Multivariate Planar Curves

• Multivariate planar curves are defined as tuples of bivariate functions mapping from [0,1] to ℝ², preserving individual shape and spatial relationships.
• They employ rigorous techniques like Procrustes minimization and tangent space projection to isolate intrinsic geometric features for statistical analysis.
• Applications include computer vision, graph drawing, and medical imaging, where precise geometric modeling and rational approximation are crucial.

A multivariate planar curve is a mathematical object or data structure comprising a finite or countable collection of closed (or possibly open) planar curves, each embedded in the Euclidean plane. In contemporary research, the analysis, construction, and application of such objects arise in disciplines spanning algebraic geometry, computer vision, geometric modeling, and metric number theory. Recent work emphasizes formalism, topology, statistics, and computation for applications ranging from rational curve topology and multiview geometry to medical image analysis and graph drawing with controlled curvature.

1. Mathematical and Statistical Formalisms

In advanced statistical shape analysis, a multivariate planar curve is formalized as a $p$-tuple of bivariate functions, $C = \mathrm{Vec}(C_1, \ldots, C_p)$, where each $C_j: [0,1] \to \mathbb{R}^2$ parameterizes a closed curve (with $C_j(0) = C_j(1)$). These curves are elements of the product Hilbert space $H^p$, equipped with the norm

$$\langle f, g \rangle_{H^p} = \sum_{j=1}^p \langle f_j, g_j \rangle_H$$

where $H = L_2([0,1]) \times L_2([0,1])$. The multivariate curve representation preserves both the geometric integrity of the individual shapes and captures their relative spatial relationships.

The deformation model used to factor out nuisance variables (scale $\rho$, rotation $\theta$, translation $T$, and parameterization/group action $\gamma$) for joint alignment leads to:

$$C_j = \rho\, O_\theta\, \widetilde{C}_j \circ \gamma_{\delta_j} + T,\qquad j=1, \ldots, p,$$

or, more compactly,

$$C = \rho (I_p \otimes O_\theta)\,[\widetilde{C} \circ \gamma] + (\mathbf{1}_p \otimes T).$$

Here, $\widetilde{C}$ lives in an infinite-dimensional unit sphere (the "pre-shape" space) subject to a centering condition, and the common group parameters enforce joint transformations across all $p$ component curves (Issam-Ali et al., 15 Aug 2025).

Such formalisms underlie data-driven or geometric modeling tasks where features (for instance, organ contours in medical images) are naturally represented as tuples of interacting curves.

2. Alignment and Tangent Space Projections in Statistical Analysis

A fundamental methodological challenge in multivariate curve analysis is alignment: removing the effects of rotation, translation, reparameterization, and (optionally) scaling so as to isolate intrinsic geometric structure. For the joint model,

• The translation $T$ and scale $\rho$ are computed via the mean position and $L_2$ norm across all curves.
• Alignment in orientation and parameterization is achieved via iterative minimization:
  • Fixing reparameterizations, the optimal rotation $\widehat{\theta}$ is found by Procrustes minimization (closed-form via root of a trigonometric equation).
  • Fixing rotation, each shift $\delta_j$ is optimized (cf. Proposition 2.1 in (Issam-Ali et al., 15 Aug 2025)), often leveraging Fourier bases to reduce this to root-finding for trigonometric equations.

Aligned shape variables are projected via the logarithmic map onto the tangent space at the (Fréchet) mean $\mu$:

$$\mathcal{L}(f) = \frac{\omega}{\sin \omega}\, (f - \cos\omega\, \mu), \qquad \omega = \arccos\langle f, \mu \rangle_{H^p}.$$

The tangent space $T_\mu$ is linear and admits classical statistical techniques (regression, classification, dimension reduction).

This alignment enables robust supervised learning when contours are viewed as predictors, as demonstrated in the detection of cardiomegaly from chest X-rays, where the relative configuration of multiple organs' contours (heart and lungs) is crucial (Issam-Ali et al., 15 Aug 2025).

3. Topology and Geometry of Rational Multivariate Planar Curves

For rational curves, the problem of computing the topology (branches, singularities, connectivity) directly from a parametrization

$$\varphi(t) = (x(t), y(t)) = \left( \frac{p_1(t)}{q_1(t)}, \frac{p_2(t)}{q_2(t)} \right)$$

avoids the need for implicitization. Algorithms construct a planar graph by:

• Computing critical parameter values from roots of a constructed polynomial $m(t)$.
• Slicing the domain into vertical "critical" and "noncritical" lines and sampling the real points of the curve on these lines.
• Establishing connections via parametric orderings (e.g., Theorem 2.7 in (0909.3248)) that use monotonicity of $x'(t)$.
• Optionally, isolated points are detected using Hermite's method.

This approach, implemented in symbolic software such as Maple, is efficient and robust for high-degree rational curve examples and forms a base for CAGD systems (e.g., Bézier curves, B-splines, NURBS) (0909.3248).

4. Multivariate Rational Approximation and Curves of Singularities

Approximation of functions with curve singularities (rather than points) motivates the generalization of rational approximation to multivariate, planar settings. Let $Q(x, y)$ define a singularity curve (e.g., zero set of a polynomial). Rational approximants take the form

$$r(x, y) = \sum_{j=1}^{N_q} \sum_{k, \ell} a_{j, k, \ell} \frac{p_j P_k(x) P_\ell(y)}{Q(x, y) - p_j} + \sum_{k, \ell} b_{k, \ell} P_k(x) P_\ell(y),$$

where $p_j$ cluster exponentially toward zero, ensuring the associated poles cluster exponentially toward the curve of singularities.

Such approximants achieve root-exponential convergence in degree for a wide class of singularity geometries (edge, circular, diagonal, or elliptic singularity curves) (Boullé et al., 2023). Techniques involve direct least-squares solutions (with regularization as needed) and algebraic structure in the denominator polynomials to capture the geometry of the singular set.

This methodology improves the numerical resolution of Green's functions and PDE solutions with singularities along complex planar sets, and generalizes 1D AAA/lightning methods.

5. Applications in Computer Vision and Graph Drawing

Multivariate planar curves arise naturally in the modeling of multiview geometry and in advanced graph visualization:

• Multiview Varieties: The projection of planar or spatial curves (such as conics or higher-degree rational curves) through pinhole cameras is codified algebraically via Chow forms and back-projected cones, forming multiview varieties whose Zariski closure describes the tuples of image curves consistent with a 3D object (Rydell et al., 3 Apr 2024). For $d=2$ (conics), the conic multiview variety in two images is cut out by a rank-1 condition on a 2×3 matrix of quadratic forms in the respective image curve parameters. The complexity of reconstructing a conic from its image projections is measured by the Euclidean distance degree (EDD), with a conjecture of EDD = 538 for two generic cameras, reflecting substantial computational challenges.
• Graph Drawing: Algorithms for planar and 1-planar graphs can replace edges with cubic Bézier curves—each edge realized as a single cubic curve—such that angular resolution at vertices is $\Theta(1/\deg(v))$ and curvature is $O(\sqrt{n})$ (where $n$ is the number of vertices), all realized in $O(n\times n)$ grid size and $O(n)$ time given combinatorial embedding (Eppstein et al., 15 Oct 2024). These results are practically significant for network visualization, circuit layout, and geometric modeling.

6. Geometric and Topological Invariants

The analysis of the complex and real topological types of multivariate planar curves is advanced by introducing invariants such as the type $t(C)$ defined via the degrees of syzygies and the Bourbaki ideal, enabling a hierarchy of types (type 0: free curves; type 1: plus-one generated; type 2: higher complexity) (Abe et al., 15 Oct 2024). For arrangements of lines and conics, explicit classification and behavior under unions are established. For real curves, maximal Mumford curves (MM-curves) composing $g+1$ ovals and $g$ tropical cycles are constructed as deformations of graph curves determined by planar trivalent graphs, with the combinatorics of double covers predicting oval counts (Kummer et al., 18 Apr 2024).

Such invariants and structural theorems bridge syzygetic, topological, and tropical perspectives, revealing deep links between singularity theory, deformations, and the combinatorial geometry of graphs and arrangements.

7. Dynamical Systems and Curvature-Driven Flows

Planar curves with position- or direction-dependent curvature, as well as curves arising as solutions to gradient flows minimizing elastic or geodesic curvature variation, are analyzed using dynamical systems techniques. For curvature prescriptions $\kappa = f(|z|)$, the classification of simple closed (Jordan) solutions reduces to the analysis of periodic orbits of an associated planar ODE (Berger, 2021), with certain regimes (e.g., $b > 3$ for $\kappa = a r^b$) admitting families of non-circular solutions parameterized by the net winding number.

In geometric regularization, $L^2$ gradient flows minimizing the integral of squared geodesic curvature variation for closed planar curves yield exponential convergence to multiply-covered circles, with global existence as long as length remains uniformly bounded. The scale-invariant quantity $L^3\|k_s\|_2^2$ is identified as critical for precluding blow-up and ensuring smooth convergence (Andrews et al., 2018).

Together, these analyses illuminate the interplay between curvature, singularities, topological class, and the asymptotic behavior of evolving curve ensembles in both geometric and application-driven contexts.

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In summary, the synthesis of multivariate planar curve theory combines rigorous geometric, algebraic, statistical, and computational frameworks to support their analysis and application. These frameworks are central to topics including rational curve topology (0909.3248), multiview geometry (Rydell et al., 3 Apr 2024), shape-based clinical prediction (Issam-Ali et al., 15 Aug 2025), rational approximation of singularities (Boullé et al., 2023), graph visualization (Eppstein et al., 15 Oct 2024), and the classification of complex and real algebraic curves (Abe et al., 15 Oct 2024, Kummer et al., 18 Apr 2024).