Tangent Intersection Rule in Geometry & Analysis

• Tangent Intersection Rule is a unifying principle that defines how tangency and collision of geometric objects determine intersection multiplicities and limit behaviors.
• It supports recursive computations in enumerative geometry, allowing for the precise enumeration of curves with prescribed tangencies and singularities.
• The rule finds broad application in variational analysis and computational geometry, notably in resolving tangent cones and planning collision-free paths.

The tangent intersection rule provides a unifying principle for computing limits of geometric, analytic, and variational intersection phenomena where tangency or higher-order contact is present. At its core, the rule encodes how intersection multiplicity, tangent cones, or tangency conditions localize or manifest at the limiting structure when two geometric objects—be they algebraic cycles, sets in function spaces, or obstacles in path planning—collide or are degenerated together. The rule appears in a wide range of mathematical settings, including algebraic geometry, geometric measure theory, set-valued analysis, and computational geometry.

1. Tangency as Collision of Intersections in Enumerative Geometry

In modern enumerative geometry, the tangent intersection rule formalizes the enumeration of plane curves with prescribed tangencies to a fixed divisor as iterated limits of ordinary (transverse) intersections. Specifically, tangency of order $m$ at a point is modeled by the collision of $m$ pairs of transverse intersection points merging to the same limit, and intersection-theoretic recursions are constructed from this viewpoint.

For plane curves $C$ of degree $d$ meeting a given line $\ell$ with tangency orders $m_1, \dots, m_k$ at distinct points $p_1, \dots, p_k$, Biswas, Choudhury, Mukherjee, and Paul develop a recursive scheme on the parameter space $M_k = D_1 \times D_d \times X^k$. The key principle is encapsulated by the formula (Theorem 3.2):

$$\pi^*[T_{m_1\cdots m_k}] \cdot \pi_{k+1}^*[T_0] = [T_{m_1\cdots m_k,0}] + \sum_{i=1}^k (m_i+1) \pi^*[T_{m_1\cdots m_k}] \cdot [\Delta_{i,k+1}]$$

where $[T_{m_1 \cdots m_k}]$ denotes the closure of the locus of curves with the prescribed tangencies, and $\Delta_{i,k+1}$ is the diagonal corresponding to collisions of points. This formula supports a fully recursive computation of intersection numbers, translating tangency conditions into combinatorial data of colliding points (Biswas et al., 2023).

The method extends to curves with singularities (nodes, cusps, tacnodes), with recursions adapting to the new classes and collision configurations.

2. Tangent Intersection Rule for Tangent Cones and Set Intersections

In variational analysis and geometric measure theory, the tangent intersection rule describes how tangent cones to intersections of sets relate to those of the individual sets. If $A, B$ are closed subsets of a Banach space $X$ with $x_0 \in A \cap B$, the central result is that, under strong tangential transversality, the Clarke tangent cone to the intersection equals the intersection of the individual Clarke tangent cones:

$T_{A \cap B}(x_0) = T_A(x_0) \cap T_B(x_0)$

Strong tangential transversality is defined via the existence of uniform tangent sets $D_A(x_0), D_B(x_0)$ generating the respective Clarke cones and a scalar $p > 0$ such that $p B \subset \operatorname{co}(D_A(x_0) - D_B(x_0))$. This result underpins subdifferential calculus and leads directly to Clarke's subdifferential sum rule for non-smooth analysis (Bivas et al., 2018).

When only metric subregularity (an error-bound property for a constraint map) holds, Durea and Strugariu show that the Bouligand and Ursescu (sequential) tangent cones satisfy intersection inclusions and, under certain regularity, equalities:

Cone Type	Intersection Rule
Bouligand ($T^B$)	$\cap$ inclusion
Ursescu ($T^U$)	$\cap$ equality

For example, $$T^U(D, \bar x) \cap T^U(E, \bar x) = T^U(D \cap E, \bar x)$$ under general conditions (Durea et al., 2011).

3. Analytic Framework: Tangent Currents and Intersection Multiplicity

In complex geometry, the tangent intersection rule appears via the theory of tangent currents and their shadows. Given positive closed currents $S, R$ on a domain, the intersection product is constructed through the tangent current $T^\infty$ of $\pi_1 S \wedge \pi_2 R$ along the diagonal $\Delta$.

The shadow $(T^\infty)_h$ of the tangent current gives a current on the submanifold encoding the intersection data. The main theorem asserts that the intersection multiplicity at a point equals the mass of the shadow:

$$\text{Intersection multiplicity of } V_1 \cap V_2 = \text{mass of the shadow of the tangent current along the diagonal}$$

For instance, for $C_1 = \{y = 0\}$ and $C_2 = \{y = x^m\} \subset \mathbb{C}^2$, the intersection multiplicity at the origin is $m$, as computed from the shadow $m \delta_0$ (Ahn, 10 Mar 2025).

King's residue formula is subsumed as a particular case, as the limit of dilated currents picks out multiplicities via the tangent current.

4. Computational Realizations and Path Planning Applications

The tangent intersection rule also finds concrete implementation in computational geometry. For UAV path planning in cluttered domains, the tangent intersection principle is used to construct collision-free sub-paths: for each obstacle, tangents from the UAV's current position and the target to the obstacle's (inflated) boundary give exactly two detour paths.

At each encounter, an elliptic-tangent graph with six vertices and two candidate sub-paths is constructed. Heuristic rules select between sub-paths based on obstacle avoidance, path length, and backtracking prevention, enabling efficient, local navigation decisions. Subsequent B-spline smoothing ensures kinematic feasibility (Liu et al., 2020).

This computational scheme generalizes the geometric principle that, at each obstruction, the family of tangents encodes the limiting behavior and the available collision-free directions, precisely aligning with the tangent intersection rule's geometric core.

5. Extensions: Singularities, Higher-Order Tangencies, and Regularity

The tangent intersection paradigm is robust under the introduction of singular strata and higher-order degeneracies. For instance, in enumerative geometry, curves with nodes or cusps are incorporated by new cycle classes and adjusted collision rules, allowing recursive enumeration of curves with both prescribed tangency and singularity types (Biswas et al., 2023).

In the analytic setting, the framework of tangent currents and their shadow accommodates cycles, generic currents, and proper as well as improper intersections. The $h$-dimension of the tangent current quantifies the residue of the intersection along the base cycle, with proper intersections yielding maximal $h$ and higher-order tangencies producing weighted shadows reflecting the multiplicity.

Metric subregularity and beyond-transversality conditions supplant traditional convexity or compactness hypotheses, enabling a broad calculus of tangent sets in geometric analysis (Durea et al., 2011, Bivas et al., 2018).

6. Illustrative Examples

• Algebraic Plane Curves: The number of rational cubics through 7 points, tangent to a given line, is computed by collision recursions, yielding 36 as the enumerative invariant (Biswas et al., 2023).
• Intersecting Hypersurfaces: Two analytic hypersurfaces $V_1, V_2$ intersecting at a point with multiplicity $m$ yield a tangent current shadow $m \delta_p$ at $p$, matching the algebraic intersection number (Ahn, 10 Mar 2025).
• Path Planning: For a UAV obstructed by an ellipse in the plane, the tangent intersection calculation produces two detour paths, one "left" and one "right," with selection based on geometric and heuristic criteria (Liu et al., 2020).

7. Significance and Theoretical Reach

The tangent intersection rule serves as the analytic, algebraic, and computational backbone for a significant class of geometric intersection problems. By reducing tangency, singularity, and higher-order intersection to the collision and limit of simpler (typically transverse) intersection data, the rule provides structure to recursive formulas, subdifferential and tangent cone calculus, and robust computational geometry strategies. Its presence in diverse mathematical areas (intersection theory, variational analysis, complex currents, path planning) underscores its foundational nature and adaptability to new analytic, algebraic, and applied settings.