spinodes

A spinode is a type of singular point that can occur on a plane algebraic curve. It is characterized by the curve crossing itself at the point in a way that the two branches have the same tangent line. This is a more specific condition than a simple node, where the branches have distinct tangent lines.

Mathematically, a spinode is a singular point where the Hessian matrix of the defining polynomial of the curve has a specific degeneracy. For a curve defined by an equation $f(x, y) = 0$, the singular points are where $\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} = 0$. At a spinode, these partial derivatives are zero, and additionally, the second partial derivatives satisfy certain conditions. Specifically, if $P = (x_0, y_0)$ is a singular point, and $f(x, y) = \sum_{i+j \ge 2} a_{ij} (x-x_0)^i (y-y_0)^j$ is the Taylor expansion of $f$ around $P$, then for a spinode, the quadratic part of this expansion, $Q(x, y) = \sum_{i+j=2} a_{ij} (x-x_0)^i (y-y_0)^j$, is a perfect square of a linear form. This means $Q(x, y) = (ax + by)^2$ for some constants $a$ and $b$.

Visually, a spinode appears as a point where the curve is tangent to itself. Unlike a crunode (where the curve crosses itself with distinct tangents) or an acnode (an isolated singular point), a spinode has a distinct "self-tangency" appearance. An example of a curve with a spinode is the semicubic parabola, $y^2 = x^3$, which has a spinode at the origin (0,0). The tangent line at this spinode is the x-axis.