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Obtuse Path in Mathematics and Applications

Updated 1 July 2025

Obtuse Path is a mathematical concept defined by non-standard trajectory angles and multi-branching properties across disciplines.
Its formulation in stochastic processes and graph theory enables precise modeling techniques in financial mathematics and network optimization.
Research on obtuse paths drives advancements in geometric probability, lattice enumeration, and computational algorithms for discrete systems.

An obtuse path, as a technical term, appears in several areas of contemporary research mathematics, probability, combinatorics, geometry, and lattice theory. The definition and role of an obtuse path depend on context, encompassing concepts from abstract stochastic processes to geometric path simplification and discrete geometry. This entry synthesizes rigorous results and frameworks in which obtuse paths or analogous constructs are central, detailing their mathematical and applied significance.

1. Stochastic Analysis and Obtuse Random Walks

In stochastic process theory, an obtuse random walk is a discrete-time, $d$ -dimensional normal martingale whose increments at each step take exactly $d+1$ values, inducing a filtration of multiplicity $d+1$ . This generalizes classical Bernoulli (binary-valued) random walks to a setting where the process explores a richer, higher-branching path space, with dependencies among increments and a combinatorially increased number of possible trajectories at each time slice.

The structure equation for the increments is

$Y_n Y_n^{T} = I_d + \sum_{k=1}^d \psi_k(n) Y_k(n)$

where $\psi_k(n)$ are predictable. This setting admits a full discrete-time analog of stochastic calculus. In particular, the chaotic representation property (CRP) ensures every square-integrable functional of the process admits a decomposition as an orthogonal sum of multiple stochastic integrals, paralleling the Wiener chaos for Brownian motion. For any $L^2$ functional $F$ ,

$F = \mathbb{E}[F] + \sum_{k=0}^\infty \langle \mathbb{E}[D_k F | \mathcal{F}_{k-1}], Y_k \rangle$

where $D_k$ is the gradient operator measuring sensitivity to increments at step $k$ , providing a discrete Clark-Ocone-type formula.

The mathematical role of obtuse paths arises in the explicit construction and predictable representation of martingale functionals in these walks, with direct applications in discrete-time option hedging and financial mathematics, where multi-asset, multi-branching (i.e., more-than-binary splitting) scenarios require such pathwise distinction. Deviation inequalities and covariance identities derived in this context provide tools for risk assessment and process concentration (1212.2324).

2. Graph Theory: Path Abstraction and Path Shortening

In graph-theoretic terms, an obtuse path emerges from the process of path abstraction, where a directed graph (digraph) is systematically simplified by detouring (bypassing) and contracting internal vertices. Given a path $u \to v_1 \to v_2 \to \dots \to v_k \to w$ in a digraph $D$ , bypassing $v_2$ (for example) results in a direct arc from its predecessor $v_1$ to its successor $v_3$ , and subsequent contractions may merge further vertices. The resulting path in the abstracted digraph is an obtuse path—a "blurred" or coarsened representative of the original, reflecting only essential steps.

The abstraction process is formalized by operations: $D \uparrow v,\quad D \upuparrows v,\quad D/\{\pi^{(1)},...,\pi^{(m)}\}$ where $\uparrow$ denotes detour, $\upuparrows$ denotes bypass, and $/$ denotes contraction. The image of each path under the abstraction map preserves connectivity while potentially omitting intermediate vertices.

Obtuse paths are critical in the analysis of network flows, data provenance, and path compression: they give minimal representatives for connectivity and allow for reasoning about "macro" transitions in machine learning, security, or dataflow systems. In random digraph models, the expected abundance and structure of obtuse paths post-abstraction are computable through explicit formulas, supporting scaling analyses in large or evolving networks (1701.07492).

3. Geometric Probability: Obtuse Paths in Random Triangles

In geometric probability, obtuse paths refer to the characterization and frequency of obtuse triangles—those containing an angle greater than $90^\circ$ —under various random models. When three points are selected randomly in a geometric domain (e.g., the unit disk, square, or, more recently, the cube), the probability that they form an obtuse triangle (i.e., that the path between them "bends" at an obtuse angle) can be computed exactly.

For triangles sampled uniformly in the shape sphere (Kendall’s approach), the probability of obtuseness is $\frac{3}{4}$ , while for three points selected in the unit cube, the exact probability is

$\eta(C_3) = \frac{323338}{385875} - \frac{13G}{35} + \frac{4859\pi}{62720} - \frac{73\pi}{1680\sqrt{2}} - \frac{\pi^2}{105} + \frac{3\pi\ln2}{224} - \frac{3\pi\ln(1+\sqrt{2})}{224}$

with $G$ Catalan’s constant, numerically about $0.54266$ (2501.11611). These results quantify the typical "bending" of random paths in space, informing questions in random mesh generation, computational geometry, and spatial approximation.

4. Discrete and Convex Geometry: Periodic Orbits and Rational Lattice Triangles

In polygonal billiards, obtuse paths describe periodic orbits in triangles or polygons with obtuse angles. The classification and existence of such paths are nontrivial in polygons with angles exceeding $90^\circ$ . Recent work establishes that every obtuse triangle with its largest angle up to $112.3^\circ$ has a periodic path—a closed billiard trajectory—constructed and verified by computational combinatorics and geometric unfolding (1808.06667).

In the paper of rational lattice triangles—triangles whose angles are rational multiples of $\pi$ —the existence of "optimal" (periodic or lattice) paths under unfolding is rare for strongly obtuse cases, with classification results showing that only two explicit infinite families (isosceles-type and a related sequence) allow for unfoldings to Veech (lattice) surfaces (2009.00174). For other obtuse rational triangles, the typical trajectories are non-periodic or chaotic, signifying the combinatorial and geometric complexity added by obtuseness.

5. Lattice Theory: Obtuse Bases and Enumeration Paths

In computational lattice theory, an obtuse basis is one in which all pairwise angles between basis vectors are at least $90^\circ$ , i.e., $\mathbf{b}_i \cdot \mathbf{b}_j \leq 0$ for $i \neq j$ . When used for lattice enumeration algorithms (e.g., in solving the shortest vector problem), any shortest vector can be written as a sum of the basis vectors with all integer coefficients of the same sign. This drastically reduces the effective search space: rather than considering all $\pm$ sign branches at each step, the algorithm may restrict to non-negative (or non-positive) combinations, exponentially accelerating computation (1912.01781, 2009.00384).

If one views each branch of the enumeration tree as a discrete path through the lattice, then enumeration using an obtuse basis follows an "obtuse path"—a sequence where the chosen steps do not double back in "acute" directions, but proceed in consistent, non-opposing directions, minimizing cross-term penalties in the vector norm and yielding efficient search.

6. Discrete Geometry: Circle Patterns and Obtuse Angles in Path Construction

In the context of circle patterns on surfaces, obtuse paths emerge in the extension of Thurston's Circle Pattern Theorem to settings where circles intersect with obtuse exterior angles ( $>90^\circ$ ). The existence and rigidity of such discrete geometric configurations are ensured under new combinatorial conditions, and are established using topological degree theory and variational principles (1703.01768).

The allowance of obtuse angles in intersection patterns enables the design of more general discrete curves and surfaces—obtuse paths—that support richer geometric and combinatorial structures on triangulated surfaces, including those relevant for discrete conformal geometry and algorithmic design in computational models.

7. Fluid Mechanics: Obtuse Angles in Dynamic Paths at Moving Contact Lines

In experimental fluid mechanics, the paper of moving contact lines at obtuse contact angles (greater than $90^\circ$ ) uncovers distinctive path and flow features. Quantitative measurements reveal that the flow field and interface speed near such contact lines differ from acute or right contact angles, with rapid velocity reduction observed near the contact point. This experimental behavior provides a physical mechanism to resolve theoretical singularities predicted by classical viscous models, affecting how dynamic wetting and dewetting paths are modeled and simulated (2502.09953).

Summary Table: Obtuse Path Across Domains

Domain	Obtuse Path Definition	Key Role
Stochastic processes	High multiplicity martingale trajectories	Predictable representation, financial modelling
Graph theory/networks	Paths after vertex deletion/contraction	Path abstraction, complexity reduction
Geometric probability	Triangles with one angle $>90^\circ$	Quantitative geometry, random mesh analysis
Polygonal billiards	Periodic orbits in obtuse polygons	Classification, dynamics, chaos vs periodicity
Lattice enumeration	Paths in enumeration with same-signed steps	Exponential speedup, search tree pruning
Discrete geometry	Paths in patterns with obtuse intersections	Discrete models, rigidity, computational geometry
Fluid mechanics	Trajectories near obtuse contact/wetting angles	Singularities, modeling correction

Obtuse paths thus serve as both objects of interest and essential technical tools in a diverse array of mathematical and applied fields, connecting stochastic representation, combinatorial network analysis, computational optimization, and geometrical structure.

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References (9)

Stochastic analysis for obtuse random walks (2012)

Path abstraction (2017)

The Probability that a Random Triangle in a Cube is Obtuse (2025)

One Hundred and Twelve Point Three Degree Theorem (2018)

Strongly Obtuse Rational Lattice Triangles (2020)

Faster Lattice Enumeration (2019)

Obtuse Lattice Bases (2020)

Circle patterns with obtuse exterior intersection angles (2017)

The study of moving contact line dynamics at obtuse contact angles: An experimental investigation (2025)