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## Two Mathematical Bugs

A few days ago, I discovered I had at least one, maybe two mathematical bugs. The first bug is in the line clipping algorithm. Here’s a quick reminder of what’s going on there:

There are six points of interest: the four intersection points, and the two endpoints of the line. We sort the six points according to their parameter, and take the two innermost points as the endpoints of the clipped line. To check if the line is outside the box we make sure that the result points’ parameters are between 0 and 1 (which correspond to the original endpoints).

Turns out this is not always true – here’s a counter example: In the counter example you can see a line, whose endpoints lie between the intersections. The algorithm described above will say this line lies completely inside the box. Clearly this is not true. How do we fix this? Simple actually. We check for each intersection point whether or not it is on the edges of the clipping rectangle. If at most one point is on – our line is outside. I hope this is the end of it. It will require more rigorous testing.

The second bug was in my solution to the random selection challenge. I wasn’t too sure of my solution, but I really liked it, as its code is rather elegant. So when meeting someone I used to work with, I told her about it. After thinking about my solution a bit, she said it is probably wrong, and gave me the reason: While the probability of a point x passing the first stage is indeed (proportional to) p(x), the probability of it passing the second stage is not 1 (or constant). I still need to think about this one, so no easy fix yet. I guess this reopens the challenge, so if anyone feels like writing a good proof or a counter example for my solution, I’d be happy to see it. I’ll be even happier to see a completely different solution.

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## Solution for the Random Selection Challenge

A few days ago, I wrote up two small Python Challenges. Several people have presented solutions for the first challenge, and I also posted my solution in the comments there.

However, the second challenge remained unsolved, and I will present a solution for it in this post.

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## Small Python Challenge No. 3 – Random Selection

This time I’ll give two related problems, both not too hard.

Lets warm up with the first:

You have a mapping between items and probabilities. You need to choose each item with its probability.

For example, consider the items [‘good’, ‘bad’, ‘ugly’], with probabilities of [0.5, 0.3, 0.2] accordingly. Your solution should choose good with probability 50%, bad with 30% and ugly with 20%.

I came to this challenge because just today I had to solve it, and it seems like a common problem. Hence, it makes sense to ask ‘what is the best way?’.

The second problem is slightly harder:

Assume a bell shaped function p(x) that you can ‘solve’. This means that given a value y, you can get all x such that p(x)=y. For example, sin(x)^2 in [0,pi] is such a function. Given a function such as Python’s random.random() that yields a uniform distribution of values in [0,1), write a function that yields a distribution proportional to p(x) in the appropriate interval.

For example, consider the function p(x) = e^(-x^2) in [-1,1]. Since p(0) = 1, and p(0.5)~0.779, the value 0 should be p(0)/p(0.5)~1.28 times more common than 0.5.

As usual, the preferred solutions are the elegant ones. Go!

note: please post your solutions in the comments, using [ python]…[ /python] tags (but without the spaces in the tags).