding – Dr. belisarius Dec 17 '15 at 20:11
• Also take a look at BrownianBridgeProcess[ ] – Dr. belisarius Dec 17 '15 at 20:16
• BrownianBridgeProcess looks promising. While looking at the manual I also found SmoothHistogram which looked even better. I would love something like that but for two points. I didn't know doing this would involve so much work. – listix Dec 17 '15 at 20:38

## 3 Answers

After lots of trial an error I found a way to do this.

I created 3 functions.

Jaggify[list_, height_] :=
Union[Flatten[
Table[{list[[x]], {(list[[x, 1]] + list[[x + 1, 1]])/
2., (list[[x, 2]] + list[[x + 1, 2]])/2. +
height RandomReal[]^5}, list[[x + 1]]}, {x, 1,
Length[list] - 1}], 1]]


This function receives a list of points. Then it iterates over pair of points and adds a new one in between. The height of the new point is in the middle of the two adjacent points plus a random value.

RandomPulsarPoints[start_, end_, y_, peaks_, height_] :=
Module[{list, seedlist},
list = Table[{start + x, y}, {x, 0, end - start, (end - start)/
peaks}];
seedlist = Table[
{list[[linum, 1]],
list[[linum, 2]] +
If[linum == 1 || linum == 2 || linum == Length[list] - 1 ||
linum == Length[list]
, 0
, RandomChoice[{1/linum^3,
1 - 1/linum^3}-> {(height^3) RandomReal[],
height (RandomReal[])^3}]]}, {linum, 1, Length[list]}];
Join[{First[seedlist]},
Jaggify[Jaggify[Most[Rest[seedlist]], height],
height/2], {Last[seedlist]}]
]


This function generates a list of random points between start and end. The first two points and the last two have the same y value to force the graph to start and end smoothly. You can define how tall and how many peaks are there going to be in that interval.

    pulsar[start_, end_, y_, height_, peaks_] :=
Module[{length = end - start},
If[length == 0 || peaks == 0, BSplineCurve[{{start, y}, {end, y}}],
BSplineCurve[RandomPulsarPoints[start, end, y, peaks, height]]]]


This final function just uses the previous functions to create a BSplineCurve.

Graphics[pulsar[0, 10, 0, 3, 10]] I hope this is useful to someone else. :)

One way to smooth things is to use an appropriate interpolating order with random points. The ends can be flattened out with the Hamming Window.

points = HammingWindow[Range[-0.7, 0.7, 0.1]] RandomReal[{0, 1}, 15];
Plot[Interpolation[points, InterpolationOrder -> 3][t], {t, 1, Length[points]}] • I havent seen that function, but the problem I see with this solution is being able to draw this starting on any random point and extend towards another random point. Besides the ends dont seem as smooth as I would like. – listix Dec 19 '15 at 3:18
• Flatten the ends more by extending the Range... maybe -0.8 to 0.8 or further. If you want to start it and end it at different values, just add a constant (or a line if the two endpoints are different). – bill s Dec 19 '15 at 4:13
• I see what you mean the problem is that if I keep adding more and more points the general figure of the function doesn't change. I think it will be useful to something else I am trying to do. – listix Dec 19 '15 at 16:48

I decided that instead of trying to use Interpolation, I would interpolate between the points using the solutions of a 4th-order ODE with appropriate boundary conditions, and then stitch them together using Piecewise. I think the results are rather appealing. I also have a random number of "peaks" which I create by using Sin.

wavePoints[n_, k_] :=
Chop@Sin[Pi*Range[0, k, k/(n - 1)]]^4*RandomReal[{1/3, 3}, {n}];

pulsarWave[n_] :=
Module[{
pairs = Partition[wavePoints[n + 1, RandomInteger[{1, 4}]], 2, 1],
x, t, sols, pw
},
sols = NDSolveValue[{x''''[t] == -x[t], x == #[], x' == 0,
x == #[], x' == 0},  x, {t, 0, 1}] & /@ pairs;

Activate[Inactive[Function][t,
Piecewise@Table[
{sols[[i + 1]][t - i], i <= t < i + 1}, {i, 0, n - 1}]]]];


Here's a sample of the output:

GraphicsColumn[
Table[With[{wave = pulsarWave},
Plot[wave[t], {t, -1, 11}, Axes -> False]],
{5}]] 