Astroide: diferència entre les revisions

{{ confuse| .3. asteroid}} ..

[[Image:Astroid.svg| .3. 240px| .3. thumb| .3. right| .3. Astroid]] ..

[[Image:HypotrochoidOn4.gif| .3. 500px| .3. thumb| .3. The construction of the astroid.]] ..

In [[mathematics]]MATHEMATICS , an '''((((( astroid''' ))))) is a particular type of [[curve]]CURVE: a [[hypocycloid]]HYPOCYCLOID with four [[cuspCUSPS. (singularity)|cusp]]s..

Astroids are also [[superellipse]]sSUPERELLIPSES: all astroids are scaled versions of the curve specified by the equation ..

: <math>x^{2/3} + y^{2/3} = 1. \ ,</math> ..

Its modern name comes from the [[Greek language|Greek]]GREEK word for "[[star]]STAR". The curve had a variety of names , including '''((((( tetracuspid''' ))))) (still used) , cubocycloid , and paracycle. It is nearly identical in form to the [[evolute]]EVOLUTE of an ellipse. ..

A [[circle]]CIRCLE of radius 1/4 rolls around inside a circle of radius 1 and a point on its circumference traces an astroid. A [[lineLINE segment]]SEGMENT of length 1 slides with one end on the [[xX-axis]]AXIS and the other on the [[yY-axis]]AXIS , so that it is tangent to the astroid (which is therefore an [[envelopeENVELOPE (mathematicsMATHEMATICS)#ExampleEXAMPLE 3|envelope]] .3. ENVELOPE). The [[parametricPARAMETRIC equation]]sEQUATIONS are ..

: <math>x=\cos^3\theta ,\qquad y=\sin^3\theta.</math> ..

The astroid is a real locus of a [[algebraicPLANE curve|planeALGEBRAIC algebraic curve]]CURVE of [[geometric genus|genus]]GENUS zero. It has the equation ..

: <math>(x^2+y^2-1)^3+27x^2y^2=0. \ ,</math> ..

The astroid is therefore of degree six , and has four cusp singularities in the real plane , the points on the star. It has two more complex cusp singularities at infinity , and four complex double points , for a total of ten singularities. ..

The [[PlückerDUAL formula|dual curve]]CURVE to the astroid is the [[cruciformCRUCIFORM curve]]CURVE with equation <math>\textstyle x^2 y^2 = x^2 + y^2.</math> ..

The [[evolute]]EVOLUTE of an astroid is an astroid twice as large. ..

An astroid created by a circle rolling inside a circle of radius <math>a</math> will have an area of <math>\frac{3}{8} \pi a^2</math> and a perimeter of 6a. ..

(==) In popular culture (==) ..

The [[Steelmark]]STEELMARK , originally created by [[U.S. Steel]]STEEL , consists of three astroids , one each of blue , yellow and red. It has been adapted as the logo for the [[PittsburghPITTSBURGH Steelers]]STEELERS on helmets and merchandise since 1962. It can also be found on many steel products and canned goods as it has been "gifted" by U.S. Steel to the [[AmericanAMERICAN IronIRON andAND SteelSTEEL Institute]]INSTITUTE for use on all steel products. ..

On the American television show ''[[The(( PriceTHE PRICE IS RIGHT )) is Right]]'', astroids have been used as set decorations behind the host , on contestants' bid booths and behind the prizes since the program's inception in 1968. ..

German [[FormulaFORMULA One]]ONE driver [[MichaelMICHAEL Schumacher]]SCHUMACHER has worn a series of race [[helmet]]sHELMETS from 1992 until the present , all of which have featured a pattern of astroids on the top. ..

A symbol composed of an astroid with a cutout tangent circle in the centre is used on [[aeronauticalAERONAUTICAL chart]]sCHARTS to signify an [[area navigation|RNAV]] [[waypoint]]WAYPOINT. ..

(==) See also (==) ..

* [[Deltoid curve|Deltoid]]DELTOID a curve with three cusps. ..

(==) References (==) ..

* {{ cite book | .3. author=J. Dennis Lawrence | .3. title=A catalog of special plane curves | .3. publisher=Dover Publications | .3. year=1972 | .3. isbn=0-486-60288-5 | .3. pages=4–5 ,34–35 ,173–174 }} ..

* {{ cite book | .3. author = Wells D | .3. year = 1991 | .3. title = The Penguin Dictionary of Curious and Interesting Geometry | .3. publisher = Penguin Books | .3. location = New York | .3. isbn = 0-14-011813-6 | .3. pages = 10&ndash ;11}} ..

(==) External links (==) ..

* {{ MathWorld | .3. urlname=Astroid | .3. title=Astroid}} ..

* [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Astroid.html "Astroid" at The MacTutor History of Mathematics archive] ..

* [http://www.2dcurves.com/roulette/roulettea.html Article on 2dcurves.com] ..

* [http://xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html Visual Dictionary Of Special Plane Curves , Xah Lee] ..

* [http://www.bama.ua.edu/~tmewes/Java/Astroid/StonerAstroid.shtml Stoner-Wohlfarth astroid applet (physics)] ..

* [http://demonstrations.wolfram.com/BarsOfAnAstroid/ Bars of an Astroid] by Sándor Kabai , [[TheTHE WolframWOLFRAM DemonstrationsDEMONSTRATIONS Project]]PROJECT. ..

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