Chorded keyboard (a writing system for the blind) uses either 6 or 8 tactile 'points' from which all letters and numbers are formed. When Louis Braille invented it, it was produced with a needle holing successively all needed points in a cardboard sheet. In 1892, Frank Haven Hall, superintendent of the Illinois Institute for the Education of the Blind, created the Hall Braille Writer, which was like a typewriter with 6 keys, one for each dot in a braille cell. The Perkins Brailler, first manufactured in 1951, uses a 6-key chord keyboard (plus a spacebar) to produce braille output, and has been
Point in polygon polygon overlaps itself. The point in polygon problem may be considered in the general repeated geometric query setting: given a single polygon and a sequence of query points, quickly find the answers for each query point. Clearly, any of the general approaches for planar point location may be used. Simpler solutions are available for some special polygons. Simpler algorithms are possible for monotone polygons, star-shaped polygons, convex polygons and triangles. The triangle case can be solved easily by use of a barycentric coordinate system, parametric equation or dot product. The dot product method extends naturally to any convex polygon. Point