Content deleted Content added
short description (?) Tags: Mobile edit Mobile app edit iOS app edit |
|||
| (6 intermediate revisions by 4 users not shown) | |||
Line 7:
==Equations==
Let {{math|''f''(''x'', ''y'', ''z'') {{=}} 0}} be the equation of a curve in [[homogeneous coordinates]] on the [[projective plane]]. Let {{math|''Xx'' + ''Yy'' + ''Zz'' {{=}} 0}} be the equation of a line, with {{math|(''X'', ''Y'', ''Z'')}} being designated its [[line coordinates]]
At a point {{math|(''p'', ''q'', ''r'')}} on the curve, the tangent is given by
:<math>x\frac{\partial f}{\partial x}(p, q, r)+y\frac{\partial f}{\partial y}(p, q, r)+z\frac{\partial f}{\partial z}(p, q, r)=0.</math>
So {{math|''Xx'' + ''Yy'' + ''Zz'' {{=}} 0}} is a tangent to the curve if
Line 21:
[[File:Dual.webm|thumb|thumbtime=0.5|458px|On the left: the ellipse {{math|({{sfrac|''x''|2}}){{su|p=2}} + ({{sfrac|''y''|3}}){{su|p=2}} {{=}} 1}} with tangent lines {{math|''xX'' + ''yY'' {{=}} 1}} for any {{mvar|X}}, {{mvar|Y}}, such that {{math|(2''X'')<sup>2</sup> + (3''Y'')<sup>2</sup> {{=}} 1}}.<br>On the right: the dual ellipse {{math|(2''X'')<sup>2</sup> + (3''Y'')<sup>2</sup> {{=}} 1}}. Each tangent to the first ellipse corresponds to a point on the second one (marked with the same color).]]
===Conic===
For example, let {{mvar|C}} be the [[Conic section|conic]] {{math|''ax''<sup>2</sup> + ''by''<sup>2</sup> + ''cz''<sup>2</sup> {{=}} 0}}. The dual is found by eliminating {{mvar|p}}, {{mvar|q}}, {{mvar|r}}, and {{mvar|λ}} from the equations
:<math>\begin{array}{c}
Line 31 ⟶ 32:
:<math>\frac{X^2}{a}+\frac{Y^2}{b}+\frac{Z^2}{c}=0.</math>
===General algebraic curve===
Consider a [[parametric equation|parametrically defined curve]] <math>(x,y) = (x(t),y(t)),</math> in projective coordinates <math>(x,y,z)=(x(t),y(t),1)</math>. Its projective tangent line is a linear plane spanned by the point of tangency and the tangent vector, with linear equation coefficients given by the [[cross product]]:<blockquote><math>(X,Y,Z) = (x,y,1)\times (x',y',0)
= (-y',x',xy'-yx'),</math></blockquote>which in affine coordinates <math>(X,Y,1)</math> is:
Line 37 ⟶ 39:
The dual of an [[inflection point]] will give a [[Cusp (singularity)|cusp]] and two points sharing the same tangent line will give a self-intersection point on the dual.
===Dual of the dual===
From the projective description, one may compute the dual of the dual:<blockquote><math>(x(x'y''-y'x''),\, y(x'y''-y'x''),\, x'y''-y'x'') = (x, \, y,\, 1)(x'y''-y'x''),</math></blockquote>which is projectively equivalent to the original curve <math>(x(t),y(t),1)</math>.
==Properties of dual curve==
Properties of the original curve correspond to dual properties on the dual curve. In the Introduction image, the red curve has three singularities – a [[Crunode|node]] in the center, and two [[Cusp (singularity)|cusps]] at the lower right and lower left. The black curve has no singularities but has four distinguished points: the two top-most points correspond to the node (double point), as they both have the same tangent line, hence map to the same point in the dual curve, while the two [[inflection point|inflection points]] correspond to the cusps, since the tangent lines first go one way then the other (slope increasing, then decreasing).
By contrast, on a smooth, convex curve the angle of the tangent line changes monotonically, and the resulting dual curve is also smooth and convex.
Further, both curves above have a reflectional symmetry: projective duality preserves symmetries a projective space, so dual curves have the same symmetry group. In this case both symmetries are realized as a left-right reflection; this is an artifact of how the space and the dual space have been identified – in general these are symmetries of different spaces.
==Degree==
If {{mvar|X}} is a plane algebraic curve, then the degree of the dual is the number of points in the intersection with a line in the dual plane. Since a line in the dual plane corresponds to a point in the plane, the degree of the dual is the number of tangents to the {{mvar|X}} that can be drawn through a given point. The points where these tangents touch the curve are the points of intersection between the curve and the [[polar curve]] with respect to the given point. If the degree of the curve is {{mvar|d}} then the degree of the polar is {{math|''d'' − 1}} and so the number of tangents that can be drawn through the given point is at most {{math|''d''(''d'' − 1)}}.
The dual of a line (a curve of degree 1) is an exception to this and is taken to be a point in the dual space (namely the original line). The dual of a single point is taken to be the collection of lines though the point; this forms a line in the dual space which corresponds to the original point.
| |||