simplest-oblique-cubic-forms

Simplest Oblique-Coordinate Equations for the Sixteen Cubic Species

Summary: §§258–259 give the simplest representative equation of each of the 16 species in oblique coordinates. The simplification uses the substitution $y=\nu u$, $x=t-\mu u$ with ${\mu }^2+{\nu }^2=1$, so the new ordinate angle is chosen (rather than fixed at perpendicular) to make as many coefficients vanish as possible. Coordinate change preserves species (§258): hyperbolic / parabolic branch character is invariant under any change of axis, origin, or obliquity, so each species has the same defining structure in oblique coordinates as in rectangular ones.

Sources: chapter10 (§§258–259).

Last updated: 2026-05-02.

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Species are coordinate-invariant (§258)

Although the equations which we have given above for each of species of third order lines were given in rectangular coordinates $x$ and $y$, the nature of the species is not changed if the coordinates are oblique. Whenever an equation in rectangular coordinates gives branches going to infinity, the same number of such branches remain if oblique coordinates are used. Nor is the nature of the branch going to infinity changed when coordinates are changed. A branch which is hyperbolic remains hyperbolic, and those which are parabolic remain parabolic. Nor does the species of either a hyperbolic or a parabolic branch change. (source: chapter10, §258)

The same statement extends degree-invariance (degree preserved under any coordinate change) and the §218 hyperbolic-and-parabolic-branches dichotomy to the species classification: every species is invariant under arbitrary affine change of coordinates. Therefore the simplest form for each species in oblique coordinates determines the same species as the rectangular form in simplest-cubic-equations.

The simplification (§259)

Substitute $y=\nu u$ and $x=t-\mu u$ with ${\mu }^2+{\nu }^2=1$, choosing the obliquity angle to absorb as many coefficients as possible. The resulting simplest forms in $(t,u)$ are:

Case 1 — single real linear factor in the principal cubic

FIRST species — $u(t^2+n^2{u}^2)+a{u}^2+bt+cu+d=0$, with $n\ne 0$ and $b\ne 0$.

SECOND species — $(t^2+n^2{u}^2)+a{u}^2+cu+d=0$, with $n\ne 0$. (As printed, this expression is degree 2 in $u,t$, not 3. The natural reading parallels the FIRST species form with $b=0$, namely $u(t^2+n^2{u}^2)+a{u}^2+cu+d=0$. The leading factor of $u$ may have dropped in printing.)

Case 2 — three distinct real linear factors

THIRD species — $u(t^2+n^2{u}^2)+a{u}^2+bt+cu+d=0$, with $n\ne 0$, $b\ne 0$, and $\pm nb+c+\frac{a^2}{4n^2}\ne 0$. (The "$+n^2{u}^2$" parallels the FIRST species form, but Case 2 requires three real factors of the principal cubic — i.e., $t^2-n^2{u}^2$ rather than $t^2+n^2{u}^2$, since the latter is irreducible. Likely the printed sign should be "$-$" to match the FOURTH and FIFTH forms below; a printing or OCR slip.)

FOURTH species — $u(t^2-n^2{u}^2)+a{u}^2+cu+d=0$, with $n\ne 0$ and $c+\frac{a^2}{4n^2}\ne 0$.

FIFTH species — $u(t^2-n^2{u}^2)+a{u}^2-\frac{a^{2}u}{4n^2}+d=0$, with $n\ne 0$.

Case 3 — double factor, third distinct

SIXTH species — $t{u}^2+a{t}^2+bt+cu+d=0$, with $a\ne 0$ and $c\ne 0$.

SEVENTH species — $t{u}^2+a{t}^2+bt+d=0$, with $a\ne 0$.

EIGHTH species — $t{u}^2+b^{2}t+cu+d=0$, with $b\ne 0$ and $c\ne 0$.

NINTH species — $t{u}^2+b^{2}t+d=0$, with $b\ne 0$.

TENTH species — $t{u}^2-b^{2}t+cu+d=0$, with $b\ne 0$ and $c\ne 0$.

ELEVENTH species — $t{u}^2-b^{2}t+d=0$, with $b\ne 0$.

TWELFTH species — $t{u}^2+cu+d=0$, with $c\ne 0$.

THIRTEENTH species — $t{u}^2+d=0$.

Case 4 — triple factor

FOURTEENTH species — $u^3+a{t}^2+cu+d=0$.

FIFTEENTH species — $u^3+atu+bt+d=0$, with $a\ne 0$.

SIXTEENTH species — $u^3+at=0$.

Reading the table

The grouping by case echoes cubic-species-classification: Cases 1–4 correspond respectively to the principal cubic having an irreducible quadratic factor + one real linear factor (Case 1, two species), three distinct real linear factors (Case 2, three species), one double + one simple real factor (Case 3, eight species), or one triple real factor (Case 4, three species).

The simplification uniformly removes any term that can be absorbed by translation along the new axes, plus any cross-term that can be absorbed by the obliquity choice $(\mu ,\nu )$. What remains is the minimal polynomial signature of the species.

Comparison with the rectangular forms

simplest-cubic-equations gives the rectangular-coordinate simplest forms (§237) — those agree term-by-term with Newton’s classical species. The §259 oblique forms are usually shorter (fewer terms), since the obliquity angle adds an extra parameter to consume.

Related pages

• chapter-10-on-the-principal-properties-of-third-order-lines
• simplest-cubic-equations — rectangular-coordinate simplest forms (§237).
• cubic-species-classification — the four-case enumeration that organizes the table here.
• hyperbolic-and-parabolic-branches — §218 species catalogue underwriting the §258 invariance claim.
• degree-invariance — analogous coordinate-invariance result for the order of a curve.
• oblique-coordinates — the substitution $y=\nu u$, $x=t-\mu u$ used here.