A couple of weeks ago, I wrote about weighted random numbers. After implementation and some experimentation, I settled on a versatile function that incorporates all the fetchers of the weighting system. Mathematically, it's a little ugly because there is an “if” statement and we end up with a piecewise function.

Where m is the minimum value, M is the maximum value, c is the center point (m ≤ c ≤ M), S is the concentration coefficient (useful range 1 ≤ S < ∞), and α and β are random numbers between 0 and 1. The core of this function is the weighting.

This has been scaled so the output is in a given range.

Here, m ≤ w_s ≤ M were as 0 ≤ w ≤ 1. From here, the body of the function is split before and after the center point. For this we require a second random number, α. This value is used to determine of the value is to the left or right of center, and the min and max of the function are adjusted accordingly.

The top graph shows the distribution of 1,000 samples, and the lower graph shows a histogram of the distribution. The average is calculated over all the samples. If the center value is half-way between min and max, the average should be the center value (or close to). The center value reflects the highest peak value in the histogram, which should always be close to the specified center.

There are some things you can do with this function that are not meaningful. Having a center value outside the min and max value will still generate values, but probably not useful for anything.

You can also use a concentration coefficient less than one and greater than zero (0 ≤ S < ≤ 1) . This has the effect of pushing the concentration away from the center point and toward the min and max values—basically the acting in the reverse of the normal algorithm. This may be useful for generating a value that is usually either one value or an other, with very little in between.

Here the min is 0, max is 100, center is 20, and the concentration coefficient is 0.1. Notice how the center point is the least populated area of the graph.

There are some ways to use this function to generate some of the other weighted functions. For example, let c = ½ (M – m) + m. This will make the function have equal distribution on both sides of the center point.

Here, the function C is a centered function, c is the center point, and s is the span that can be deviated from the center.

For a simple left or right weighted version of the function, simply set the center point to the min value (left weighted) or max value (right weighted).

Using a concentration coefficient of one (S =1) results in just random uniform random data (assuming β is random). Small values of S are harder to notice in this demo, but become pronounced when more samples are used.

Here is an example of a center at 70, min of 0, max of 100, and a concentration coefficient of 2. At 1000 samples it is not apparent there is any concentration, but at 100,000 samples it is easier to see. The higher sample set also makes the histogram more clear. Notice how the histogram falls to around 100 on both sides, but more rapidly to the right of center. This is necessary because of the uneven weight. So a 0 or a 100 are both equally likely (or unlikely as the case may be), but a 60 and 80, despite being equal distance from the center point are not both as likely as one an other (higher likelihood of 60 over 80).

I've written articles about weighted random number in the past, but today I ran into a use I've been meaning to explain for a long time.

For example when rolling two dice, the mostly likely number to roll is 7. With 4 dice, it's 14. These are weighted rolls in the context of this article as the likely outcomes are not evenly distributed, but tend toward some center point.

One of the weighting algorithm I've written about in the past is Banded Inverse Root Nonuniform Scatter. This is the function:

Where α₁ and α₂ are random numbers between 0 and 1, and S is the “scatter coefficient”. The root of this function is the banding part.

This weights the roll toward 0. The larger the value of S, strong the pull toward 1. Using two of these functions together give a range the function a peak centered at 0 that goes both positive and genitive. Note that the last part of the function normalizes the output so it is between 0 and 1. The process will be explained in a bit, but this function will be called n_b(S). So in parts, the full function is:

This function can be simplified if the square root is removed. The root makes the curve more gradual, but this isn't needed.

The trick to this function is the use of the -1, +1 in the denominator. This allows the scatter coefficient to have a defined range between negative infinity and positive infinity (i.e. -∞ < S < +∞), although the useful range is 0 ≤ S < +∞.

The normalized function looks like this:

Rebuilding the center-weighted function results in:

So g( α₁, α₂, S ) is our weighted function. α₁ and α₂ are random numbers between 0 and 1. S is the scatted coefficient 0 ≤ S < ∞. The larger the value of S, the more weighted the output is toward 0.

The graph above shows the histogram for distribution for various scatter values and illustrates how as the scatter coefficient increases, the concentration toward the center increases. Note that this function does not create a bell curve (or normal distribution). Instead it has a sharp point at the center. This means that for larger values of S the likelihood of being away from the center point diminishes very rapidly—much more than it would with a function that has normal distribution. So the function favors the center point more strongly than those producing normal distribution.

Now some of the function's versatility. The function is normally used to generate some range.

Here, M is a scale factor (magnitude) and c is an offset that allows the function to have a range such that -(M + c) < v < (M + c). Now a function can be defined to return a value in a given range with some weight.

Where v_min < w( v_min, v_max, S ) < v_max. The floor function makes sure the values are integer numbers, and can be omitted if real number are desired. The center point will always be half way between v_min and v_max.

This function can be modified slightly to simulate a dice roll. Let n be the number of dice, and s be the number of sides on each die. Then v_min = n, v_max = n * s. The scatter coefficient (S) can be varied, but the distribution will not be identical to that of an actual dice roll.

Here the floor function is required. n < d( n, s, S ) < n*s.

In this histrogram, the difference in distribution can be seen between an actual dice roll (in this case, five 6-sides die) and the simulated function d( n, s, S ) where S = 3. Note they both peak at the same location (between 17 and 18) with roughly the same likelihood for these numbers. However, the chances for rolling a 15 are greater with a true dice roll, and less in the simulated. Likewise, rolling an 8 is less likely with dice, and more likely simulated. Keep in mind that the simulated dice roll can do something an actual dice roll can not: produce fractional results. If the floor function part of d( n, s, S ) is removed, any real number in the range can be returned. So while an exhaustive check for every dice roll is possible, every simulated roll is not. Thus, the graph above used one million samples to produce the simulated histrogram.

There are some additional way the function g( α₁, α₂, S ) can be used. If an uncentered value desired, the random input can be fixed.

These histrogram show the output of 10,000 samples of the function, where α is a random number (0 ≤ α ≤ 1). Note how in both cases, when S = 1 the distribution is uniform for all values. This is because when S = 1, the weighting function is doing nothing, and the random value α is being returned.

I was doing some work in Google Sketchup, and started experimenting with star polygons. I was drawing an 8-point star when it dawned on me was more than one configuration that could be used to draw such a star. After a little reading, I discovered the nomenclature on this topic. Using the Schläfli symbol, I discovered what I had normally been drawing when I made stay polygons was of the form {p / %u230Ap / 2%u230B-1}. Schläfli symbol is of the form {p / q}, where p is the number of points in the star, and q is the number of points between connecting lines. For example, an octagon shape is {8 / 1} as it has 8 points, and each line is connected to the very next point. A pentagram is {5 / 2}, having 5 points, and each line is connected to the 2^nd closest point to either side. What I had been drawing was a star with the distance between points always %u230Ap / 2%u230B-1, or having the connecting points as far away from one an other as possible for the star.

After learning this, I decided to create a little web application to demonstrate this.

The demo is done using Scalable Vector Graphics (SVG), with some Javascript used manipulate the image. The math is quite simple. First we get the distance between vertices (points on edge). The distance is in degrees (or radians). For example, an 8-point star is 360º / 8º = 45º degrees between points. To draw an octagon, we simply start a 0º and draw a point to 45º, and then 45º to 90º, ext. In order to be coordinates, we need a distance from the center—the radius—which depends on the height and width of the image. The SVG image uses standard computer coordinate—that is (0,0) is the upper left part of the screen. To convert from the polar coordinates requires first knowing where the center of the view port is located. This is half the width and height of the view port. Polar coordinates typically start with 0º on the right side, but I wanted it like a clock—0º on the top. So the polar to screen conversions are as follows: x = centerX – radius * sin( angle ), y = centerY – radius * cos( angle ).

The only trick comes when drawing star figures. For example, a {6 / 2} star is actually two {3 / 1} stars (the notation is 2 {3 / 1}), and not a single continuous path. For this case, we need only to know how many smaller polygons this figure is made from, and draw each of them offset one point from the previous. For example, a {15 / 6} is the same as 3 {5 / 2}. This means there are 3 pentagrams, each offset 24º (360º / 15 sides). So the first pentagram would be drawn with it's tip at 0º, the second at 24º, ext.

Ubuntu 12.04 was released today. After it was, and I managed to get on their website, I started doing a torrent download of, well, all of them. If nothing else, it's a good test of our bandwidth. Our connection has been holding around 5.2 Mbit/sec, peaking out around 5.36 Mbits/sec. I don't even know what speeds our ISP say we should have, but it's nice to give them a workout from time to time.

Why do I need all the flavors of Ubuntu? Well, my main computer is a 64-bit machine, but I have several virtual machines setup, and several other computers that run Ubuntu as their primary OS. So having each of the types (desktop, server, and alternate both i386 and AMD64) will save me a step in the future. Otherwise I always find I need the version I don't have downloaded.

The first question someone might ask is why one would want to have a Linux system that starts Firefox full-screen on boot. A web browser makes a good cross-platform human-machine interface. While there are still issues with web pages looking different in different browsers, these issues have become less and less as browser developers comply to standards. Javascript with AJAX and server-side scripting allow for the developer to create rich applications. So designing a user-interface with a web browser is sometimes a great option. One example would be a controller for home automation. After some back-end scripting to handle calls the server would make to change settings, making an interface is as simple as making a web page. On a low-cost touch screen might server this setup from multiple locations in a house. I've implemented an interface to a large engine controller using Firefox. So there are reasons one might want such a setup.

I want to share something I learned and had wanted to explore for a long time. What is minimum required to have a Linux box boot into Firefox? This was a question I first asked sometime in 2005, when I chopped down a Debian install of Linux to make it small enough to fit on a compact flash to be used on single board computer. I recall I did get the system to fit on a 512 MB compact flash, and the boot setup optimized from power-on to fully running happened in 30 seconds. The next part of the project would have been to get Firefox to start full screen on this setup, but unfortunately never happened as the project was ended before we reached this phase.

I have much more experience with Linux as a text-based server than I do with it as a desktop. DrQue.net has been running on a Linux setup since 2003, but it wasn't until around 2008 I was regularly using Ubuntu on my laptop. How X Windows (X11), the windows manager that sits atop it, and applications that run in the GUI fit together is still rather fuzzy to me. However, I knew that it all starts with X11—so I had to have that. But the windows manager, user log-in screen and all the other things I'm use to seeing on a Ubuntu desktop setup I was less clear about. Turns out for Firefox, you only need X11 and Firefox. No windows manager, no log-in handler, or anything else.

So after install just the bare-bones system, you only need get fetch two additional packages. I added a third, unclutter, so I could turn the mouse off after some timeout period (I'll address that latter).

apt-get install xorg firefox unclutter –no-install-recommends

To start Firefox from the command prompt, first start X11, and then start Firefox:

startx & firefox –display=:0.0

Firefox needs to be told where the display is, but that's the only caveat. To get this to happen on system start up requires a couple of additions. First, add a line to /etc/rc.local

su <user name> -c startx

This will start X11 under some user name. By default, most users are not allowed to start X11. This can be changed be editing /etc/X11/Xwrapper.config and modifying :

allowed_users=anybody

This will let anyone start X11. There is a security risk here as everyone recommends setting this value back, but my system is a single user system with no keyboard/mouse. So I'm not going to be terribly concerned about it.

Now that X11 is setup to start when the system does, it's time to add Firefox to the mix. For this, create ~/.xinitrc with the line

firefox <url>

This will cause Firefox to start with X11 for the specified user, and go to some URL. My setup goes to a local web page so that Firefox begins to view some AJAX web page.

In order to get the setup to run full-screen, you will need a plug in for Firefox called “autohide”. I e-mailed the developer of this plug-in to thank him for his work, and he stated that he no longer does anything with this project. So while it works now, it isn't going to be maintained. The developer also states it was only tested on a windows-based machine, but it does work fine under Linux. What this plug-in will allow is the command line option “-fullscreen” which will cause Firefox to start full screen.

In my setup I found the X11 places a resize triangle in the bottom right corner, even when Firefox is full-screen. Since I want nothing on the screen but the content of my web page, I was not pleased with this artifact. While I didn't find a way to remove it, I did find a workaround. On the command line, you can specify the height and width of the window. Making the height 24 pixels longer than the screen resolution places the triangle out of view.

The last item I had to change was the mouse cursor. My system doesn't normally have a mouse connected, so having the cursor on the screen is rather moot. The package “unclutter” can take care of this. One of the parameters is a delay for how long of a pause to allow before turning off the mouse cursor. Setting this delay time to 1 second makes the mouse go away quickly on start. So the ~/.xinitrc becomes something like this:

unclutter -idle 1 &

firefox -height 1048 -width 1280 -fullscreen 127.0.0.1

Where 1024x1280 is the screen resolution for my system.

The last item, and one I found really frustrating to get functional is disabling the screen saver. By default after 10 minutes of no keyboard/mouse activity X11 blanks the screen. Since there is no keyboard or mouse on my setup, this is an issue. I found no better way to stop the screen saver than by doing this in the X11 configuration. I created the file /etc/X11/xorg.conf and added the following lines:

Section "ServerLayout"

         Identifier "Default Layout"

         Option "BlankTime"   "0"

         Option "StandbyTime" "0"

         Option "SuspendTime" "0"

         Option "OffTime"     "0"

EndSection

Of all the methods to disable screen blanking I tried, this was the only one that worked. Other methods involved using the command xset, but they didn't work for me.

That's it, and I hope someone finds this useful.