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→Equations: Explain dual of parametric curve Tags: nowiki added Visual edit |
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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.
If {{mvar|X}} is smooth
For curves with singular points, these points will also lie on the intersection of the curve and its polar and this reduces the number of possible tangent lines. The degree of the dual given in terms of
==Polar reciprocal==
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{{Main article|Dual polygon}}
The dual curve construction works even if the curve is [[piecewise linear curve|piecewise linear]]
In the case of a polygon, all points on each edge share the same tangent line, and thus map to the same vertex of the dual, while the tangent line of a vertex is ill-defined, and can be interpreted as all the lines passing through it with angle between the two edges. This accords both with projective duality (lines map to points, and points to lines), and with the limit of smooth curves with no linear component: as a curve flattens to an edge, its tangent lines map to closer and closer points; as a curve sharpens to a vertex, its tangent lines spread further apart.
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