Laguerre–Forsyth Invariant¶

A cubic differential extracted from the normalized third-order equation of a nondegenerate projective plane curve.

Version: v1 · 2026-10-31 · History
Domain-specific #: 14527
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Projective Differential Geometry → Mathematics
Aliases: Laguerre-Forsyth invariant

Core Idea¶

The Laguerre–Forsyth invariant is a cubic differential calculated from a nondegenerate local projective plane curve.^[1] Write the curve's normalized coordinate equation as \(z'''+3P_2z'+P_3z=0\).^[1] The invariant is \(P\,dt^3\), with \(P=P_3-\tfrac32P_2'\).^[1] The differential is the named object; \(P\) alone is its coefficient in the chosen parameter.^[1] A quadratic curve gives a vanishing case.^[1]

How would you explain it like I'm…

No faithful explanation at this level. All three agree a child-level picture turns the invariant into a single curviness score attached to a curve, whereas it is a cubic differential, not an isolated number.

Perspective-Proof Curve Measure

Mathematicians study curves in a kind of geometry where you're allowed to look at shapes through any slanted window, like how a shadow can stretch a circle into an oval. The Laguerre–Forsyth invariant is a measurement that runs along a curve, bit by bit, and stays meaningful no matter how you trace the curve. You get it from a special equation that the curve's position numbers obey. For curves like circles and ovals, the measurement is zero everywhere, while some spirals give something that isn't zero.

Projective Cubic Differential

In projective geometry, the Laguerre–Forsyth invariant attaches a quantity to a nondegenerate curve in the projective plane. You write the curve's homogeneous coordinates as solutions of a normalized third-order differential equation, z''' + 3P₂z' + P₃z = 0. From the coefficients you compute P = P₃ − (3/2)P₂', and the invariant is P dt³, a 'cubic differential'. The important point is that it's this differential, not P alone written in one particular parameter, that stays invariant. The same recipe works for any admissible curve. For a conic (quadratic curve), P is zero; for curves like non-quadratic logarithmic spirals or exponential graphs, it can be nonzero.

The Laguerre–Forsyth invariant assigns to a nondegenerate local curve in the projective plane a cubic differential. The homogeneous coordinates of the curve satisfy a third-order linear ODE, which can be normalized to the form z''' + 3P₂z' + P₃z = 0. From the coefficients one forms P = P₃ − (3/2)P₂', and the invariant is the differential P dt³. It is the differential, not the coefficient P expressed in one parameter, that the primary source identifies as invariant. The construction is a general recipe: choose an admissible local curve, find its normalized coordinate equation, and apply the same combination of coefficients. For a quadratic curve P vanishes, giving a boundary case, while nonquadratic logarithmic spirals and exponential graphs are settings where the result can be nonzero.

Scope of Application¶

This construction concerns local projective plane curves whose homogeneous-coordinate triple is nondegenerate.^[1] The source lists logarithmic spirals and exponential graphs among distinct homogeneous curve families on which it can be evaluated.^[1] A local value does not establish a global count of zeros or a theorem about a closed curve.

Clarity¶

Keep three stages apart: the curve is the input, its third-order equation is a way to compute, and \(P\,dt^3\) is the output. The correction \(-\tfrac32P_2'\) matters; \(P_3\) by itself is not the stated formula.^[1] Nondegeneracy and normalization must be checked before interpreting a coefficient.

Manages Complexity¶

Different coordinates can make the same curve look different. The normalized equation and cubic-differential formula give a repeatable test without comparing raw coordinate lists.^[1] Report the parameter and local domain with any calculated coefficient; a zero of the coordinate Wronskian can invalidate the displayed normalization.

Abstract Reasoning¶

For a proposed curve, calculate the coordinate Wronskian, obtain the common third-order equation, remove its second-derivative term, and read off \(P_2\) and \(P_3\). Then compute \(P_3-\tfrac32P_2'\) and report the cubic differential.^[1] A nonzero value proves nonvanishing near that point; it does not by itself settle the curve's global behavior.

Knowledge Transfer¶

The same calculation applies to different admissible projective plane curves. The paper lists both a logarithmic spiral and \(Y=Xe^X\) as curve families.^[1] Their displayed parameters lead to different equations, but the roles of curve, normalization and cubic output stay the same. Another kind of differential invariant needs its own formula; a resemblance in name is insufficient.

Relations¶

This construction is a strict case of Differential Invariant: its coefficient has a finite-jet relative-invariant expression of weight three under the prolonged projective action, while the cubic differential is invariant.^[2] It separately presupposes a Curve as its carrier.^[1] The isolated official graph check confirmed both relations without a cycle.

Examples¶

Logarithmic spiral. For \([1,e^t\cos t,e^t\sin t]\), direct calculation gives nonzero Wronskian \(2e^{2t}\).^[1] The normalized equation has \(P_2=2/9\) and \(P_3=20/27\), so \(P=20/27\).^[1] Mapped back: curve → independent coordinates → normalized coefficients → nonzero cubic differential.^[1] These values are derived from the paper's formula and listed spiral family, not quoted from the paper.

Exponential graph. For \([1,t,te^t]\) near \(t=0\), the Wronskian is \((t+2)e^t\) and the same construction gives \(P(0)=-5/12\).^[1] Mapped back: a different curve family → locally independent coordinates → normalized coefficients → another nonzero cubic differential.^[1] The source lists \(Y=Xe^X\); the numeric value is derived here.^[1]

The conic \([1,\cos t,\sin t]\) instead gives \(P=0\), matching the source's quadratic boundary.^[1]

Relationships to Other Abstractions¶

Current abstraction Laguerre–Forsyth Invariant Domain-specific

Parents (2) — outgoing typed relations

Laguerre–Forsyth Invariant is a kind of Differential invariant Domain-specific

The cubic form has a weight-three relative finite-jet representative under the projective group action.
Laguerre–Forsyth Invariant presupposes Curve Domain-specific

A nondegenerate local projective plane curve is required to supply the invariant's coordinate equation.

Hierarchy paths (3) — routes to 2 parentless roots

Laguerre–Forsyth Invariant → is a kind of Differential invariant → is a kind of Invariance

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Laguerre–Forsyth Invariant sits in a sparse region of the domain-specific corpus (90^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2835 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

Not to Be Confused With¶

The curve itself: it supplies the input, while the invariant is a derived differential.
An osculating curve: that construction selects an approximating curve by local contact; this one computes a coefficient combination.
A global zero count or sextactic theorem: those claims need separate hypotheses and sources; this entry establishes only the stated local construction and examples.
The frozen alternate coefficient formula: its equivalence to the primary paper's presentation has not been verified.

References¶

[1] Franki Dillen, Takeshi Sasaki, and Luc Vrancken, “The Classification of Projectively Homogeneous Surfaces II,” Osaka Journal of Mathematics 35, no. 1 (1998): 117–146, Appendix §1, 143–144, equations (a.1)–(a.3), curve families [P1]–[P3], and Proposition A1. https://doi.org/10.18910/11131. Open published PDF. The numerical examples are independent symbolic calculations from the paper's formula and listed families. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s

[2] Boris Doubrov and Maciej Dunajski, “Co-calibrated G2 Structure from Cuspidal Cubics,” Annals of Global Analysis and Geometry 42 (2012): 247–265, §5, equations (5.21)–(5.23) and footnote 3. https://doi.org/10.1007/s10455-012-9310-8. Published author copy. registry ↩