Useful if caught in overly simplistic space combat

One of my favorite games, and by far the one I used the most time creating content for during my youth is Escape Velocity Nova. The Escape Velocity series are 2D space RPGs, which means there is lots of 2D space combat. In these types of games you typically have three main types of weapons, non-turreted, turreted and homing. Non-turreted and homing weapons are not really that complicated. The non-turreted fire in one direction (typically forward), while the homing weapon typically turns as hard as it can towards the target at maximum speed. Turreted weapons were always a puzzle to me though, they somehow knew where you would be, and fired to intercept your ship. I always wondered how this was done. Some years later during a visualization class (on how to find intersections between geometric primitives) I was wondering if I could use geometric equations to figure out where a spaceship needs to aim its torpedo, if it knows the position and velocity of the target. Below is the solution I came up with. It assumes constant speed for the target and the torpedo. The positions of something moving at constant speed, can be described by this equation where t denotes time, \vec{v} the current velocity and \vec{p_o} is the starting position. This restricts the position of the target to a line, where the position on the line is determined by t.

(1)   \begin{equation*} \vec{p} = \vec{p_o} + \vec{v}t \end{equation*}

For 2 dimensions this is:

(2)   \begin{equation*} \begin{pmatrix}p_x\\p_y\end{pmatrix} = \begin{pmatrix}p_{xo}\\p_{yo}\end{pmatrix} + \begin{pmatrix}v_x\\v_y\end{pmatrix}t \end{equation*}

The possible locations p of a constant speed (s) bullet fired from point c in any direction, are given by this equation, where again time is denoted by t. This restrict the possible positions to a circle which expands with time.

(3)   \begin{equation*} (st)^{2} = (\vec{p} - \vec{c})^{2} \end{equation*}

For 2 dimensions this is:

(4)   \begin{equation*} (st)^2 = (p_{x} - c_x)^2 + (p_{y} - c_y)^2 \end{equation*}

Now inserting the restrictions on p_x and p_y from 2 into 4 gives a polynomial that only depends on t:

(5)   \begin{equation*} (st)^2 = ((p_{xo}+v_xt) - c_x)^2 + ((p_{yo}+v_yt) - c_y)^2 \end{equation*}

Rearranging gives:

(6)   \begin{equation*} s^2t^2 = (p_{xo}-c_x + v_xt)^2 + (p_{yo}-c_y + v_yt)^2 \end{equation*}

Simplifying by setting constants i = p_{xo}-c_x and j = p_{yo} - c_y and moving towards standard form:

(7)   \begin{equation*} s^2t^2 = (i)^2 + 2iv_xt + v_x^2t^2 + (j)^2 + 2jv_yt + v_y^2t^2 \end{equation*}

Rearranging to standard form:

(8)   \begin{equation*} 0 = (v_y^2 + v_x^2 - s^2)t^2 + 2(iv_x + jv_y)t + (i)^2 + (j)^2 \end{equation*}

This is a quadratic equation, with coefficients:

(9)   \begin{equation*} a = v_y^2 + v_x^2 - s^2 \end{equation*}

(10)   \begin{equation*} b=2(iv_x + jv_y) \end{equation*}

(11)   \begin{equation*} c=i^2 + j^2 \end{equation*}

Solving the quadratic equation gives two solution for t, if an answer is positive and has no imaginary part, there is a solution to the problem. Inserting a solution for t in equation 1 gives the x,y coordinates where the torpedo should be aimed, and where it will hit the target. Below is an implementation of this, where the constant speed of the torpedo can be set. If there are two solutions, the smallest t is chosen. If there is no solution (typically because the target it too fast for the torpedo to catch) the torpedo is never launched.  
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As you can see in some cases, this does not take into account the size of the target, only the targets center. This can be remedied by using a representative selection of boundary points of the target instead, and firing if one of them would connect. The solution does also not take into account how fast the target can change direction, or when the torpedo runs out of fuel. This can be handled by not firing if the time from the torpedo is fired to the impact is too long. This method can also quite easily be expanded for an accelerating weapon or target. This will result in a higher degree polynomial though, which are mostly only possible to solve using numerical methods like Newtons's method for example.

The van der Corput sequence

Lately I have been looking for a solid way to create a robust implementation moving a spaceship from one point to another. This turned out to be a hard problem, and led me to a lot of new literature, which included Steven M. LaValle’s Planning Algorithms. Randomly reading from it I found a small gem called the van der Corput sequence.

The van der Corput sequence is useful for sampling in an interval. LaValle discusses it as an option to random sampling within the context of sampling for planning algorithms. I had thought about similar issues in the context of root finding, where I always wondered how to predictably generate a sequence that would distribute samples evenly over an interval, as well as do so in a manner that would not “favor” parts of the interval over others.

One naive way to sample over an interval, is to split the interval in X pieces and do the samples in order. If X is 8 and this method is used the interval would be explored as shown to the left in the figure below (clearly this means the the early parts of the interval will be explored first). Using the van der Corput sequence would result in exploration as seen on the right (which explores the interval much more evenly).

Van der Corput
Naive sampling on the left. Van der Corput sampling on the right. Grey circles show performed samples.

Generating the van der Corput sequence is surprisingly simple and elegant, just flip the bits of the naive sequence as seen in the binary column above. I find this very peculiar, since mirrored donkeys do not normally turn into cheetahs.

The great properties of this sequence is that whenever you end it, it will have explored the interval pretty evenly. A second nice property is that you can easily continue the sequence beyond an initial size, by creating the naive sequence at the position you want to start from, and keep reversing the results.

This method will definitely go into my toolbox of things to consider whenever I am thinking about using random sampling.

Comments and HTML5 scatterplots

To try and learn some Javascript and use of the HTML5 canvas, I decided to make a visualization tool. After some googling I decided on scatterplots that allow linking and brushing. My idea was to allow such plots to easily be added to any webpage or blog. At the moment they only work well in Webkit based browsers (Chrome, Safari) though.

If you are not familiar with visualization lingo, brushing basically means that you can select datapoints in some fashion. Linking means selections are propagated across two linked views. So if I select some datapoints in one plot the same datapoints are selected in a linked plot. Below is an example of the canvas based scatterplot implementation I ended up with. To try the linking and brushing scroll down to here.

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As you might have understood from the axis names, instead of generating a dataset, I wanted to apply this to some real world data. The idea was inspired by "kommentarfeltdugnad", which aim to not let extremism run unchallengde in public newspaper discussions. I have a hard time finding anything I want to do less then argue in a newspaper comment section, instead i figured studying the people doing this would be more interesting. Using visualization for this was inspired by Correll et al's excellent paper on text visualization. If possible I would like to implement several more of their approaches later on.

Kommentarfeltdugnad also got me thinking about alternative approaches to discussion and forum moderation, especially wondering if it is possible to profile writers over time based on their writing, and ban or even hellban users which pass a certain threshold of badly argued or impossible to decipher posts.

The dataset I'm using in this post was derived from this article due to the amount of comments. All HTML is stripped away and descriptive parameters derived from the remaining comment data.

The aim of the initial exploration of this dataset is to see if it is possible to connect some derived parameters, and possibly tie these to certain usage of some words. You are of course compelled to explore the datasets yourself using the interactive scatterplots below.

I quickly figured out that a large percentage of comments were to short to make out any meaningful statistic. I decided that comments below 50 words were not to be included. Even though 50 is quite arbitrary and decimal oriented.

From the remaining posts I derived several parameters describing each post:

  • Percent of capitals.
  • Percent of punctuation.
  • Amount of words.
  • Percentage of occurrences of specific words (islam, muslim, sosialism, …).
  • Letters in post.

Initially I thought I would start with something obvious as a test of my data data. I think it reasonable to assume a very strong correlation between the amount of letters and words in a post. This can be seen in this plot.

Less obvious is whether there is a correlation between the percent of capitals and an excessive amount of punctuation in a post. Is it so that excessive capitals typers are more likely to either forgo or misuse punctuation?

Even more hypothetical is whether there is a correlation between mentions of islam or muslims and excessive use of capitals?

Below three scatterplots are linked together, with the first showing percentage of capitals and punctuation, the second showing word count and letters in post, the third shows percentage of capitals and percentage of words that are (islam/muslim). This allows for exploration of some of the questions above, as well as some other questions.

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There seems to be a small correlation between the amount of capitals and punctuation in a post. It would be very interesting to compare this dataset to a dataset with properly typed text. I also think that punctuation is very broad, and just focusing on exclamation and question marks might also be interesting.

The percentage of capitals and occurrences of islam seems not really related, though curiously, very long posts with a high amount of capitals do not seem to mention islam for some reason.

I guess that concludes my very superficial analysis of this dataset. Feel free to do whatever with the dataset, and once cleaned up I'll link a zip with the JS source for convenience. My next posts will detail how to scrape the data, and how to add scatterplots to your own webpage or wordpress blog.

Finding matrix position from vector address

I have a symmetric matrix, which is stored as a vector containing only the needed elements. Is there an easy way to go from vector address to positions in the matrix?

The same operation on a square matrix stored in this way is done by:

(1)   \begin{equation*} x = v \mod{n} \end{equation*}

and

(2)   \begin{equation*} y = \frac{v}{m} \end{equation*}

Where v is the position in the vector, and m and n matrix columns and rows. This also conveniently reduces to bit shifts and masking if m and n are powers of 2. Operations are of course Integer operations.

My question is whether there exists a similar simple operation on a vector containing just one of the parts of a symmetric matrix.

 

Example of problem
Symmetric matrix stored in vector, not necessarily the best way to store it but one of many possibilities.

One opportunity is to use that the amount of elements needed to store a symmetric matrix is given by:

(3)   \begin{equation*} \frac{n(n+1)}{2} \end{equation*}

Using this one can iteratively search over n until the vector address is contained in the current block of inspected data. Then the remaining elements along with the amount of removed elements give the position. The problem is that this is n operations in a linear search, or log(n) operations if binary search is used.

so I’m wondering if there is an easier way to do this. Basically what I want is a very quick algorithm from vector position to matrix position.

Possibly useful observations are that the result from. n(n+1)/2 can always be divided by the odd component part of n and n+1, as well as the even component divided by two.

Also the resulting triangle can be divided into a structured amount s of square areas of values w:

(4)   \begin{equation*} s = 2^i \end{equation*}

(5)   \begin{equation*} w = \lfloor\frac{(n+2^i)}{2^{(i+1)}}^2\rfloor \end{equation*}

Where i is the squares where 0 is the largest square, and larger i gives the next smaller one. n is from the original matrix m and n. When the result is zero no more squares are needed to describe the area.

How this can be turned into a storage and addressing scheme, I do not know, and I wonder if there is no faster way (the binary search) to recover a position (without storing them beforehand) when something is stored in this manner?

Note that I’m not discussing packing and unpacking the entire matrix here (which would be O(n) using the linear search method from above), but finding the position of a single element from its position in the vector.