Absolute First-Order Joint Projective Differential Invariants

• Absolute first-order joint projective differential invariants are rational functions that remain unchanged under PGL(3,R) actions, providing robust descriptors for point and derivative configurations.
• They are constructed using a moving frame approach that normalizes point configurations and ensures algebraic independence through closed-form generating sets.
• These invariants underpin applications in geometry, image analysis, and invariant theory by offering stable, efficient descriptors even when higher-order derivatives are unstable or unavailable.

Absolute first-order joint projective differential invariants are rational invariants under the action of the projective group $PGL(3,\mathbb{R})$ on configurations of $n$ points in the plane, together with first derivatives, that transform with weight zero under the prolonged group action. These invariants play a central role in the algebraic and geometric analysis of shapes, fields, and features that are preserved under projective transformations, especially in contexts where higher-order derivatives are either unstable or unavailable. The field of absolute first-order joint projective invariants supports the construction of robust, closed-form projectively invariant descriptors and integrals, foundational to applications in geometry, image analysis, and invariant theory (Bedratyuk, 17 Jul 2025, Bedratyuk, 23 Nov 2025).

1. Group Actions and the Definition of Joint Projective Invariants

For $n$ distinct points in the real projective plane, each given in affine coordinates as $(x_i,y_i)$ with first derivatives $p_i = \partial u/\partial x|_{A_i}$, $q_i = \partial u/\partial y|_{A_i}$, the diagonal action of $T = [G] \in PGL(3,\mathbb{R})$ sends each $A_i$ to $G A_i$ in homogeneous coordinates. The associated transformation of first derivatives follows by the chain rule.

A first-order joint projective invariant is a rational function $F(\{x_i, y_i, p_i, q_i\})$ such that

$$F(T(x_i), T(y_i), \widetilde{p}_i, \widetilde{q}_i) = J(T)^{\omega} F(x_i, y_i, p_i, q_i)$$

holds for all $T$, with $\omega \in \mathbb{Q}$ and $J(T)$ the total Jacobian multiplier. Absolute invariants correspond to $\omega = 0$ (unchanged by the group), while relative invariants have $\omega \neq 0$, picking up a scalar factor (Bedratyuk, 17 Jul 2025, Bedratyuk, 23 Nov 2025).

2. Construction of the Generating Set and Algebraic Independence

For $n \geq 3$, the field $\mathcal{I}_{n,0}$ of absolute invariants has transcendence degree $4n-8$, matching the dimension $4n$ of the space of points and jets minus the 8 dimensions of $PGL(3,\mathbb{R})$.

An explicit, algebraically independent generating set is constructed by first normalizing three base points (affine frame) and their tangent data, and then expressing the remaining $4n-8$ degrees of freedom in terms of basic invariants:

• For each pair/triple, determinant expressions $\zeta_{ij}^{(k)} = \det(\ell_i, \ell_j, L_k)$ where $\ell_i = (p_i, q_i, -p_i x_i - q_i y_i)$ encodes the directional derivative at $A_i$, and $L_{ij} = A_i \times A_j$ is the line through $A_i$ and $A_j$.
• For $n=3$, the minimal basis is $(\zeta_{12}, \zeta_{23}, \zeta_{13}, \tau)$, with $\tau$ involving triple determinants.
• For $n=4$, one includes $(\zeta_{12}, \ldots, \zeta_{34}, \tau, \sigma)$, with $\sigma$ a higher degree invariant in the gradients and positions.
• For $n\geq 7$, blocks of invariants $Z_k$ are defined for indices $k = 2, \ldots, n$, combined into a complete generating family $\mathcal{G}_n = \bigsqcup_{k=2}^n Z_k$ (Bedratyuk, 17 Jul 2025).

Algebraic independence is established by choosing a normal form for the first three points and showing the Jacobian matrix of the $4n-8$ generators with respect to the remaining variables is generically invertible.

3. Structure of the Field of Relative and Absolute Invariants

The field of all relative first-order joint projective invariants, $\mathcal{I}_n$, is a simple (in fact, rational) algebraic extension of the absolute field $\mathcal{I}_{n,0}$. There exists an explicit primitive relative invariant $z_n$ of weight $-1/g$ (with $g = \gcd(n,3)$):

$$z_n = \begin{cases} \dfrac{(\prod_{i=5}^{n} \Delta_{12i})^3 (\Delta_{134} \Delta_{234})^{n-3}}{(\Delta_{123} \Delta_{124})^{2n-9}} & g = 1 \ \dfrac{(\prod_{i=5}^{n} \Delta_{12i}) (\Delta_{134} \Delta_{234})^{n/3-1}}{(\Delta_{123} \Delta_{124})^{2n/3-3}} & g=3 \end{cases}$$

with $\Delta_{ijk}$ denoting determinants of the gradient and point data. Every relative invariant can be written as a rational function in the absolute invariants and the primitive generator $f_n = (z_n)^g$, which has weight $-1$. Thus,

$\mathcal{I}_n = \mathcal{I}_{n,0}(f_n)$

(Bedratyuk, 17 Jul 2025, Bedratyuk, 23 Nov 2025).

4. Explicit Formulas and Cases for Small $n$

For $n = 2$:

• The field of absolute invariants is generated by $$\zeta_{12} = (p_1(x_1-x_2) + q_1(y_1-y_2))(p_2(x_1-x_2) + q_2(y_1-y_2))$$.
• A relative invariant of weight $-1$ is $f_2 = \Delta_{12} = \det(\ell_1, \ell_2, L_{12})$.

For $n = 3$:

• $$\mathcal{I}_{3,0} = \mathbb{R}(\zeta_{12}, \zeta_{23}, \zeta_{13}, \tau)$$.
• The primitive relative invariant is $z_3 = \Delta_{123}$ of weight $-1/3$, and $f_3 = z_3^3$ is of weight $-1$ (Bedratyuk, 17 Jul 2025).

5. Methodologies: Moving Frames and Invariantization

The moving frame approach provides a systematic, algorithmic technique:

1. Impose normalization (cross-section) constraints on coordinates and derivatives to fix the group action.
2. Solve for the frame parameters, yielding a normalization map $\rho$.
3. Invariantize coordinate functions by evaluating at the normalized configuration.
4. The essential invariants are precisely the remaining variables after normalization, ensuring functional independence (Bedratyuk, 23 Nov 2025).

This construction extends to symbolic and numerical algorithms, with care required for degenerate configurations and numerical stability. The moving frame construction guarantees that the set of absolute invariants is minimal, closed-form, and robust for computation.

6. Relationship to Classical Projective Differential Invariants

Classical projective differential invariants, e.g., the Schwarzian derivative in one dimension, appear only at third order owing to the dimensionality of the projective group and the invariance structure. In particular, for the action of $PGL(2,\mathbb{R})$ (Möbius group) on functions of one variable, no nonconstant first- or second-order absolute invariants exist; the first is the third-order Schwarzian $$S[f](x) = f'''(x)/f'(x) - \frac{3}{2}\left( f''(x)/f'(x) \right)^2$$ (Euler et al., 14 May 2025, Blázquez-Sanz et al., 2014).

In contrast, the absolute first-order joint invariants constructed for $n>1$ points in the plane exploit the joint data of multiple points and their first derivatives, circumventing the absence of low-order absolute invariants in the classical, univariate setting. These invariants yield robust, algebraic expressions applicable to sampled image data and geometric configurations without the numerical instability of higher jets (Bedratyuk, 17 Jul 2025).

7. Connections to Broader Invariant Theory and Applications

The absolute first-order joint projective invariants embody principles from classical invariant theory, group cohomology, and the algebraic geometry of jet spaces. Their explicit minimal generating sets and algebraic relations inform practical algorithms for invariant feature detection, energy functionals, and integrals in fields such as computer vision, image analysis, and symbolic geometry (Bedratyuk, 17 Jul 2025, Bedratyuk, 23 Nov 2025). The formal structure, including the contracting homotopy for associated cochain complexes, ensures the vanishing of higher cohomologies and clarifies the extension structure of the full field of invariants.

Applications leverage the robustness of first-order joint invariants to noise and discrete sampling, in contrast to high-order jet invariants. Their closed-form rational dependence on point and gradient data facilitates efficient and predictable numerical implementation.