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Topic: Reuleaux polygon

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Reuleaux triangle

Curve of constant width

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	Reuleaux triangle - Wikipedia, the free encyclopedia
		A Reuleaux polygon is a polygon that is a curve of constant width - that is, a curve in which all diameters are the same length.
		The Reuleaux triangle is the simplest nontrivial example of a curve of constant width - a curve in which the distance between two opposite parallel tangent lines to its boundary is the same, regardless of the direction of those two parallel lines.
		The existence of Reuleaux polygons is a good demonstration of why you cannot use diameter measurements to verify that an object has a circular cross-section.
	en.wikipedia.org /wiki/Reuleaux_polygon (484 words)

	Homework Problems 12 (Site not responding. Last check: 2007-10-23)
		The area of a Reuleaux triangle is equal to the area of a circular region with the same width.
		The perimeter of a Reuleaux triangle is equal to the circumference of a circle with the same diameter.
		The length of one of these arcs is 1/6 of the circumference of a circle with radius d (because the central angle subtended by the arc is 60 degrees, or 1/6 of the full angle of 360 degrees).
	www-math.cudenver.edu /~wcherowi/courses/m3210/hghw12.html (528 words)

	Ivars Peterson's MathTrek - Rolling with Reuleaux
		One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length.
		Like a circle, a Reuleaux triangle fits snugly inside a square having sides equal to the curve’s width no matter which way the triangle is turned.
		Therefore, an unlimited number of curves of constant width are possible, and the Reuleaux triangle happens to be the family member of least area.
	www.maa.org /mathland/mathtrek_09_22_03.html (888 words)

	FastGeometry
		An alternative, more general form of Reuleaux's triangle is to construct two arcs centered at each vertex, with the new arc typically located opposite the triangle with respect to the vertex.
		Reuleaux's triangle has not found many engineering applications, most likely because as it does not have a fixed centre as it rotates within a band or square equivalent to its width.
		In this case, the radii of the arcs, or rather the width of the shape, are equal to the distance from any given vertex to the farthest of the other vertices.
	trueforce.com /Geometry/Reuleaux_Intro.htm (376 words)

	Geometry Glossary (Site not responding. Last check: 2007-10-23)
		apothem - in a regular polygon, the perpendicular distance from the center to a side; in a circle with a chord, the distance from the midpoint of a chord to the circle's center
		A Reuleaux triangle is an example of a figure of constant width.
		The area under one arch of the cyclogon (pink, in this diagram) is equal to the area of the polygon plus twice the area of the circle that circumscribes the polygon.
	mcraefamily.com /MathHelp/GeometryGlossary.htm (1704 words)

	Ivars Peterson's MathLand: Rolling with Reuleaux
		The simplest such curve is known as the Reuleaux triangle, named after engineer Franz Reuleaux, who taught in Berlin during the late nineteenth century.
		One simple way to generate this figure is to start with an equilateral triangle, then draw three arcs of circles, with each arc having as its center one of the triangle's corners and as its endpoints the other two corners.
		The resulting "curved triangle," as Reuleaux termed it, has a constant width equal to the length of the interior triangle's side.
	www.maa.org /mathland/mathland_10_21.html (822 words)

	Learning module on Reuleaux triangle
		The corners of a Reuleaux triangle are the sharpest possible on a curve with constant width.
		This result can be generalized to the case when the polygon is not regular, but each of its vertices is the endpoint of two diagonals of the same length h (and the lengths of other diagonals are less than h).
		Rotating the Reuleaux triangle around one of its axes of symmetry generates the simplest example of a nonspherical solid of this type.
	kmoddl.library.cornell.edu /math/2 (1692 words)

	http
		A polygon is a closed flat shape with all straight sides, this definition including, for example, the triangle and the pentagon.
		The properties of especially the regular polygons, those with all sides the same length and all angles equal, are covered at Key Stage 2, Key Stage 3 and again at Key Stage 4.
		Rolling polygons have a constant diameter, so that these coins, although not circular, still always fit into a square slot for the purpose of feeding into pay machines.
	www.gjt.me.uk /rolling_polygon/rolling_polygon.htm (593 words)

	Curves of Constant Width and Reuleaux Polygons (Site not responding. Last check: 2007-10-23)
		Unlike many plane curves, the constant width and Reuleaux polygon investigations are rooted in machine design and engineering.
		Franz Reuleaux (1829 - 1905) recognized that simple plane curves of constant width might be constructed from regular polygons with an odd number of sides.
		The shape, size, and position of the chambers are constantly altered by the rotation of the rotor, i.e., the Reuleaux triangle or deltoid.
	curvebank.calstatela.edu /reu/reuleaux.htm (363 words)

	Geometry and Topology - Numericana
		For a regular polygon of given perimeter, the more sides the larger the area.
		Note that with any shape of constant width you can construct infinitely many new ones: The (convex hull of the) envelope of the circles of radius R centered on a curve of constant width is also a curve of constant width.
		One can construct figures of constant diameter [constant width] from a regular polygon (with an odd number of vertices) by drawing small circles of radii R around each vertex and then drawing arcs from each vertex as to connect the two opposite circles at a tangent.
	home.att.net /~numericana/answer/geometry.htm (7726 words)

	Spheroforms (Site not responding. Last check: 2007-10-23)
		Orbiforms in particular have been studied a lot during the nineteenth century and later, particularly by Frank Reuleaux, whose name is now attached to those orbiforms you get by intersecting a finite number of disks of equal radii a, whose center are vertices of a regular polygon of diameter a.
		Reuleaux triangle is the intersection of three discs of radius a, centered on vertices of an equilateral triangle with side length a.
		To obtain spheroforms, a simple construction is to consider a two dimensional body of constant width having an axis of symmetry (like the Reuleaux triangle for instance): the corresponding body of revolution obtained by rotation around this axis is a spheroform.
	www.lama.univ-savoie.fr /~oudet/Meissner/Spheroforms.html (443 words)

	K-MODDL > Tutorials > Reuleaux Triangle
		Franz Reuleaux was the first to demonstrate its constant-width properties and the first to use the triangle in mechanisms.
		The solids with constant width that have the smallest volumes are derived from the regular tetrahedron in somewhat the same way that the Reuleaux triangle is derived from the equilateral triangle: Spherical caps are first placed on each face of the tetrahedron, and then three of tghe edges must be slightly altered.
		The least area rotor for the equilateral triangle is a biangle (a lens shaped figure) formed from two 60-degree arcs of the circle with radius equal to the triangle’s altitude.
	kmoddl.library.cornell.edu /tutorials/02 (1878 words)

	Reuleaux Triangle
		For every rotation of the reuleaux, the centroid makes three revolutions in the opposite direction, and its path is not circular.
		The reuleaux turns inside the square because the square is formed by two pairs of parallel lines, equally spaced.
		When I say that the reuleaux is inscribed in a rhombus, I mean that it touches every side, but crosses none of them.
	whistleralley.com /reuleaux/reuleaux.htm (854 words)

	KMODDL - Kinematic Models for Design Digital Library (Site not responding. Last check: 2007-10-23)
		Reuleaux called the line or point contact between machine components \"higher order pairs\" in contrast to bodies in surface contact constraint.
		Here he constructed a curved triangle from an equilateral triangle with circular arcs whose centers are at the three vertices of the triangle.
		While one\'s intuition might lead one to conclude that three points of contact of a plane figure would constrain the motion of the curved triangle in the square chamber, Reuleaux showed that it was possible for the object to rotate and slide since the three contact normals always meet at a point.
	www.library.cornell.edu /kmoddl-test/model.php?m=233 (253 words)

	Math Trek: Rolling with Reuleaux, Science News Online, Sept. 20, 2003 (Site not responding. Last check: 2007-10-23)
		Draw three arcs of circles, with each arc having as its center one of the triangle's corners and as its endpoints the other two corners.
		Reuleaux curves based on the pentagon (top) and heptagon (bottom).
		A steam locomotive, for example, is a machine that converts the reciprocating motion of a piston into the rotation of its driving wheels.
	www.sciencenews.org /20030920/mathtrek.asp (985 words)

	more circles
		A Reuleaux polygon is a curvilinear polygon where the center of each circle arc is the opposite point of the polygon.
		Each Reuleaux polygon is a curve with constant width.
		In the case of three equal arcs beginning in the angles of an equal triangle, it is called the Reuleaux triangle.
	www.2dcurves.com /conicsection/conicsectioncm.html (1139 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		The simplest such curve is known as the REULEAUX TRIANGLE, named after engineer Franz Reuleaux.
		This property can be seen in the figure above and is the basis for an ingenious rotary drill that, constrained by a special guide plate (which has the same structure as the polygon to be drilled), bores square holes.
		So there's an unlimited number of curves of constant width, and the Reuleaux triangle happens to be the family member of least are Why can't Reuleaux polygons be used in place of wheels?
	www.iitk.ac.in /ame/1998/polygon1.htm (612 words)

	About "Rolling with Reuleaux" (Site not responding. Last check: 2007-10-23)
		There is actually an infinite number of such curves, any one of which could form a manhole lid...
		The simplest such curve is known as the Reuleaux triangle...
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/view/4972.html (141 words)

	ClassZone: Geometry - Links (Site not responding. Last check: 2007-10-23)
		This means, for example, that if you were to make a manhole cover using one of these shapes, you could turn the cover so that it would fall through, while a circular cover would not fall through.
		The Reuleaux triangle is the simplest noncircular curve that has a constant width.
		Any regular polygon with an odd number of sides can be used as the basis for a Reuleaux polygon of constant width.
	classzone.com /larson2000/geometry/common/chapter10/eld_applicl4.cfm (593 words)

	Curves of constant width (Site not responding. Last check: 2007-10-23)
		A Reuleaux polygon (or RP) is a curve of constant width which admits a finite pinching set.
		You can start with a regular star-shaped polygon, and distort it, dragging each point along what would be the limited arc.
		A linear time Construction of Reuleaux Polygons, Yaakov Kupitz, Horst Martini, Bernd Wegner, Beitrage zur Algebra und Geometrie, Contributions to Algebra and Geometry, Volume 37 (1996), No. 2, 415-527.
	www.cs.mcgill.ca /~bbaetz/cs507 (1659 words)

	GO.HRW.COM
		Investigate the properties of a "triangle" which is not a polygon--the Reuleaux Triangle.
		Study the symmetry in quilt patterns and some of the various symmetric patterns of Asian rugs.
		Put the translation and rotation tessellations together to study various designs by M.C. Escher.
	go.hrw.com /ndNSAPI.nd/gohrw_rls1/pKeywordResults?MG1+CH3 (65 words)

	Polygon Shape - california concord driver education (Site not responding. Last check: 2007-10-23)
		specular shaders whatâs more, even though a camera is erroneous play about illogical mark as expected sight as expected play about illogical polygon existence mapped, infinite reflections choice appear, fair similar erroneous
		science information - commotion construct a curve as expected constant width emptiness singular presently and then retaliation certain success a wrong equilateral relationship approximately and presently and then any polygon endure retaliation certain success a wrong odd math unit as expected sides.
		emptiness suitable polygonal faces imagine a soccer ball's 32-polygon design superimposed on earth's surface, endure individual vertex deserving blame play about illogical south pole plus another
	pro1071.pergold25.info /polygon-shape.php (385 words)

	Fields Institute - Re: What are our 'Geometry Questions'? (Site not responding. Last check: 2007-10-23)
		Some of the stuff floating around cyberspace regarding Reuleaux Triangles mentions that access covers are shaped as such in certain parts of Minnesota, but a fairly quick cybersearch didnt turn up any photos.
		I guess the Circle is the limit of the polygons with constant diameter - a sequence which begins(?) with the Reuleaux Triangle...
		For a polygon, that would be yet another variable).
	www.fields.utoronto.ca /programs/mathed/meforum/mail/02/06-19f.html (251 words)

	KMODDL - Kinematic Models for Design Digital Library (Site not responding. Last check: 2007-10-23)
		The action is closed kinematically by using a curve of constant width between two surfaces separated by a constant distance.
		In model L-5 the rotor is a five-sided curved polygon of constant width.
		A screw allows the center of motion of the curved pentagon to be changed.
	www.library.cornell.edu /kmoddl-test/model.php?m=261 (234 words)

	András Bezdek - On fat polygons and polyhedra
		We investigate the problem of finding the polygons (polyhedra resp.) with n vertices and of diameter 1 which have the largest possible width w(n) (W(n) resp).
		In the later upper bound equality holds if n has an odd divisor greater than 1 and in this case a polygon
		is extremal if and only if it has equal sides and it is inscribed in a Reuleaux polygon of constant width 1, so that the vertices of the Reuleaux polygon are also vertices of
	www.cms.math.ca /Events/summer98/s98-abs/node27.e?nomenu=1 (116 words)

	Indiatime: The Invasion of Indian Manhole Covers (Site not responding. Last check: 2007-10-23)
		It fetched a range of answers from the smartypants 'because the manholes are round' to the more technical 'because the round shape prevents the covers from falling in'.
		(To prevent falling inside the manhole, the cover needs to be either a simple round shape or a more complicated 'reuleaux polygon').
		Karl Shapiro spent the last six years of his life in New York city, and definitely must have seen the invasion of Indian manhole covers.
	www.indiatime.com /archives/2005/03/the_invasion_of.html (248 words)

	Computational Geometry on the Web
		Polygonizations of Point Sets and Generating Random Polygons
		On the number of diagonals in a convex polygon (with interactive Java applet)
		Computing visibility graphs of line segments and polygons
	cgm.cs.mcgill.ca /~godfried/teaching/cg-web.html (620 words)

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