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Given a rectilinear polygon (whose sides meet at right angles) R in the plane, a set S of points in R, and a set of identical squares, the goal is to find the largest number of non-overlapping squares that can be packed in points of S. Suppose that, for each point p in S, we put a square centered at p. Let G S be the intersection graph of these ...
The convex hull of a simple polygon is divided by the polygon into pieces, one of which is the polygon itself and the rest are pockets bounded by a piece of the polygon boundary and a single hull edge. Although many algorithms have been published for the problem of constructing the convex hull of a simple polygon, nearly half of them are ...
An ear of a polygon is defined as a triangle formed by three consecutive vertices ,, of the polygon, such that its edge lies entirely in the interior of the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears. [1]
Codeforces (Russian: Коудфорсес) is a website that hosts competitive programming contests. [1] It is maintained by a group of competitive programmers from ITMO University led by Mikhail Mirzayanov. [2] Since 2013, Codeforces claims to surpass Topcoder in terms of active contestants. [3] As of 2019, it has over 600,000 registered users ...
In computational geometry, the point-in-polygon (PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon. It is a special case of point location problems and finds applications in areas that deal with processing geometrical data, such as computer graphics , computer vision , geographic ...
In this setup, back-face may be regarded as a test of whether the points in the polygon appear in clockwise or counter-clockwise order when projected onto the screen. If the user has specified that front-facing polygons have a clockwise winding, but the polygon projected on the screen has a counter-clockwise winding then it has been rotated to ...
For two convex polygons P and Q in the plane with m and n vertices, their Minkowski sum is a convex polygon with at most m + n vertices and may be computed in time O(m + n) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given and the direction, say, counterclockwise, along the ...
When is a prime number and and are members of a commutative ring of characteristic, then (+) = +.This can be seen by examining the prime factors of the binomial coefficients: the nth binomial coefficient is