Bow curve

From Wikipedia, the free encyclopedia

A bow curve
A 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 - \frac{1}{2}x^2 + \frac{3}{8}x^3 - \frac{1}{2}x^4 + \cdots y_3 = x - \frac{1}{2}x^2 - \frac{3}{8}x^3 - \frac{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.

Singularity link of the bow curve
Singularity link of the bow curve