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Suppose l > t.In this case, integrating the joint probability density function, we obtain: = = (), where m(θ) is the minimum between l / 2 sinθ and t / 2 .. Thus, performing the above integration, we see that, when l > t, the probability that the needle will cross at least one line is
The lines connecting the remaining houses with the utilities divide the plane into three regions. Whichever region the deleted house is placed into, the similarly shaded utility is outside the region. By the Jordan curve theorem, a line connecting them must intersect one of the existing lines.
If r and t cross, add that crossing point as a potential future event in the event queue. If p is the crossing point of two segments s and t (with s below t to the left of the crossing), swap the positions of s and t in T. After the swap, find the segments r and u (if they exist) that are immediately below and above t and s, respectively.
The "nine dots" puzzle. The puzzle asks to link all nine dots using four straight lines or fewer, without lifting the pen. The nine dots puzzle is a mathematical puzzle whose task is to connect nine squarely arranged points with a pen by four (or fewer) straight lines without lifting the pen or retracing any lines.
Intersections of Lines and Planes Algorithms and sample code by Dan Sunday; Robert Pless. Lecture 4 notes. Washington University in St. Louis, CS 506: Computational Geometry (cached copy). Line segment intersection in CGAL, the Computational Geometry Algorithms Library "Line Segment Intersection" lecture notes by Jeff Erickson.
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The missionaries and cannibals problem, and the closely related jealous husbands problem, are classic river-crossing logic puzzles. [1] The missionaries and cannibals problem is a well-known toy problem in artificial intelligence , where it was used by Saul Amarel as an example of problem representation.