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CU-1: Notes on the General Cubic

General Approach to Cubics

Many papers have been written about cubic curves, often enriched with examples involving triangles, quadrilaterals, special points, and other geometric configurations.But what are the intrinsic geometric properties of a cubic when viewed purely as a third-degree polynomial curve?

In 2023, Eckart Schmidt, Bernard Keizer, and Chris van Tienhoven (the author of this Encyclopedia) began a renewed exploration into the fundamental nature of cubics. Having collaborated for many years on the geometry of triangles, quadri-figures, and n-figures, we encountered numerous remarkable properties of cubic curves. Eventually, we decided it was time to focus exclusively on the cubic itself—stripped of external references—to uncover its core characteristics as a third-degree polynomial curve.

The central question became:What are the general properties of the general cubic?

Historical Context

Many well-known books and websites are written about cubics. Here are a few examples:

  • George Salmon, Arthur Cayley. A Treatise on the Higher Plane Curves: Intended as a sequel to A Treatise on Conic Sections.
  • George Samon. TraitĂ© de gĂ©omĂštrie analytique (courbes planes). ISBN: 1-4212-0859-8
  • Julian Lowell Coolidge, A treatise on Algebraic Curves. ISBN: 0-486-49576-0
  • Heinrich Edward Schroeter, Die Theorie de ebenen Kurven dritter Ordnung. ISBN: 9783743307506.
  • Bernard Gibert made a tremendous contribution by cataloging many types of cubics, most often using a triangle as reference. See [17].
  • Roger Cuppens, Faire de la GĂ©omĂštrie supĂ©rieure en jouant avec Cabri-GĂ©omĂštre II. Tome II. APMEP Brochure no 125, ISBN: 2-912846-38-2. See [63].

Undoubtedly, I’ve omitted many names of researchers who have made significant contributions in the ongoing quest to understand the cubic, each from their own unique perspective.

A Different Approach

In the Encyclopedia of Polygon Geometry, we considered a cubic as a curve defined by nine points—or seven, in the case of a circular cubic.

Let us now define a cubic simply as a third-degree polynomial curve, abbreviated as CU.

Let 9P represent a set of nine distinct points in the complex projective plane.

When comparing the cardinality of the set of all 9P’s to the set of all CU’s, we observe that: |9P| > |CU|,

where |X| denotes the cardinality (i.e., the number of elements) in set X.

This observation reveals two key facts:

  • Given a 9P, there exists a unique cubic CU passing through it.
  • Given a CU, there are many possible 9P’s lying on it.

This implies that a cubic cannot be uniquely described by a single 9P.Therefore, neither the BG-style nor the QPG-style fully captures the essence of a general cubic.In our search for general properties, we must remain aware of the cardinality relationships between cubics and sets like |3P|, |4P|, |7P|, |9P|, etc.

Cardinality in Transformations

We encounter similar issues with the Scimemi Transformation CO–Tf3.

Originally introduced by Benedetto Scimemi as a 5P-transformation, it was later recognized as a conical transformation. It is now described as CO–Tf3, with properties valid for any set of five points on a reference conic.

Another example of cardinality’s importance is the relationship between Quadrangles (QA), Quadrilaterals (QL), and Quadrigons (QG):  |QA| < |QG| and |QL| < |QG|.

One cannot properly describe quadrilaterals without accounting for these cardinality rules.

Complex Projective Plane

In EPG, we work within the Complex Projective Plane, where we distinguish between finite and infinite points, as well as between real and imaginary points.

Main Results

Some key findings about general cubics include:

  • A cubic has at least 1 real Asymptote (out of 3).
  • A cubic has at least 2 real Anallagmatic Points (out of 12).
  • A cubic has at least 1 real Miquel point (out of 9).
  • A cubic has at least 1 real Möbius transformation (out of 9).
  • A cubic has at least 1 real Central Conic (QF-conic) (out of 3).
  • A cubic has at least 1 real Diametral Conic (out of 3).
  • A cubic has 3 real Flexpoints (out of 9) and 3 real Harmonic Polars (out of 9.
  • A cubic has 4 real Flexlines (out of 12) and 4 real Harmonic Polar Crosspoints (out of 12).

All of these elements can be constructed for any cubic.This list is not exhaustive.

Note

The concepts of Anallagmatic points, Miquel point, and Möbius transformation are treated here as quasi-versions of their classical definitions, meaning that there is a new ‘extended’ definition, implied by the cubic environment. For clarity, we refer to them as such when associated with general cubics.

The specific meaning of each quasi-version is explained in detail within the Encyclopedia.



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