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Topic: Curve of constant width


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  Curve of constant width - Wikipedia, the free encyclopedia
In geometry, a curve of constant width is a convex planar shape whose width, measured by the distance between two opposite parallel tangent lines to its boundary, is the same regardless of the direction of those two parallel lines.
One defines the width of the curve in a given direction to be the perpendicular distance between the tangents perpendicular to that direction.
The circle is obviously a curve of constant width.
en.wikipedia.org /wiki/Curve_of_constant_width   (553 words)

  
 Curve
Cubic curve In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 in X,Y, and Z. The...
Curve of pursuit A curve of pursuit is a point or points which represents pursuers and pursuees, and the curve of pursui...
Koch curve of the Koch snowflake]] The Koch curve is a snowflake.
www.brainyencyclopedia.com /topics/curve.html   (1028 words)

  
 Reuleaux triangle - Wikipedia, the free encyclopedia
The Reuleaux triangle is a constant width curve based on an equilateral triangle.
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 - that is, a curve in which all diameters are the same length.
en.wikipedia.org /wiki/Reuleaux_triangle   (459 words)

  
 Curve of constant width -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-08)
One defines the width of the curve in a given direction to be the perpendicular distance between the tangents (A Gothic style in 14th and 15th century England; characterized by vertical lines and a four-centered (Tudor) arch and fan vaulting) perpendicular to that direction.
The (Ellipse in which the two axes are of equal length; a plane curve generated by one point moving at a constant distance from a fixed point) circle is obviously a curve of constant width.
A simple example of this would be a circle with width ((The length of a straight line passing through the center of a circle and connecting two points on the circumference) diameter) d having a perimeter of πd.
www.absoluteastronomy.com /encyclopedia/c/cu/curve_of_constant_width.htm   (616 words)

  
 curve ball - Hutchinson encyclopedia article about curve ball
It is one of the family of curves called conic sections that also includes the circle, ellipse, and hyperbola.
Other common geometrical curves are the ellipse, parabola, and hyperbola, which are also produced when a cone is cut by a plane at different angles.
Many curves have been invented for the solution of special problems in geometry and mechanics – for example, the cissoid (the inverse of a parabola) and the cycloid.
encyclopedia.farlex.com /curve+ball   (189 words)

  
 Curve of constant width   (Site not responding. Last check: 2007-10-08)
One defines the widthof the curve in a given direction to be the perpendicular distance between the tangents perpendicular to that direction.
A basic result on curves of constant width is Barbier'stheorem, which asserts that the perimeter of any curve of constant width a isequal to πa.
Famous examples of a curve of constant width are the British 20p and 50p coins.Their heptagonal shape with curved sides means that an automated coin machine will always measure the correct diameter throughthe middle, no matter which angle it takes its measurement from.
www.therfcc.org /curve-of-constant-width-188433.html   (417 words)

  
 [No title]
It is a trumpet-shaped curve that is the locus of points P such that the square of distance of P from the origin is inversely proportional to the angle theta that p makes with the horizontal axis.
The curve t->epitrochoid3d[a,b,h,0][t] is a (planar) epitrochoid and the curve epitrochoid3d[a,b,h,Pi][t] is a (planar) hypotrochoid.
viviani[a][t] is a Viviani curve on a sphere of radius 2*a.
www.ma.umist.ac.uk /kd/mmaprogs/CURVES.m   (5260 words)

  
 Math Trek: Rolling with Reuleaux, Science News Online, Sept. 20, 2003   (Site not responding. Last check: 2007-10-08)
The resulting "curved triangle," as Reuleaux termed it, has a constant width equal to the length of the interior triangle's side.
Moreover, a curve of constant width need not be symmetrical or even consist of circular arcs.
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.sciencenews.org /articles/20030920/mathtrek.asp   (998 words)

  
 Ivars Peterson's MathLand: Rolling with Reuleaux
This width is defined as the distance between a pair of parallel lines touching the curve on opposite sides.
There is actually an infinite number of such curves, any one of which could form a manhole lid or the cross section of a roller that gives as smooth a ride as a cylinder.
So there's an unlimited number of curves of constant width, and the Reuleaux triangle happens to be the family member of least area.
www.maa.org /mathland/mathland_10_21.html   (822 words)

  
 Geometry and Topology - Numericana
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.
Once you have a solid of constant width, you may build infinitely many others, since, for any D>0, the set of all points within a distance D of some given solid of constant width is also a solid of constant width...
home.att.net /~numericana/answer/geometry.htm   (7724 words)

  
 Integral
Formulating the area under a curve is the first step toward developing the concept of the integral.
The area under the curve formed by plotting function f(x) as a function of x can be approximated by drawing rectangles of finite width and height f equal to the value of the function at the center of the interval.
If the width of the rectangles is made smaller, then the number N is larger and the approximation of the area is better.
hyperphysics.phy-astr.gsu.edu /hbase/integ.html   (346 words)

  
 Curve of constant width - Definition up Erdmond.Com   (Site not responding. Last check: 2007-10-08)
For closed convex planar bodies whose boundary is a smooth curve, one notes that there are exactly two parallel tangent_lines to the boundary curve in any given direction.
A basic result on curves of constant width is Barbier's_theorem, which asserts that the perimeter of any curve of constant width a is equal to πa.
The generalization of the the definition of bodies of constant width to convex bodies in R3 and their boundaries leads to the concept of surface_of_constant_width.
www.erdmond.com /Curve_of_constant_width.html   (456 words)

  
 AIPS HELP file (version 31DEC05) for GAL
Other possibilities are the width of the profile, the skewness of the profile, or - if the velocity field was determined using XGAUS - the goodness of fit.
APARM(6) The height of the rotation curve can be guessed by halving the difference of two points on opposite ends of the major axis, and applying an cosec(i) correction.
CPARM(2) If 1, the observed rotation curve is plotted (by integrating the circular velocities in annuli), as well as the fitted model curve.
www.aoc.nrao.edu /cgi-bin/aips/ZXHLP2.PL?GAL   (2026 words)

  
 Curve of constant width - The Jiggies Reference Guide   (Site not responding. Last check: 2007-10-08)
To construct this, take an equilateral triangle ABC and draw the arc BC on the circle with radius A, the arc CA on the circle with radius B, and the arc AB on the circle with radius C. The resulting figure is of constant width.
A basic result on curves of constant width is Barbier's theorem, which asserts that the perimeter of any curve of constant width a is equal to πa.
Δ curve, or curves which can be rotated in the equilateral triangle, have many similar properties to curves of constant width.
www.jiggies.com /reference/Curve_of_constant_width   (449 words)

  
 Learning module on Reuleaux triangle
Another interesting result about curves of constant width is that inscribed and circumscribed circles of an arbitrary figure of constant width h are concentric and the sum of their radii is equal to h.
The solids of constant width that have the smallest volumes are derived from the regular tetrahedron in somewhat the same way the Reuleaux triangle is derived from the equilateral triangle.
Since all curves of the same constant width have the same perimeter, it might be supposed that all solids of the same constant width have the same surface area.
kmoddl.library.cornell.edu /math/2   (1692 words)

  
 K-MODDL > Tutorials > Reuleaux Triangle
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.
Since all curves with the same constant width have the same perimeter, it might be supposed that all solids width the same constant width have the same surface area.
Hermann Minkowski that all the shadows of solids with constant width are curves of the same constant width (when the projecting rays are parallel and the shadow falls on a plane perpendicular to the rays).
kmoddl.library.cornell.edu /tutorials/02   (1878 words)

  
 Constant Acceleration in 1 Dimension
Of course since we are working with constant acceleration, the area of each time slice is exactly equal to dt times the a at any point in the slice.
In the case where the acceleration is constant we derived an expression for the velocity and position as functions of time using the fact that the average acceleration and instantaneous acceleration have the same values.
The value of that constant near the surface of the Earth is approximately 9.8 meters per second per second.
www.mcasco.com /p1consta.html   (3188 words)

  
 Curves of constant width   (Site not responding. Last check: 2007-10-08)
The diameter of C is the largest width for the curve.
In a curve of constant width, it is clear that the (constant) width must be equal to the diameter.
Formally, a pinching set V is a set of points on the boundary of a curve C of constant width such that every diameter of C is incident with at least one point on V.
www.cs.mcgill.ca /~bbaetz/cs507   (1659 words)

  
 mtq24.htm, Vehicle Track, Highway, Railway Circular and Transition Curves   (Site not responding. Last check: 2007-10-08)
On a circular curve at constant speed on a track with constant cross slope there is a steady side acceleration (force) away from the centre.
The common transition curve is a spiral that increases radius from infinity to equal the circular curve radius at a constant rate of change with distance or angular change in direction.
The transition curves are also used to change the cross slope of the track from tangent conditions to the desirable superelevation on the circular curve.
www.unb.ca /web/transpo/mynet/mtq24.htm   (452 words)

  
 KMODDL - Kinematic Models for Design Digital Library   (Site not responding. Last check: 2007-10-08)
Here he constructed a curved triangle from an equilateral triangle with circular arcs whose centers are at the three vertices of the triangle.
This figure is known as a "curve of constant width".
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)

  
 Review of Simple Integrals for PHY 207 / Dr. Miner   (Site not responding. Last check: 2007-10-08)
Since the velocity is constant its value on the graph will be a horizontal line at 50 mi/h.
The product of 50 mi/h (the height) times 3 h (the width) is the area of a box on the graph (the area under the curve).
This is the area under the curve with a non-constant velocity.
www.udayton.edu /~physics/gkm/sintegrals.htm   (370 words)

  
 [No title]   (Site not responding. Last check: 2007-10-08)
Key parts of the analysis extend to the higher--dimensional situation, where the convex body of given constant width and minimal volume is unknown.
Reuleaux polygons with any odd number of sides likewise enjoy the property of constant width It has long been known that among all two-dimensional convex bodies of constant width, the Reuleaux triangle has the smallest area.
Another reason may be the rigidity of the condition of constant width.
www.math.gatech.edu /~harrell/Pubs/reul.texold   (1920 words)

  
 curve - Hutchinson encyclopedia article about curve   (Site not responding. Last check: 2007-10-08)
The Kangaroo immediately adjourned, tracing against the sunset sky a parabolic curve spanning seven provinces, and evanished below the horizon.
She went on around the curve of the veranda, where she found herself alone.
Barbicane took fresh observations on the inclination of the projectile, but to his annoyance it had not turned over sufficiently for its fall; it seemed to take a curve parallel to the lunar disc.
encyclopedia.farlex.com /curve   (277 words)

  
 patterns for curve stitching
Introduction to Curve Stitching: Line Designs (SMILE) A lesson designed to develop students' awareness that straight line segments can produce the illusion of a curve, and their ability to...
I recently read "Curve Stitching - the art of sewing beautiful mathematical patterns" by Jon Millington, in which he discusses curve stitching as an art form...
curve -- like the curve from a neckline up to the ear-- you just snip straight in toward the stitching line...
www.1st-in-needlepoint.com /61/patterns-for-curve-stitching.html   (594 words)

  
 curve stitching pictures
In using curve stitching, the main aim should be to encourage the students to...
Curve stitching is a creative, practical activity with strong mathematical background...
The Knitting and Stitching Show is the definitive event in both the uk and ireland for all aspects of needlecraft...
www.1st-in-needlepoint.com /57/curve-stitching-pictures.html   (459 words)

  
 Round Manhole Covers, or: If Richard Feynman applied for a job at Microsoft :: hebig.org/blog
Still, obviously, a circle can't fit through a hole of the same geometry but slightly smaller size, because it has a constant width - but it is not the only curve of constant width, like the desired answer suggests.
There is an infinite number of such curves, and used as a manhole cover, none would fit through the hole.
As a simple example, the Wankel rotary engine has a curved rotor shaped like the Reuleaux triangle - a curve of constant width (that can basically also be used to drill square holes).
www.hebig.org /blogs/archives/main/000962.php   (871 words)

  
 MayaMan Model Attributes Dialog   (Site not responding. Last check: 2007-10-08)
Curve data is stored in maya as a regular scalar or vector array attribute on the shape, the transform which is the immediate parent of the shape or the attribute node itself.
User attributes whose names start with "rman_" and are found on a shape node, transform node that is the immediate parent of a shape, or a mayaman model attributes node will be placed in the rib file as primitive data that can then be interpreted by the shader.
This data is only output for nurbs and polygons as both curves and particles already have their own data output mechanisms already.
animallogic.com /research/mayaman/onlinedocs/permodel/modelatts.html   (2615 words)

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