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Topic: Star polygon

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Regular polygon

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	Polygon - LoveToKnow 1911
		The term regular polygon is usually restricted to "convex" polygons; a special class of polygons (regular in the wider sense) has been named "star polygons" on account of their resemblance to star-rays; these are, however, concave.
		The Arabian geometers of the 9th century showed that the heptagon required the solution of a cubic equation, thus resembling the Pythagorean problems of "duplicating the cube" and "trisecting an angle." Edmund Halley gave solutions for the heptagon and nonagon by means of the parabola and circle, and by a parabola and hyperbola respectively.
		Mystical and magical properties were assigned to them at an early date; the Pythagoreans regarded the pentagram, the star polygon derived from the pentagon, as the symbol of health, the Platonists of well-being, while others used it to symbolize happiness.
	www.1911encyclopedia.org /Polygon (1465 words)

	Construction Sheet and Flag Variants (Jordan)
		The star is quite a lot smaller than is often seen in many actual flags and lies at the intersection of the three lines that bisect the angles of the corners of the triangle.
		A hexagram is the star polygon {6/2} and a pentagram is {5/2}.
		The star is to be inscribed in a circle with diameter 1/7th of hoist, be of density 3, and set in the point where the bisectors of the angles of the triangle cross.
	areciboweb.50megs.com /fotw/flags/jo'.html (1091 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-12)
		Date: 07/07/2005 at 05:34:43 From: Louisa Subject: Star Polygon If you arrange a set of points roughly around a circle or an oval, and then you connect each point to the next with segments, you should get a convex polygon.
		Stars with even numbers of points are a bit odd, since they are made of two distinct paths, so you have to analyze them as two "stars" with half the paths.
		The six-pointed "Star of David" is actually two triangles, so you'd look at it as the angles in two triangles, et cetera.
	mathforum.org /library/drmath/view/68402.html (660 words)

	POV_Ray Star Polygons
		Star Polygons: With two user-defined parameters, N and S, we may create a variety of both regular convex polygons and star polygons, with cylinders on their sides and spheres on their edges.
		With two additional parameters, F and E1, we may construct star polygons in which every Fth vertex is at distance E1 from the center of the polygon, otherwise, at distance 1.
		Hence an irregular star 24-gon is created, in which six of the vertices are at distance 1.5 (and lie on the vertices of a regular hexagon of circumradius 1.5), while the rest are at distance 1, from the center of the polygon.
	home.inreach.com /rtowle/POV/STAR_POLYGON.html (310 words)

	star
		The stars are classified according to their relative brightness in what are known as magnitudes, the first being the brightest and the sixth the faintest visible to the naked eye.
		In Astronomy is: A common proper motion of stars in the same region of the heavens: noticed in close groups of stars and in pairs of widely separated stars.
		Star of David: A hexagram used as a symbol of Judaism.
	ourworld.compuserve.com /homepages/art_lewis/star.htm (1253 words)

	Polygon Summary
		Polygons are named to indicate the number of their sides or number of noncollinear points present in the polygon.
		If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.
		Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g.
	www.bookrags.com /Polygon (1346 words)

	A Unified Theory of Proportion
		Through many cultures, star polygons were used as sacred symbols with the star of David and the Sri Yantra Hindu patterns shown in Figure 1a and 1b as two examples.
		I have previously shown that star polygons are also related to the chaotic dynamics of the logistic equation [Kappraff 2002].
		A regular star polygon is denoted by the symbol {n/k} where n is the number of vertices and edges while k indicates that each vertex is connected to the k-th vertex from it in a clockwise direction.
	www.mi.sanu.ac.yu /vismath/kappraff1 (3000 words)

	Activity 7
		Star polygons are constructed by connecting evenly-spaced dots on a circle.
		A 5-pointed star can be formed by connecting every second dot on a circle with 5 evenly-spaced dots, as is shown at the right.
		Describe each as a star polygon, giving the number of points, or as a regular polygon, giving the number of sides.
	homepage.mac.com /efithian/Geometry/Activity-08.html (588 words)

	Making Stars
		Some caution must be exercised if you wish perfectly straight end points to your star, but all this requires is the use of the CTRL key for contstraint when moving a node either with the cursor of by selecting it and "nudging" it with the keyboard arrow keys.
		The second "dancing" star is a series of 4-sided polygon star shadows,colored white, and arranged irregularly and randomly at various intervals along the timeline (with one base design maintained when random changes are not made to the object positions).
		The third "whorling" star is a 16-sided star polygon with half its vertices stretched out and rotated to "thin" the rays.
	www.corelmag.com /0703/star/makestar.html (1036 words)

	star figure (Site not responding. Last check: 2007-10-12)
		The star figure is a variation on the star polygon.
		Star figure or star polygon is sometimes called a polygram.
		a=6, b=2: the hexagram, also named star of David or Jews' star (a=6, b=2), named to the Jewish king David, is the symbol of the Jews.
	www.2dcurves.com /line/linesf.html (126 words)

	CG & CG Lab (Director: Martin Held) - Random Objects
		This need for polygonal test data motivated Thomas Auer and me to study the generation of random polygons on a given set of vertices.
		Since no polynomial-time solution is known for the uniformly random generation of simple polygons, we have focused on heuristics that offer a good time complexity and still generate a rich variety of different polygons.
		Roughly, we start with generating a random polygon on the set of vertices and then cut-off some of the pockets of the polygon by inserting parallel bridge edges, thus generating polygonal holes within a polygonal outer boundary.
	www.cosy.sbg.ac.at /~held/projects/rpg/rpg.html (641 words)

	Point in Polygon Strategies
		The polygon is treated as a fan of triangles emanating from one vertex and the point is tested against each triangle by computing its barycentric coordinates.
		The bounding box surrounding the polygon is split into a number of horizontal bins and the parts of the edges in a bin are kept in a list, sorted by the minimum X component.
		Random polygons tend to be somewhat unlikely (no one ever uses 1000 edge random polygons for anything except testing), while regular polygons are more orderly than a "typical" polygon; normal behavior tends to be somewhere in between.
	www.acm.org /pubs/tog/editors/erich/ptinpoly (4418 words)

	Stars
		The proportions relevant to a system of proportionality make their appearance as a pair of ratios: 1) the ratio in which the edges of a regular star polygon intersect, and 2) the ratio of the longest diagonal of a regular polygon to the length of the side.
		A star polygon with n vertices in which each vertex connects to the p-th vertex rotated from it in a clockwise direction is denoted by {n,p}.
		If n and p are relatively prime, the star polygon can be drawn in a single stroke without taking pencil off paper.
	members.tripod.com /vismath/kappraff/kap2.htm (252 words)

	CS 506: Assignments
		Problem 1: a simple polygon is called star-shaped if there exists a point q in the interior of the polygon from which every interior point is visible (that is, such that, for any point p within the polygon, the segment pq does not cross the perimeter).
		Show that, given a star-shaped polygon and its special point q, we can store the polygon in such a way that the query "is point r=(x,y) within the polygon" can be answered in O(log n) time.
		If the result is empty, then the claim that the polygon was star-shaped was false; otherwise, any point within the convex polygon that is the intersection is a valid choice.
	www.cs.unm.edu /~moret/cs506/hwk3_sol.html (708 words)

	Inkscape tutorial: Shapes
		Stars are the most complex and the most exciting Inkscape shape.
		In geometry, a polygon is a shape with straight line edges and sharp corners.
		As you can see, while a rounded rectangle has straight line segments in its sides and circular (generally, elliptic) roundings, a rounded polygon or star has no straight lines at all; its curvature varies smoothly from the maximum (in the corners) to the minimum (mid-way between the corners).
	www.inkscape.org /doc/shapes/tutorial-shapes.html (2699 words)

	A New Class of Tilings with Two Prototiles
		Reflections are needed with any star polygon to ensure transitivity between the edges either side of the points of the star.
		If this regular polygon is small, it does not extend to meet any other vertex (other than A which it replaces), then we have tiling VA1.
		On the other hand, if the regular polygon is large, is must extend to C and therefore be a 6-pointed star, that is tiling NEW95.
	www.mi.sanu.ac.yu /vismath/PUB/TWOPUBS.HTM (1600 words)

	Covering Orthogonal Polygons with Star Polygons: The Perfect Graph Approach \| EECS at UC Berkeley
		A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q.
		Since weakly triangulated graphs are perfect, we obtain the following duality relationship: the minimum number of star polygons needed to cover an orthogonal polygon P without holes is equal to the maximum number of points of P, no two of which can be contained together in a covering star polygon.
		In the case where the polygon has at most three dent orientations, we show that the polygon covering problem can be reduced to the problem of covering a triangulated (chordal) graph with a minimum number of cliques.
	www.eecs.berkeley.edu /Pubs/TechRpts/1987/6218.html (277 words)

	2D Polygon Scan-line Algorithm
		In addition to submitting the code of your implementation of the polygon fill algorithm, you are to turn in a written report that answers questions about your work, given at the end of this assignment description.
		If two polygons share an edge, then there should be no gaps between the polygon, and no point should be used by both polygons.
		This will require augmenting your scan-line algorithm to deal with patterns, where a solid polygon is the special case of a solid pattern, and adding a routine that responds to the `p' key by switch the pattern from solid to something else.
	www-static.cc.gatech.edu /classes/cs4390_97_summer/assign1.html (896 words)

	Shape Blending Using the Star-Skeleton Representation (Site not responding. Last check: 2007-10-12)
		We present a novel approach to the problem of generating a polygon sequence that blends two simple polygons given a correspondence between their boundaries.
		We present algorithms for creating equivalent star-skeletons for the two polygons and for blending two star-skeletons by interpolating their skeletons and unfolding intermediate polygons from the skeletons.
		The intrinsic reasons for the good results obtained are the fact that the interiors of the polygons are considered, not only the boundaries, and that the star-skeleton explicitly models an interdependence between all the vertices of the polygons.
	www.cs.huji.ac.il /~arir/starskl-abs.html (202 words)

	Computational Geometry. Homework Assignments, Exams, Lecture Notes
		The task is to generate a random polygon using different approaches: incremental, divide-and-conquer, onion peeling, flipping edges, gift-wrapping/monotone algorithms.
		A star shaped polygon may be generated from the input set of points.
		One possibility is to have two buttons: button “Star-Shaped Polygon” for generating star-shaped polygons based on the input set of points, and button “Convex Hull” for generating the convex hull based on the polygon.
	carbon.cudenver.edu /~vzakaria/CSC5803/projects (606 words)

	Creating Objects in Illustrator 10 > Creating Complex Objects (Site not responding. Last check: 2007-10-12)
		Radius 1 is the distance from the center of the star to the end of a point.
		Radius 2 is the distance from the center of the star to the angle at the base of two points.
		The only difference between the two stars is the length of the second radius, the distance from the center of the star to the base of the points.
	www.samspublishing.com /articles/article.asp?p=26355&seqNum=3 (1064 words)

	Computer Graphics - JAVA Applets by Meiko Rachimow
		A "Star Polygon" is labeled with {p/q}(means every qth point is connected).
		A vertex figure is a polygon formed by connecting the midpoints of all adjacent sides of a vertex.
		The p describes the polygons, and the q the count of that polygons at each vertex.
	www.pentatope.de (214 words)

	Measurement Session 4, Part C: GeoLogo
		Examine the polygons you drew where your turns were all in one direction (either all to the right or all to the left).
		Use your drawings to determine the relationship between the measure of central angles and the measure of exterior angles in regular polygons.
		In order to draw a non-intersecting star polygon, you must direct your pencil to turn both right and left.
	www.learner.org /channel/courses/learningmath/measurement/session4/part_c/noflash.html (325 words)

	VB Helper: HowTo: Draw "stars" inside regular polygons in VB .NET
		For example, a normal five-pointed star is drawn inside a pentagon by connecting every 2nd vertex.
		This drawing scheme will visit every vertex in the polygon before repeating if and only if the number of vertices N and the number K is relatively prime (i.e.
		It makes a new array containing the polygon's vertices in the proper order to draw the "star," translates the Graphics object to position the "star," and then draws it.
	www.vb-helper.com /howto_net_draw_stars.html (673 words)

	Drawing basic polygons and stars
		With the Polygon tool, you can draw any equilateral polygon or star, from a triangle to a polygon or star with 360 sides.
		To constrain a polygon's orientation to increments of 45˚, hold down Shift as you draw.
		To constrain a star's orientation to increments of 45˚, hold down Shift as you drag.
	livedocs.macromedia.com /fireworks/8/fwhelp/04_vect4.htm (219 words)

	Proof: Polygons with Holes
		Most cases are trivial except for the one where i and j in the figure could be covered by one star polygon.
		Since no star polygon can contain more than one even numbered point in our staircase structure, and since there are K even distinguished points, every cover must contain K stars to cover those even points.
		Finally, if r is the number of numbered points in a variable loop, the number of star polygons needed to cover that loop is r/2.
	cgm.cs.mcgill.ca /~cwu25/proj507/holeproof.html (798 words)

	Stars and Rosettes
		The pattern is one of the oldest in the Islamic tradition.
		When sixfold stars are arranged as on the left side of Figure 3, a higher-level structure emerges: every star is surrounded by a ring of regular hexagons.
		The rosette, a central star surrounded by hexagons, appears frequently in Islamic art.
	members.tripod.com /vismath4/kaplan/node2.html (394 words)

	A Thing In Common
		Draw an {N/D} star polygon - a polygon whose N vertices are uniformly distributed on a circle, but each vertex is connected to its D-th neighbor.
		When N and D have a common factor, there are conflicting view points as to whether the configuration of nodes and edges represents a single polygon or a collection of polygons with fewer vertices.
		However, for the puzzle we shall assume that N and D are mutually prime.
	www.maa.org /editorial/knot/CommonThing.html (1117 words)

	Adonai - Living in the Light - The Number 7
		Every circle has 360°, so we divide 360 by 9 = 40°, and therefore we know that each of the 9 points on the circle is in 40° distance to the next point.
		A nonagram (nona, from Latin = nine) is a nine-pointed star drawn with nine straight strokes.
		The star polygon built from 3 equilateral triangles, {9/3} star polygon, is known as the Star of Goliath.
	www.adonim.com /numbers/number9.html (884 words)

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