Bow Curve

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Bow Curve

A bow curve is a Quartic Plane Curve with the equation:
:

x^4=x^2y-y^3 \,

The bow curve has a single triple point at x=0, y=0, and consequently is a rational curve, with Genus zero. Expanding around zero, the three branches have series

y_1 = x^2+x^4+3x^6+12x^8+55x^{10} \cdots

y_2 = x - rac{1}{2}x^2 + rac{3}{8}x^3 - rac{1}{2}x^4 + \cdots

y_3 = x - rac{1}{2}x^2 - rac{3}{8}x^3 - rac{1}{2}x^4 - \cdots

The nature of any singularity of a plane algebraic curve is defined by its link; the link for the singularity, an ordinary triple point, of the bow curve is illustrated below.