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| {{Unreferenced|date=February 2007}}
| | Hi [http://www.publicpledge.com/blogs/post/7034 online psychics] there, I am Andrew Berryhill. Office supervising is where my primary income comes from but I've usually wanted my own company. The preferred hobby for him and his kids is to play lacross and he'll be starting something else alongside with it. Alaska is the only place I've been residing in but now I'm considering other choices.<br><br>Also visit my website :: [http://help.ksu.edu.sa/node/65129 email psychic readings] chat online ([http://www.familysurvivalgroup.com/easy-methods-planting-looking-backyard/ www.familysurvivalgroup.com]) |
| In [[computational geometry]], the '''intersection of a polyhedron with a line''' is the problem of computing the [[intersection (set theory)|intersection]] of a [[convex polyhedron]] and a [[ray (geometry)|ray]] in [[Euclidean space]]. This problem has important applications in [[computer graphics]], [[optimization (mathematics)|optimization]], and even in some [[Monte Carlo methods]].
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| ==Statement of the problem==
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| In general, a convex [[polyhedron]] is defined as the intersection of a finite number of [[Half-space (geometry)|halfspace]]s. That is, a convex polyhedron is the set of solutions of a system of [[inequation]]s of the form
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| :<math>Ax \le b. </math> | |
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| The formal statement of our problem is to find the intersection of the set <math>\{x\in\mathbb{R}^n|Ax \le b\}</math> with the line defined by <math>c+tv</math>, where <math>t\in\mathbb{R}</math> and <math>c, v\in\mathbb{R}^n</math>.
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| ==General solution==
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| To this end, we would like to find <math>t</math> such that <math>A(c+tv)\le b</math>, which is equivalent to finding a <math>t</math> such that
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| :<math>[Av]_it \leq [b-Ac]_i</math>
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| for <math>i=1,\ldots,n</math>.
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| Thus, we can bound <math>t</math> as follows:
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| :<math>t \leq \frac{[b-Ac]_i}{[Av]_i} \;\;\;{\rm if}\;[Av]_i > 0</math>
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| :<math>t \geq \frac{[b-Ac]_i}{[Av]_i} \;\;\;{\rm if}\;[Av]_i < 0 </math>
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| :<math>t {\rm\; unbounded} \;\;\;{\rm if}\;[Av]_i = 0, [b-Ac]_i > 0 </math>
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| :<math>t {\rm\; infeasible} \;\;\;{\rm if}\;[Av]_i = 0, [b-Ac]_i < 0 </math>
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| The last two lines follow from the cases when the direction vector <math>v</math> is parallel to the halfplane defined by the <math>i^{th}</math> row of <math>A</math>: <math>a_i^Tx \leq b_i</math>. In the second to last case, the point <math>c</math> is on the inside of the halfspace; in the last case, the point <math>c</math> is on the outside of the halfspace, and so <math>t</math> will always be infeasible. | |
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| As such, we can find <math>t</math> as all points in the region (so long as we do not have the fourth case from above)
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| :<math>\left[ \max_{i:[Av]_i\leq 0}\frac{[b-Ac]_i}{[Av]_i}, \min_{i:[Av]_i\geq 0}\frac{[b-Ac]_i}{[Av]_i}\right],</math> | |
| which will be empty if there is no intersection.
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| ==External links==
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| *[http://adrianboeing.blogspot.com/2010/02/intersection-of-convex-hull-with-line.html Intersection of convex hull with a line with pseudo code]
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| {{DEFAULTSORT:Intersection Of A Polyhedron With A Line}}
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| [[Category:Euclidean geometry]]
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Hi online psychics there, I am Andrew Berryhill. Office supervising is where my primary income comes from but I've usually wanted my own company. The preferred hobby for him and his kids is to play lacross and he'll be starting something else alongside with it. Alaska is the only place I've been residing in but now I'm considering other choices.
Also visit my website :: email psychic readings chat online (www.familysurvivalgroup.com)