Content deleted Content added
→Properties of dual curve: Move section |
|||
Line 35:
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.
==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 node in the center, and two 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 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 symmetrties 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 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)}}.
Line 48 ⟶ 54:
| publisher= MacMillan | location=London | pages=[https://archive.org/details/in.ernet.dli.2015.109607/page/n194 176]| year=1892
|url=https://archive.org/details/in.ernet.dli.2015.109607}}</ref> The conic {{mvar|Q}} is nearly always taken to be a circle, so the polar reciprocal is the [[Inverse curve|inverse]] of the [[Pedal curve|pedal]] of {{mvar|C}}.
▲==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 node in the center, and two 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 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 symmetrties 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.
==Generalizations==
| |||