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I am an essentially self-trained programmer with little mathematical background. I do not quite know where to start for this problem, and do not know the terminology to help me get conclusions researching the answer myself. My efforts have lead me to Wikipedia pages that I do not understand enough to know whether I am reading the right material (hyperplane arrangement? lattice?). All this is to say, please be gentle.

I am a new computer science graduate student working with clustering and instance generation as part of my research. The clustering technique I am using constructs planes in k-dimensions (are these called hyperplanes?) that divide the space into regions where separate regions are composed of vectors that are classified as more similar to each other than the instances in the other regions. As regards my research, some of these regions are more interesting than others. I want to explore them further by randomly generating vectors within the interesting regions.

The problem is that I do not know how to do this.

Intuitively, the limits of regions are determined by the intersections of the bounding planes. These intersections form lines that intersect with the bounds of the problem space, so I think I can calculate minimum and maximum bounds for each of the k attributes by determining where the lines formed by pairs of planes intersection intersect with each other or the bounding space limts. As I generate random values for each attribute, I can recalculate the bounds of the other variables by treating the values chosen for the previous attributes as new planes. At least, I think this would work.

Is this the correct approach? If so, how do I calculate the line that forms the intersection between the two planes? How would I then calculate these k-dimensional lines' intersections?

If this is not the correct approach, what is? I'd appreciate an algorithmic approach that explains what each step is and what it means if possible.

I would also appreciate recommendations as to what I could study, and in what order, to determine into what I have stumbled. My current background only goes as far as some rusty single-variable calculus.

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