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A *nodal curve* is any curve with a node (i.e., a nodal singularity). The “usual” ℝ^{2} (real-valued) nodal cubic is defined by the equation [2]

** y^{2} – x^{2}(x + 1) = 0**,

or equivalently

** y^{2} – x^{3} – x^{2} = 0**.

Like all cubics, the nodal curve is defined by a polynomial of degree 3.

For complex-valued equations, the nodal cubic is a limit of elliptic curves.

*y*^{2} = *x*(*x *+ ε)(*x *– 1).

## Properties of the Nodal Cubic

- The usual form has one double point at the origin, where the curve crosses over itself.
- Tangents are at the node along the x- and y-axes.
- The curve is homeomorphic to a figure 8 (i.e., it has a one-to-one continuous mapping in both directions).
- It isn’t a union of other curves, so it is
*irreducible*(the polynomials that define the cubic are by definition, also irreducible). - Nodal cubics are
*nondegenerate*, meaning that they cannot be expressed as a finite union of conics, lines, and points [3]. - The cubic is a member of the Folium class of curves under projective equivalence. However, it belongs to a separate class under affine equivalence [4].

## Cayley’s Nodal Cubic

Cayley’s nodal cubic surface has four double points, which is the largest possible number of double points; every cubic with four double points is isomorphic to Cayley’s surface. The surface is defined by the equation [5]

*wxy *+ *wxz *+ *wyz *+ *xyz *= 0.

## References

Cayley’s cubic image: Salix alba,

[1] Desmos.com. Nodal.

[2] Nodal and cuspidal curves. Retrieved March 3, 2022 from: https://www.math.purdue.edu/~arapura/graph/nodal.html

[3] Ding, A. Plane Algebraic Curves. Retrieved March 3, 2022 from: https://staff.math.su.se/shapiro/UIUC/DingPlaneCurves.pdf

[4] Holme, A. A Royal Road to Algebraic Geometry. Springer Berlin Heidelberg.

[5] Baez, J. (2016). Cayley’s Nodal Cubic Surface. Retrieved March 3, 2022 from: https://blogs.ams.org/visualinsight/2016/08/15/cayleys-nodal-cubic-surface/

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