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Dual vs. polar reciprocal
The 'Classical construction' section actually describes the reciprocal polar which is closely related to but not the same as the dual. The dual is a curve in the dual plane, really the collection of lines that are tangent to the curve, while the reciprocal polar is a collection of points in the original plane. With respect to a given conic, a line can be made to correspond to a point via the pole-polar relationship, but this doesn't mean a line and a point are the same thing. Similarly, the pole-polar relationship defines a correspondence between the dual curve and the reciprocal polar, but that doesn't mean they are the same object. For example the reciprocal polar is the inverse of the pedal curve, but it would be somewhat nonsensical to make a similar statement about the dual curve. Also, there is only one dual for a curve but there are different (though projectively equivalent) reciprocal polars depending on the choice of conic. I'm thinking the material on the polar reciprocal should be moved, perhaps expanded into it's own article.--RDBury (talk) 04:48, 18 February 2012 (UTC)
- Agree here, I've removed the section. Do the articles Polar curve, Pole and polar fit with polar reciprocal.--Salix (talk): 10:14, 18 February 2012 (UTC)
Considering some recent edits, there seems to be some confusion about the use of homogeneous coordinates in this article. First of all, while it is not explicitly stated, this discussion is taking place in a projective plane (the use of homogeneous coordinates (p,q,r) for points says so). Every line in the plane has an equation of the form aX + bY + cZ = 0 which can be interpreted as the set of points with coordinates (X,Y,Z) that are orthogonal to the fixed point (a,b,c). However, the same linear equation can also be interpreted as saying that this line is completely determined by the triple (a,b,c) (up to a proportionality factor). Thus, (a,b,c) may also be thought of as homogeneous coordinates of a line (and called line coordinates when this is done). A completely parallel theory to the usual point coordinate theory can be developed using line coordinates. An easy way to describe a duality, via coordinates, is to map a point with point coordinates (a,b,c) to a line with line coordinates (a,b,c) and this is what is being done in the article. Notice that the curve is described by f(p,q,r) = 0, so p,q and r are variables and thus, for fixed X,Y,Z the equation Xp + Yq + Zr = 0 describes a line in point coordinates that has (X,Y,Z) as line coordinates. The equation remains algebraically valid whether you are considering it as a point coordinate or a line coordinate expression, which is why it is the correct expression in the article. This material can be found in most projective geometry books under the heading of line coordinates (in a plane). Bill Cherowitzo (talk) 18:40, 20 October 2014 (UTC)