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If {{mvar|X}} is smooth (no [[Singular point of a curve|singular points]]) then the dual of {{mvar|X}} has maximum degree {{math|''d''(''d'' − 1)}}. This implies the dual of a conic is also a conic. Geometrically, the map from a conic to its dual is [[one-to-one correspondence|one-to-one]] (since no line is tangent to two points of a conic, as that requires degree 4), and the tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require an inflection point in the original curve, which requires degree 3).
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 [[Plücker formula]]s give the degree of the dual
==Polar reciprocal==
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