Geometric interpretation of cubic curve?

Question

Lines and conics have clear geometric meanings that are coordinate-free, but cubics seem to rely entirely on cubic equations and coordinate systems. Are there ways to define cubic curves without cubic equations?

I thought about this question by trying to generalize tangent lines and osculating circles. Since a tangent line is defined by taking the limit of two points on the curve, and an osculating circle is defined by taking the limit of two tangent lines on the curve, I thought about taking the limit of two osculating circles on the curve, but after playing around with evolutes and involutes, I couldn't come up with anything.

Answer 1

I'm not entirely clear on what's meant by the phrase

a clear geometric meaning that is coordinate free

Whether or not a curve is a line is not intrinsic to that curve; it depends on the embedding, and hence on the coordinate system. In the context of algebraic geometry, lines and circles are isomorphic to one another, and you can't talk about a circle as being "the set of all points $r$ units from some point $P$" because there's no such thing as distance.

That said, given that lines satisfy your criteria, you should also agree that cubic curves do, because a plane cubic is simply a plane curve that intersects all but finitely many lines in the plane in exactly three points.

Answer 2

A similar question has been asked and answered on Mathoverflow:

It is a "classical" fact that any nonsingular plane cubic curve $C$ can be projectively generated by means of a pencil of lines and a pencil of conics.

The starting point of the construction is the observation that the lines defined by any $g_2^1$ on $C$ all pass through the same point $p$ of $C$, that following Sylvester is called the coresidual point.

Now, take four points $q_1, \ldots, q_4$ on $C$, such that any three of them are not collinear. Let $p \in C$ be the coresidual point with respect to the $g_2^1$ on $C$ cut by the pencils of conics through $q_1, \ldots, q_4$. Therefore such a pencil of conics and the pencil of lines through $p$ projectively generate $C$.

All of this is explained in the paper by N. Fraser Kötter's synthetic geometry of algebraic curves, Proceedings of the Edinburgh Mathematical Society 7, 46–61 (1888). However, the language is rather old-fashioned so the article is not easy to read nowadays.

A modern treatment can be found in Dolgachev's book Classical Algebraic Geometry, Section 3.3 (this is the googlebook link).

See also Bunch, On the Straight Line Construction of Unicursal Cubics. Here the construction uses a 1-2 correspondence between two pencils of lines.

And for something more down to earth: Matheq, A General Construction for Circular Cubics