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| {{About|a concept in algebraic geometry|concept in aviation that goes by that name|Polar curve (aviation)|curves given in polar coordinates|Polar coordinate system#Polar equation of a curve}}
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| [[Image:PolarEllipticCurvePlot.svg|thumb|300px|right|The [[elliptic curve]] ''E'' : 4''Y''<sup>2</sup>Z = ''X''<sup>3</sup> − ''XZ''<sup>2</sup> in blue, and its polar curve (''E'') : 4''Y''<sup>2</sup> = 2.7''X''<sup>2</sup> − 2''XZ'' − 0.9Z<sup>2</sup> for the point ''Q'' = (0.9, 0) in red. The black lines show the tangents to ''E'' at the intersection points of ''E'' and its first polar with respect to ''Q'' meeting at ''Q''.]]
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| In [[algebraic geometry]], the '''first polar''', or simply '''polar''' of an [[algebraic plane curve]] ''C'' of degree ''n'' with respect to a point ''Q'' is an algebraic curve of degree ''n''−1 which contains every point of ''C'' whose tangent line passes through ''Q''. It is used to investigate the relationship between the curve and its [[Dual curve|dual]], for example in the derivation of the [[Plücker formula]]s.
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| ==Definition==
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| Let ''C'' be defined in [[homogeneous coordinates]] by ''f''(''x, y, z'') = 0 where ''f'' is a [[homogeneous polynomial]] of degree ''n'', and let the homogeneous coordinates of ''Q'' be (''a'', ''b'', ''c''). Define the operator
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| :<math>\Delta_Q = a{\partial\over\partial x}+b{\partial\over\partial y}+c{\partial\over\partial z}.</math>
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| Then Δ<sub>''Q''</sub>''f'' is a homogeneous polynomial of degree ''n''−1 and Δ<sub>''Q''</sub>''f''(''x, y, z'') = 0 defines a curve of degree ''n''−1 called the ''first polar'' of ''C'' with respect of ''Q''.
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| If ''P''=(''p'', ''q'', ''r'') is a [[Singular point of an algebraic variety|non-singular point]] on the curve ''C'' then the equation of the tangent at ''P'' is
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| :<math>x{\partial f\over\partial x}(p, q, r)+y{\partial f\over\partial y}(p, q, r)+z{\partial f\over\partial z}(p, q, r)=0.</math>
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| In particular, ''P'' is on the intersection of ''C'' and its first polar with respect to ''Q'' if and only if ''Q'' is on the tangent to ''C'' at ''P''. Note also that for a double point of ''C'', the partial derivatives of ''f'' are all 0 so the first polar contains these points as well.
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| ==Class of a curve==
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| The ''class'' of ''C'' may be defined as the number of tangents that may be drawn to ''C'' from a point not on ''C'' (counting multiplicities and including imaginary tangents). Each of these tangents touches ''C'' at one of the points of intersection of ''C'' and the first polar, and by [[Bézout's theorem]] theorem there are at most ''n''(''n''−1) of these. This puts an upper bound of ''n''(''n''−1) on the class of a curve of degree ''n''. The class may be computed exactly by counting the number and type of singular points on ''C'' (see [[Plücker formula]]).
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| ==Higher polars==
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| The ''p-th'' polar of a ''C'' for an natural number ''p'' is defined as Δ<sub>''Q''</sub><sup>''p''</sup>''f''(''x, y, z'') = 0. This is a curve of degree ''n''−''p''. When ''p'' is ''n''−1 the ''p''-th polar is a line called the ''polar line'' of ''C'' with respect to ''Q''. Similarly, when ''p'' is ''n''−2 the curve is called the ''polar conic'' of ''C''.
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| Using [[Taylor series]] in several variables and exploiting homogeneity, ''f''(λ''a''+μ''p'', λ''b''+μ''q'', λ''c''+μ''r'') can be expanded in two ways as
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| :<math>\mu^nf(p, q, r) + \lambda\mu^{n-1}\Delta_Q f(p, q, r) + \frac{1}{2}\lambda^2\mu^{n-2}\Delta_Q^2 f(p, q, r)+\dots</math> | |
| and
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| :<math>\lambda^nf(a, b, c) + \mu\lambda^{n-1}\Delta_P f(a, b, c) + \frac{1}{2}\mu^2\lambda^{n-2}\Delta_P^2 f(a, b, c)+\dots .</math>
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| Comparing coefficients of λ<sup>''p''</sup>μ<sup>''n''−''p''</sup> shows that
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| :<math>\frac{1}{p!}\Delta_Q^p f(p, q, r)=\frac{1}{(n-p)!}\Delta_P^{n-p} f(a, b, c).</math>
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| In particular, the ''p''-th polar of ''C'' with respect to ''Q'' is the locus of points ''P'' so that the (''n''−''p'')-th polar of ''C'' with respect to ''P'' passes through ''Q''.<ref>Follows Salmon pp. 49-50 but essentially the same argument with different notation is given in Basset pp. 16-17.</ref>
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| ==Poles==
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| If the polar line of ''C'' with respect to a point ''Q'' is a line ''L'', the ''Q'' is said to be a ''pole'' of ''L''. A given line has (''n''−1)<sup>2</sup> poles (counting multiplicities etc.) where ''n'' is the degree of ''C''. So see this, pick two points ''P'' and ''Q'' on ''L''. The locus of points whose polar lines pass through ''P'' is the first polar of ''P'' and this is a curve of degree ''n''−''1''. Similarly, the locus of points whose polar lines pass through ''Q'' is the first polar of ''Q'' and this is also a curve of degree ''n''−''1''. The polar line of a point is ''L'' if and only if it contains both ''P'' and ''Q'', so the poles of ''L'' are exactly the points of intersection of the two first polars. By Bézout's theorem these curves have (''n''−1)<sup>2</sup> points of intersection and these are the poles of ''L''.<ref>Basset p. 20, Salmon p. 51</ref>
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| ==The Hessian==
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| For a given point ''Q''=(''a'', ''b'', ''c''), the polar conic is the locus of points ''P'' so that ''Q'' is on the second polar of ''P''. In other words the equation of the polar conic is
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| :<math>\Delta_{(x, y, z)}^2 f(a, b, c)=x^2{\partial^2 f\over\partial x^2}(a, b, c)+2xy{\partial^2 f\over\partial x\partial y}(a, b, c)+\dots=0.</math>
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| The conic is degenerate if and only if the determinant of the [[Hessian matrix|Hessian]] of ''f'',
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| :<math>H(f) = \begin{bmatrix}
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| \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x\,\partial y} & \frac{\partial^2 f}{\partial x\,\partial z} \\ \\
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| \frac{\partial^2 f}{\partial y\,\partial x} & \frac{\partial^2 f}{\partial y^2} & \frac{\partial^2 f}{\partial y\,\partial z} \\ \\
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| \frac{\partial^2 f}{\partial z\,\partial x} & \frac{\partial^2 f}{\partial z\,\partial y} & \frac{\partial^2 f}{\partial z^2}
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| \end{bmatrix},</math>
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| vanishes. Therefore the equation |''H''(''f'')|=0 defines a curve, the locus of points whose polar conics are degenerate, of degree 3(''n''−''2'') called the ''Hessian curve'' of ''C''.
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| == See also ==
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| * [[Polar hypersurface]]
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| * [[Pole and polar]]
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| ==References==
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| {{reflist}}
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| *{{cite book |title=An Elementary Treatise on Cubic and Quartic Curves
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| |first=Alfred Barnard |last=Basset|publisher=Deighton Bell & Co.|year=1901|pages=16ff.
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| |url=http://books.google.com/books?id=yUxtAAAAMAAJ}}
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| *{{cite book |title=Higher Plane Curves
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| |first=George|last=Salmon|publisher=Hodges, Foster, and Figgis|year=1879|pages=49ff.
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| |url=http://www.archive.org/details/treatiseonhigher00salmuoft
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| |authorlink=George Salmon}}
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| *Section 1.2 of Fulton, ''Introduction to intersection theory in algebraic geometry'', CBMS, AMS, 1984.
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| *{{springer|title=Polar|id=P/p073400|last=Ivanov|first=A.B.}}
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| *{{springer|title=Hessian (algebraic curve)|id=H/h047150|last=Ivanov|first=A.B.}}
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The name of the writer is Jayson. Mississippi is where her house is but her spouse desires them to transfer. The preferred hobby for him and his kids is style and he'll be beginning some thing else alongside with it. For years she's been working as a travel agent.
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