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y2 – x2(x + 1) = 0,
y2 – x3 – x2 = 0.
Like all cubics, the nodal curve is defined by a polynomial of degree 3.
y2 = 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 .
- The cubic is a member of the Folium class of curves under projective equivalence. However, it belongs to a separate class under affine equivalence .
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 
wxy + wxz + wyz + xyz = 0.
Cayley’s cubic image: Salix alba,
 Desmos.com. Nodal.
 Nodal and cuspidal curves. Retrieved March 3, 2022 from: https://www.math.purdue.edu/~arapura/graph/nodal.html
 Ding, A. Plane Algebraic Curves. Retrieved March 3, 2022 from: https://staff.math.su.se/shapiro/UIUC/DingPlaneCurves.pdf
 Holme, A. A Royal Road to Algebraic Geometry. Springer Berlin Heidelberg.
 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/
Stephanie Glen. "Nodal Cubic" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/nodal-cubic/
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