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A simple polygonal chain A self-intersecting polygonal chain A closed polygonal chain. In geometry, a polygonal chain [a] is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (,, …,) called its vertices. The curve itself consists of the line segments connecting the ...
In mathematics, a stuck unknot is a closed polygonal chain in three-dimensional space (a skew polygon) that is topologically equal to the unknot but cannot be deformed to a simple polygon when interpreted as a mechanical linkage, by rigid length-preserving and non-self-intersecting motions of its segments.
In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
For closed polygonal chains the same result holds with the integral of curvature replaced by the sum of angles between adjacent segments of the chain. By approximating arbitrary curves by polygonal chains, one may extend the definition of total curvature to larger classes of curves, within which the Fáry–Milnor theorem also holds ( Milnor ...
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which ...
Essentially, this folding process is a time-reversed version of the problem of convexifying a polygon of length smaller than π, but on the surface of a sphere rather than in the Euclidean plane. This result was extended by Panina & Streinu (2010) for spherical polygons of edge length smaller than 2π.
Given a polygonal chain (often called a polyline), the algorithm attempts to find a similar chain composed of fewer points. Points are assigned an importance based on local conditions, and points are removed from the least important to most important. In Visvalingam's algorithm, the importance is related to the triangular area added by each point.
The family of lines formed by the sides of a regular polygon together with its axes of symmetry, and; The sides and axes of symmetry of an even regular polygon, together with the line at infinity. Additionally there are many other examples of sporadic simplicial arrangements that do not fit into any known infinite family. [22]