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Topic: Fundamental polygon

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	GameDev.net - The Polygon Primeval
		Filling a polygon is fundamentally a rasterization process involving drawing all of the horizontal lines within the polygon's boundaries.
		Rasterization of convex polygons is easily done by starting at the top of the polygon and tracing down the left and right sides, one scan line (one vertical pixel) at a time, filling the extent between the two edges on each scan line, until the bottom of the polygon is reached.
		Polygons are skewed toward the top and left edges, which not only introduces drawing error relative to the ideal polygon but also means that a filled polygon won't match the same polygon drawn unfilled.
	www.gamedev.net /reference/articles/article367.asp (2500 words)

	Uniform Tessellations and Polyhedra (Site not responding. Last check: 2007-10-25)
		The result is a tessellation of the sphere; it is "uniform" in that each face is a regular spherical polygon and the vertices are all alike in the same sense as for polyhedra.
		The reflections in two edges of the fundamental triangle generate a dihedral group of rotations and reflections; the rotations are centered at the intersection of the two sides.
		The edges coming from an original vertex have their midpoints arranged in a circle, and in fact they form a regular polygon; the dual edges also form a regular polygon which is a face of the dual polyhedron.
	www.monmouth.com /~chenrich/UniformTessellations.html (2796 words)

	Fundamental polygon - Wikipedia, the free encyclopedia
		This fundamental polygon is convex in that the geodesic joining any two points of the polygon is contained entirely inside the polygon.
		By contrast, the metric fundamental polygon is six-sided, a hexagon.
		The area of the standard fundamental polygon is 4π(n − 1) where n is the genus of the Riemann surface (equivalently, where 4n is the number of the sides of the polygon).
	en.wikipedia.org /wiki/Fundamental_polygon (932 words)

	Efficient 2D Geometric Operations
		The linked list must be circular because polygons are closed sequences of vertices, and operations on pairs or triplets of consecutive vertices may not be limited by the fact that a vertex is the last or the first in the chain.
		Unless otherwise specified, polygons are assumed to be simple (i.e., no edges intersect, and no two vertices are the same point), and, when it is required to assume an orientation, it will be assumed that the vertices are oriented in counter-clockwise sense.
		The orientation of a polygon is often known in advance, since it may be guaranteed by the mechanism used to obtain the polygon.
	mochima.com /articles/cuj_geometry_article/cuj_geometry_article.html (4256 words)

	Linda Keen Abstract (Site not responding. Last check: 2007-10-25)
		Fricke defined the concept of a canonical fundamental polygon for a finitely generated Fuchsian groups.
		His polygon is not canonical in the present technical sense of the word and his proof is complicated.
		The strengthened definition determines the polygon uniquely in terms of a standard system of generators of G. An independent proof of the existence of this new polygon is given.
	www.agnesscott.edu /lriddle/women/abstracts/keen_abstract.htm (62 words)

	Fundamental domain - Wikipedia, the free encyclopedia
		In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern.
		For example, for wallpaper groups the fundamental domain is a factor 1, 2, 3, 4, 6, 8, or 12 smaller than the primitive cell.
		In practice the main use of a fundamental domain may be to compute integrals on G/Γ, in which case the set of measure zero is mentioned only to keep straight the pedantic assertion that D is exactly a set of coset representatives, and may quickly be forgotten.
	en.wikipedia.org /wiki/Fundamental_domain (902 words)

	Tiling the plane with congruent tiles.
		A certain class of tiles which allows such a tiling presents itself readily, namely the fundamental areas of the cover transformation groups of the space R^n, since a fundamental area together with its images under the group obviously form a cover of the required type.
		E' is the chosen neighboring polygon of E. The fact that a polygon can be a fundamental area of a group means that it is possible in some way to select the transition transformations from an tile to an adjacent tile so that each transformation can appear in a two-dimensional discontinuous group of cover transformations.
		Neither of the given glide reflections fulfills this condition however, for their repetition is the translation of ±2 along the y-axis, and this is not a covering transformation.
	www.angelfire.com /mn3/anisohedral/heesch35.html (577 words)

	The Incredible Hull
		For a set of two-dimensional objects, their convex hull is the convex polygon of least area that completely encloses all of the objects.
		In the figure below, the three solid-lined objects on the left are enclosed in a dotted-line convex polygon, but that polygon is not their convex hull; on the right the same three objects are enclosed by their convex hull.
		Any vertex that is part of an input polygon that is also part of the convex hull should appear in the output.
	acm.uva.es /p/v5/596.html (553 words)

	GRASS to ARC/INFO Data Conversion
		A polygon is defined as an ordered set of connected arcs, with the constraint that the first and last arcs must connect (area-definition).
		A polygon is defined by a series of arcs comprising its border and by a label point positioned inside the polygon.
		There is one record in the PAT for each polygon, which is related to the polygon using the polygon's internal sequence number.
	www.cecer.army.mil /techreports/DIL_GRAS/dil_gras-02.htm (2515 words)

	Dr. Dobb's \| Efficient 2-D Geometric Operations, Part 1 \| April 15, 2003
		In addition to this fundamental operation and the basic arithmetic (including the scalar product), two other operations complete the basis for all the algorithms to be presented: a test for segment intersection, and a test for point inclusion in a triangle.
		In this case (as with the Polygon class, yet to be presented), the draw function is platform independent, since it is implemented in terms of the Segment class draw member function.
		The linked list must be circular because polygons are closed sequences of vertices, and operations on pairs or triplets of consecutive vertices must not be limited by a vertex being the last or the first in the chain.
	www.ddj.com /dept/cpp/184403573?pgno=8 (1977 words)

	Dissection Tiling
		It has long been known that for any two polygons of equal area (or groups of polygons adding to the same area) there is a collection of smaller polygonal pieces that can be arranged to form either of the two polygons; this partition into smaller pieces is known as a dissection.
		Tilings have been used directly for constructing dissections (by overlaying two tilings with the same fundamental domain), but they also are useful for understanding n-to-one dissections in the limit as n grows large -- the number of pieces can be approximated by kn+O(sqrt n) where k is the average pieces/polygon in a dissection tiling.
		This sort of averaging allows different copies of a polygon to be cut differently; for instance we can cut two pentagons into a dissection tiling averaging 1.5 pieces/polygon whereas the best one could do with a single polygon would be 2.
	www.ics.uci.edu /~eppstein/junkyard/distile (1634 words)

	Dr. Dobb's \| Graphics Programming \| July 22, 2001 (Site not responding. Last check: 2007-10-25)
		Convex polygons include what you'd normally think of as "convex" and more; as far as we're concerned, a convex polygon is one for which any horizontal line drawn through the polygon encounters the right edge exactly once and the left edge exactly once, excluding horizontal and zero-length edge segments.
		The basic principle of polygon filling is decomposing each polygon into a series of horizontal lines, one for each horizontal row of pixels, or scan line, within the polygon (a process I'll call scan conversion), and drawing the horizontal lines.
		When filling a polygon, we want to draw the pixels within the polygon, but a standard vertex-to-vertex line-drawing algorithm will draw many pixels outside the polygon, as shown in Figure 2.
	www.ddj.com /184408505?pgno=4 (3863 words)

	Polygon.net - FAQs
		When a company applies for Polygon membership, their name is broadcast to the full membership to obtain information from those who might know them and to check for possible credit problems.
		Polygon members include retail jewelry stores, Diamond Trading Company siteholders, AGS guild stores, colored stone dealers, finished jewelry manufacturers/wholesalers, refiners, estate jewelry specialists, appraisers, and just about anyone else professionally involved in the jewelry industry.
		We are dedicated to ensuring your success on Polygon, whether you are an experienced computer user or a beginner.
	www.polygon.net /cgi/en/PogInfo/PogIndexFaqs.htm (792 words)

	: Class RPolygon
		This fundamental class has two functions that it serves the shape of the spores as well as the shape of the claims.
		The remembered side is the only side that is kept to locate the polygon.
		the order of the one and two are important since it is used to determine where the centre of the polygon is. The heading is relative to this side.
	www.csd.uwo.ca /~morey/new/docs/morey/spore/RPolygon.html (932 words)

	[No title]
		Consider the triangulation of the convex polygon that results from joining every second vertex (remember the vertices of P are given as an ordered sequence) to a fixed vertex and then joining every second vertex with an edge.
		A Palm-Shaped Polygon with respect to a point O is a polygon such that the shortest geodesic path from O to any point is either right turning or left turning.
		Given that hint, the trick simply lies in constructing the polygon with an edge between x(i) and x(k) such that the vertices of that edge (or edges connecting it to the rest of the polygon) do not violate the "empty" conditions in the intuition.
	www.ics.uci.edu /~eppstein/junkyard/godfried.toussaint.html (2501 words)

	Graphical output
		Given a sequence of polygons, each of which is defined by a sequence of points in the upper half plane, produce the postscript drawing of these polygons, and write the result to the named file.
		This colour is used for drawing the outline of the polygon.
		In the following example a function is defined to determine the colouring of the polygons in terms of which Farey sequence pairs of end points of the polygons belong to.
	www.umich.edu /~gpcc/scs/magma/text465.htm (1770 words)

	Hyperbolic Planar Tesselations
		The dual of the fundamental tiling is composed of "Schwarz polygons" denoted (p
		We can color each vertex of the uniform tiling (or Schwarz polygon of the dual) "even" or "odd" depending on whether it is generated as an even or odd number of reflections of a fixed initial vertex (or Schwarz polygon of the dual, respectively), i.e.
		Starting with the fundamental tiling, add edges connecting each pair of "even" vertices that share an "odd" neighbor vertex, then remove all "odd" vertices along with all the original edges.
	www.plunk.org /~hatch/HyperbolicTesselations (1397 words)

	Utilities
		These determine whether a polygon is convex, whether a point is inside a given polygon, and the area of a given polygon.
		An antipodal pair is an edge of a polygon and a corresponding vertex, for which a line through the edge and another line through the vertex that is parallel to the edge do not cross through any part of the polygon, but remain tangent.
		The polygon must be convex; this guarantees that an antipodal pair exists for each edge.
	documents.wolfram.com /applications/signals/Utilities.html (1662 words)

	Advanced Guide to UV Mapping - Tutorials - LightWave 3D®
		UVs have to come from somewhere, and for existing polygonal models, the choices are pretty much limited to setting the texture coordinates for each point in the object manually, or applying some 'projection' which automatically generates the 2D texture coordinates from the given 3D point positions.
		The texture on this polygon is mapped from something like 0.94 to 0.12, so the entire image is squished, backwards onto this polygon.
		The fundamental reason for this is that the nice continuous projection is actually discontinuous at this 'seam', and a vertex at that point would have to be double-valued, using a 1.0 for one polygon, and a 0.0 for the other.
	www.newtek.com /products/lightwave/tutorials/uvmapping/uv_mapping/advanced_uv_mapping.html (1517 words)

	Fast Polygon Triangulation based on Seidel's Algorithm (Site not responding. Last check: 2007-10-25)
		This divides the polygon into trapezoids (which can degenerate into a triangle if any of the horizontal segments of the trapezoid is of zero length).
		A monotone polygon is a polygon whose boundary consists of two y-monotone chains.
		These polygons are computed from the trapezoidal decomposition by checking whether the two vertices of the original polygon lie on the same side.
	www.whisqu.se /per/docs/math13.htm (946 words)

	ESRI News -- ArcNews Summer 2002 Issue -- ArcGIS 8.3 Brings Topology to the Geodatabase
		The geodatabase follows the fundamental relational data model in which each object and its attributes are stored as a row in a table.
		The ability to store the complete geometry of a simple feature (such as a parcel polygon) is one of the advantages of the geodatabase model, as the feature is always available for display and analysis.
		Example rules include polygons must not overlap, lines must not have dangles, points must be covered by the boundary of a polygon, polygon class must not have gaps, lines must not intersect, and points must be located at an endpoint.
	www.esri.com /news/arcnews/summer02articles/arcgis83-brings.html (2242 words)

	Spacesimulator.net - vectors, normals and OpenGL lighting
		A polygon is always uniformly illuminated by the ambient environment, it doesn't matter what orientation or position it has in the space.
		The position of the observer is not used for this and the polygon is always uniformly illuminated.
		To calculate the normal vector (also called only "normal") of a polygon it is enough to take the vector product of two vectors that are co-planar to the polygon (we can take for example two sides of the polygon) and then normalize the result.
	www.spacesimulator.net /tut5_vectors_and_lighting.html (2959 words)

	Polygon Triangulation
		Computing the triangulation of a polygon is a fundamental algorithm in computational geometry.
		In computer graphics, polygon triangulation algorithms are widely used for tessellating curved geometries, as are described by splines [Kumar and Manocha 1994].
		A monotone polygon can be triangulated in linear time by using a simple greedy algorithm which repeatedly cuts off the convex corners of the polygon [Fournier and Montuno 1984].
	www.cs.unc.edu /~dm/CODE/GEM/chapter.html (837 words)

	Gamasutra - Features - "GDC 2002: Polygon Soup for the Programmer's Soul: 3D Pathfinding" [04.05.02]
		One of the fundamental goals of an AI system is to avoid making the unit appear "dumb." At the root of this challenge lies one of the hardest problems to overcome efficiently and believably: pathfinding.
		This paper addresses the pitfalls of attempting to pathfind the arbitrary world we call "polygon soup." It covers automatic data generation and compression, a run-time distributed algorithm, organic post-process modifiers, and solutions for tricky situations such as doors, ladders, and elevators.
		The only tools we'll need are a good collision detection system and the knowledge of a single point in the polygon soup where a unit can stand.
	www.gamasutra.com /features/20020405/smith_pfv.htm (4240 words)

	Article Details
		Triangles are the preferred polygon type because they are always convex, and they are always planar—two conditions that are required of polygons by the renderer.
		A polygon is convex if a line drawn between any two points of the polygon is also inside the polygon.
		Each face in a mesh has a perpendicular face normal vector whose direction is determined by the order in which the vertices are defined and by whether the coordinate system is right- or left-handed.
	www.web-helper.net /IBasicDev/Article_Details.asp?category_id=55&art_id=40 (1115 words)

	Torus Summary
		Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA
		The first homology group of the torus is isomorphic to the fundamental group (since the fundamental group is abelian).
		The fundamental group of an n-torus is a free abelian group of rank n.
	www.bookrags.com /Torus (951 words)

	PostgreSQL: Documentation: Manuals: PostgreSQL 7.4: Geometric Types
		The most fundamental type, the point, forms the basis for all of the other types.
		Polygons should probably be considered equivalent to closed paths, but are stored differently and have their own set of support routines.
		where the points are the end points of the line segments comprising the boundary of the polygon.
	www.postgresql.org /docs/7.4/static/datatype-geometric.html (261 words)

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