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| [[Image:MaclaurinTrisectrix.SVG|right|thumb|300px|The Trisectrix of Maclaurin showing the angle trisection property]]
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| In [[geometry]], the '''trisectrix of Maclaurin''' is a [[cubic plane curve]] notable for its [[trisectrix]] property, meaning it can be used to trisect an angle. It can be defined as locus of the points of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a [[sectrix of Maclaurin]]. The curve is named after [[Colin Maclaurin]] who investigated the curve in 1742.
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| ==Equations==
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| Let two lines rotate about the points <math>P = (0,0)</math> and <math>P_1 = (a, 0)</math> so that when the line rotating about <math>P</math> has angle <math>\theta</math> with the ''x'' axis, the rotating about <math>P_1</math> has angle <math>3\theta</math>. Let <math>Q</math> be the point of intersection, then the angle formed by the lines at <math>Q</math> is <math>2\theta</math>. By the [[law of sines]], | |
| :<math>{r \over \sin 3\theta} = {a \over \sin 2\theta}\!</math>
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| so the equation in [[polar coordinate]]s is (up to translation and rotation)
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| :<math>r= a \frac{\sin 3\theta}{\sin 2\theta} = {a \over 2} \frac{4 \cos^2 \theta - 1} {\cos \theta} = {a \over 2} (4 \cos \theta - \sec \theta)\!</math>.
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| The curve is therefore a member of the [[Conchoid of de Sluze]] family.
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| In [[Cartesian coordinate]]s the equation this is
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| :<math>2x(x^2+y^2)=a(3x^2-y^2)\!</math>.
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| If the origin is moved to (''a'', 0) then a derivation similar to that given above shows that the equation of the curve in polar coordinates becomes
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| :<math>r = \frac{a}{2 \cos{\theta \over 3}}\!</math>
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| making it an example of an [[epispiral]].
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| ==The trisection property==
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| Given an angle <math>\phi</math>, draw a ray from <math>(a, 0)</math> whose angle with the <math>x</math>-axis is <math>\phi</math>. Draw a ray from the origin to the point where the first ray intersects the curve. Then, by the construction of the curve, the angle between the second ray and the <math>x</math>-axis is <math>\phi/ 3</math>
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| ==Notable points and features==
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| The curve has an [[root of a function|x-intercept]] at <math>3a \over 2</math> and a [[double point]] at the origin. The vertical line <math>x={-{a \over 2}}</math> is an asymptote. The curve intersects the line x = a, or the point corresponding to the trisection of a right angle, at <math>(a,{\pm {1 \over \sqrt{3}} a})</math>. As a nodal cubic, it is of [[Genus (mathematics)|genus]] zero.
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| ==Relationship to other curves==
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| The trisectrix of Maclaurin can be defined from [[conic sections]] in three ways. Specifically:
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| * It is the [[inverse curve|inverse]] with respect to the unit circle of the [[hyperbola]]
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| ::<math>2x=a(3x^2-y^2)</math>.
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| * It is [[cissoid]] of the circle
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| ::<math>(x+a)^2+y^2 = a^2</math>
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| :and the line <math>x={a \over 2}</math> relative to the origin.
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| * It is the [[Pedal curve|pedal]] with respect to the origin of the [[parabola]]
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| ::<math>y^2=2a(x-\tfrac{3}{2}a)</math>.
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| In addition:
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| * The inverse with respect to the point <math>(a, 0)</math> is the [[Limaçon trisectrix]].
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| * The trisectrix of Maclaurin is related to the [[Folium of Descartes]] by [[affine transformation]].
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| ==References==
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| * {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=36,95,104–106 }}
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| * {{MathWorld|title=Maclaurin Trisectrix|urlname=MaclaurinTrisectrix}}
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| * [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Trisectrix.html "Trisectrix of Maclaurin" at MacTutor's Famous Curves Index]
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| * [http://www.2dcurves.com/cubic/cubictr.html "Trisectrix of MacLaurin" on 2dcurves.com]
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| * [http://xahlee.org/SpecialPlaneCurves_dir/TriOfMaclaurin_dir/triOfMaclaurin.html "Trisectrix of Maclaurin" at Visual Dictionary Of Special Plane Curves]
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| * [http://www.mathcurve.com/courbes2d/maclaurin/maclaurin.shtml "Trisectrice de Maclaurin" at Encyclopédie des Formes Mathématiques Remarquables]
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| ==External links==
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| * [http://www.jimloy.com/geometry/trisect.htm#curves Loy, Jim "Trisection of an Angle", Part VI]
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| [[Category:Curves]]
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Let me first start by introducing myself. My name is Boyd Butts although it is not the name on my birth certification. My day job is a librarian. North Dakota is her beginning location but she will have to move one day or an additional. He is really fond of performing ceramics but he is having difficulties to find time for it.
Visit my website ... http://www.gaysphere.net/