Self-Crossing Avoidance: Methods & Applications

• Self-crossing avoidance is a constraint that prevents trajectories, curves, or state-space paths from intersecting by enforcing exclusion zones or penalties.
• Different models apply hard geometric constraints, infinite-energy barriers, and probabilistic penalties to achieve noncrossing in contexts ranging from cell migration to graph drawing.
• These strategies fundamentally influence long-term dynamics and optimization, enabling robust superdiffusion in biological systems and efficient design in variational and combinatorial settings.

Self-crossing avoidance denotes the suppression of intersections between a trajectory, curve, or geometric structure and its prior history or itself. In recent research, the notion appears in mechanistic models of Physarum polycephalum migration, in probabilistic models of human scan paths, in variational optimization of plane and space curves, in extremal constructions for geometric graphs, and, in a distinct non-spatial sense, in the elimination of self-intersections of hysteretic current–voltage loops in graphene emitters (Tröger et al., 2024, Szorkovszky et al., 25 Aug 2025, Yu et al., 2020, Aloupis et al., 2024, Gorodetskiy et al., 2020). The common thread is geometric or state-space exclusion, but the formal realization varies sharply across domains: hard path exclusion, infinite-energy barriers, likelihood penalties, constructive noncrossing embeddings, and parameter regimes that remove loop intersections.

1. Formalizations of crossing and noncrossing

Self-crossing is not defined identically across the relevant literatures. In saccadic eye movements, a new segment $s_i=(Q_{i-1}\to Q_i)$ crosses an earlier segment $s_j=(Q_{j-1}\to Q_j)$ when the two open line segments intersect in the plane; the corresponding indicator is then counted over a finite recent history window. In geometric graphs, a drawing is noncrossing when no two straight-line edges have interior points in common. In variational curve design, self-intersection is prevented indirectly by an energy that diverges as two spatially close points remain far apart along the curve parameter. In Physarum, avoidance is implemented as a path-memory constraint requiring a trial step to remain outside an exclusion zone around the past trajectory. In memristive field emission, “self-crossing” refers instead to an intersection of the ascending and descending branches of an $I$–$V$ loop at the same voltage (Szorkovszky et al., 25 Aug 2025, Aloupis et al., 2024, Yu et al., 2020, Tröger et al., 2024, Gorodetskiy et al., 2020).

Setting	Object	Crossing or avoidance criterion
Physarum migration	Trial step $r(t)\to r^*$	Require $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$
Human scan paths	Saccade segment $s_i$	Crossing if two open line segments intersect
Geometric graphs	Straight-line edges	Noncrossing if open interiors do not intersect
Curve optimization	Arclength-parametrized curve $\gamma$	Tangent-point energy gives an infinite barrier to self-intersection
Graphene emitter hysteresis	Up/down $I$–$V$ branches	Self-crossing when $s_j=(Q_{j-1}\to Q_j)$0

These definitions delimit different mathematical objects. A trajectory model asks whether future motion may revisit prior geometry; a variational curve method assigns cost to all nonlocal pairwise interactions; a combinatorial construction engineers an embedding in which the longest admissible object is noncrossing; and a memristive model treats self-crossing as a dynamical feature of a response loop rather than of a spatial path. This distinction is essential, because results in one setting do not automatically transfer to another.

2. Self-avoiding run-and-tumble migration in macroscopic unicellulars

In "Size-dependent self-avoidance enables superdiffusive migration in macroscopic unicellulars" (Tröger et al., 2024), each P. polycephalum plasmodium is modeled by position $s_j=(Q_{j-1}\to Q_j)$1, heading $s_j=(Q_{j-1}\to Q_j)$2, phase $s_j=(Q_{j-1}\to Q_j)$3, and remaining dwell-time $s_j=(Q_{j-1}\to Q_j)$4. At each $s_j=(Q_{j-1}\to Q_j)$5, the agent attempts a step of length $s_j=(Q_{j-1}\to Q_j)$6 in direction $s_j=(Q_{j-1}\to Q_j)$7, with data-derived parameters for run and tumble dwell-time laws, phase-specific speeds, run-phase heading-change scale, and a self-avoidance width $s_j=(Q_{j-1}\to Q_j)$8. The geometric rule is explicit: no new step may approach the past trajectory closer than an avoidance radius $s_j=(Q_{j-1}\to Q_j)$9. If a trial step violates

$I$0

the step is rejected and a new heading is drawn during runs until the constraint is met.

The run and tumble durations follow broken power laws in CCDF form, while turning-angle PDFs are Laplace-like during runs and uniform during tumbles. Within this two-state kinesis, the mean-squared displacement is defined by

$I$1

with $I$2 corresponding to superdiffusion. On plain agar, the instantaneous exponent $I$3 starts near $I$4 at $I$5 min, indicating ballistic motion, and decays to $I$6 for $I$7 h, identified in the paper as the Flory exponent for a 2D self-avoiding walk. Simulations reproduce both $I$8 and the orientational correlation $I$9 with the same parameters.

The central mechanistic claim is that long-term superdiffusion is controlled by self-avoidance determined by cell size. Measured $V$0 scales with cell area $V$1: small cells with $V$2 have $V$3, whereas large cells with $V$4 have $V$5. Small plasmodia occasionally cross old trails, with a memory time of about $V$6 h, and then $V$7 at long times. Large plasmodia maintain full-length avoidance and sustain $V$8 up to days. The phase diagram in $V$9 identifies the regime $r(t)\to r^*$0 step-size and $r(t)\to r^*$1 run durations as self-avoiding, with $r(t)\to r^*$2, whereas $r(t)\to r^*$3 or small $r(t)\to r^*$4 recovers standard run-and-tumble behavior with $r(t)\to r^*$5.

This establishes a concrete biological instance in which self-crossing avoidance is not a secondary regularizer but the dominant determinant of long-time transport. The design principles stated in the paper are correspondingly direct: use a two-state run/tumble kinesis with power-law dwell times to tune short-time persistence, impose a size-dependent exclusion zone around the path, ensure that the exclusion radius exceeds the typical step length, and retain memory longer than the mean run time. These ingredients yield robust superdiffusion and maximize area coverage for sparse-target search.

3. Self-crossing avoidance as a saccadic search bias

"Saccade crossing avoidance as a visual search strategy" (Szorkovszky et al., 25 Aug 2025) formulates self-crossing avoidance within a step-selection framework. Each observed fixation-to-fixation displacement is treated as a case, control candidates are sampled from a baseline distribution, and a logistic regression assigns relative weight to predictors that include center bias, horizontal bias, length-change effects, inhibition of return, and self-crossing count. The generative baseline is a memoryless nonparametric model: the next-step displacement density is estimated by a Gaussian kernel density estimate in $r(t)\to r^*$6, a cubic spline rescales saccade lengths over the 45 s trial, and fixation durations are sampled independently from the empirical marginal distribution. A parametric extension then introduces a self-crossing penalty,

$r(t)\to r^*$7

where $r(t)\to r^*$8 is the avoidance strength.

History truncation for the crossing count was optimized over $r(t)\to r^*$9, yielding $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$0, and $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$1 when jointly optimizing other time constants. Maximum-likelihood fitting and held-out comparisons showed that the full model with $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$2 fit better than the memoryless baseline. The empirical mean number of crossings per saccade was $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$3 in human data, $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$4 under the memoryless nonparametric model, and $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$5 under the parametric model with self-crossing penalty. The reduction from $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$6 to $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$7 is reported as about $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$8 of potential crossings avoided. The estimated $\min_{\tau<t}\|r^*-r(\tau)\|\ge \bar w$9 was of the same order of magnitude as the inhibition-of-return coefficient; one example given is $s_i$0.

The paper further reports significant participant-level variation in almost all predictors, especially log-length change and self-crossing. Intraclass Correlation Coefficients were highest for length change ($s_i$1) and self-crossing ($s_i$2). The detailed summary states that the participant random effect for self-crossing avoidance was positively correlated with mean fixation duration and negatively correlated with mean saccade length, whereas the abstract states that stronger avoidance was associated with smaller saccade lengths and shorter fixation durations. Both statements agree on the negative association with saccade length, but they differ on fixation duration. This discrepancy is part of the record and should be kept distinct from the more stable findings that the effect is strongest over approximately $s_i$3 s of recent path history, is most evident for small to medium amplitudes below about $s_i$4 pixels, and is comparable in magnitude to inhibition of return.

In this setting, self-crossing avoidance functions as a local orienting bias rather than as a hard constraint. The model does not forbid crossings; instead it lowers their probability conditional on recent history. That probabilistic formulation is suited to visual search, where scan paths remain stochastic and history dependence is partial rather than absolute.

4. Variational self-repulsion of curves

"Repulsive Curves" (Yu et al., 2020) develops a variational framework for plane and space curves based on the generalized tangent-point energy

$s_i$5

with kernel

$s_i$6

for $s_i$7 and $s_i$8. Setting $s_i$9 recovers the classic tangent-point energy. As $\gamma$0 while $\gamma$1 along the curve stays bounded away from $\gamma$2, the energy diverges, producing an infinite repulsive barrier to self-intersection.

A naive $\gamma$3-gradient flow is described as extremely stiff because the first variation contains high-order, nonlocal terms. The paper instead equips the curve space with a fractional Sobolev–Slobodeckij inner product of order $\gamma$4, defines the Sobolev gradient by solving $\gamma$5, and thereby removes the remaining spatial derivatives from the flow. The practical consequence is large, resolution-independent time steps. For scalability, the method combines an axis-aligned 6D Barnes–Hut BVH for approximate edge-pair interactions, a block-cluster tree with rank-1 far-field approximation for $\gamma$6 matrix–vector products, and a geometric multigrid hierarchy using Braess–Sarazin’s strategy for the saddle-point systems that arise under constraints.

The mathematical and empirical consequences are explicit. The tangent-point kernel tends to infinity precisely when two points become arbitrarily close in space while remaining far apart along the curve, so attempted local crossings incur infinite energy cost. The paper cites Blatt–Reiter (2015) for regularity and non-self-intersection of local minima. Empirically, the reported descents never exhibit self-intersections, and the outputs remain isotopic to the inputs. The framework also accommodates hard constraints such as fixed endpoints, total length, surface contact, and prescribed tangents, as well as soft penalties for length, obstacle repulsion, and field alignment.

Case studies include knot untangling, non-crossing spline interpolation, graph drawing, robotic path planning, and streamline visualization. Quantitatively, the $\gamma$7-descent is reported as $\gamma$8–$\gamma$9 faster than $I$0, $I$1, $I$2, or L-BFGS methods on knot-untangling tasks, with $I$3 success within minutes on the cited datasets, and a typical $I$4-vertex curve network converges in $I$5 min on a single CPU. In this literature, self-crossing avoidance is thus a global nonlocal optimization principle rather than a local collision-detection heuristic.

5. Noncrossing longest paths, cycles, and matchings

"Noncrossing Longest Paths and Cycles" (Aloupis et al., 2024) addresses a common geometric intuition: although many shortest structures are inherently noncrossing, longest structures are often expected to contain crossings. The paper gives a negative answer to the question whether the longest spanning path on any finite planar point set must contain a crossing, and it refutes the conjecture that the longest spanning cycle must contain a crossing. For every integer $I$6, it constructs an $I$7-point set in the Euclidean plane whose longest perfect matchings, longest Hamiltonian paths, and longest Hamiltonian cycles are all noncrossing and, in fact, unique.

The construction proceeds in two phases. First, points are placed exactly on the $I$8-axis so that every longest one-dimensional structure has a prescribed combinatorial form. Second, the points receive tiny vertical perturbations obeying a strict hierarchy

$I$9

with the last coordinate $V$0. These perturbations are chosen so that among all one-dimensional longest structures only one gains the maximum extra length when lifted to the plane. Because the chosen path or cycle is strictly $V$1-monotone, it is noncrossing. Supporting lemmas on the real line characterize longest paths and cycles by the property that every edge crosses the median.

The significance of the result is twofold. First, it overturns the expectation that maximal total Euclidean length forces crossings. Second, it isolates a constructive mechanism for noncrossing extremality: encode the desired combinatorics in one dimension, then use a rapidly decaying hierarchy of perturbations to break ties. The paper also states clear limitations. The vertical coordinates must shrink extremely rapidly, making the construction numerically delicate, and the result does not imply a polynomial-time algorithm for finding a noncrossing longest path or cycle on an arbitrary planar point set; those optimization problems remain NP-hard or of unknown complexity in the plane.

This line of work is distinct from dynamical or variational avoidance. No evolution equation or repulsive term is used. Instead, self-crossing is precluded by the geometry of the instance itself, so that the extremal object is already noncrossing at optimum.

6. Avoiding self-crossing in hysteretic graphene-emitter $V$2–$V$3 loops

"A memristive model for graphene emitters: hysteresis and self-crossing" (Gorodetskiy et al., 2020) uses the term in a different sense. The object of interest is not a spatial trajectory but a hysteretic current–voltage loop whose ascending and descending branches may intersect. The two-stage model couples a modified Fowler–Nordheim law,

$V$4

to threshold dynamics of an internal state $V$5 representing the degree of graphene delamination. State $V$6 corresponds to a flat, unpeeled sheet, and $V$7 to a fully peeled-off, high-$V$8 geometry. Switching occurs upward for $V$9 and downward for $s_j=(Q_{j-1}\to Q_j)$00, with rates $s_j=(Q_{j-1}\to Q_j)$01 and $s_j=(Q_{j-1}\to Q_j)$02, or equivalently times $s_j=(Q_{j-1}\to Q_j)$03 and $s_j=(Q_{j-1}\to Q_j)$04.

Within this framework, self-crossing of the $s_j=(Q_{j-1}\to Q_j)$05–$s_j=(Q_{j-1}\to Q_j)$06 loop arises from finite switching rates, mechanical inertia, overshoot $s_j=(Q_{j-1}\to Q_j)$07, and delayed relaxation. The crossing condition is stated as $s_j=(Q_{j-1}\to Q_j)$08 for some $s_j=(Q_{j-1}\to Q_j)$09. The paper then identifies parameter regimes that move or remove the crossing. In the adiabatic limit $s_j=(Q_{j-1}\to Q_j)$10 and $s_j=(Q_{j-1}\to Q_j)$11, where $s_j=(Q_{j-1}\to Q_j)$12 is the inverse time scale of the voltage sweep, $s_j=(Q_{j-1}\to Q_j)$13 remains near quasi-equilibrium and overshoot disappears. In a refined model with maximal overshoot $s_j=(Q_{j-1}\to Q_j)$14, reducing inertia so that $s_j=(Q_{j-1}\to Q_j)$15 suppresses loop crossing. The strength of crossing also depends on the contrast between the two emission states; the paper states that if $s_j=(Q_{j-1}\to Q_j)$16, or in practice

$s_j=(Q_{j-1}\to Q_j)$17

then $s_j=(Q_{j-1}\to Q_j)$18 is monotonic and the up/down branches do not intersect. Bringing $s_j=(Q_{j-1}\to Q_j)$19 close to $s_j=(Q_{j-1}\to Q_j)$20 narrows the hysteresis window and likewise prevents a resolvable crossing.

The design guidelines are operational: keep the sweep rate low enough that $s_j=(Q_{j-1}\to Q_j)$21, engineer the graphene–substrate interface to increase damping, limit the maximum bias so the emitter is not driven deep into the peeled regime, reduce the field-enhancement contrast $s_j=(Q_{j-1}\to Q_j)$22, and bring the thresholds $s_j=(Q_{j-1}\to Q_j)$23 and $s_j=(Q_{j-1}\to Q_j)$24 as close as feasible. The paper also notes trade-offs, including slower device speed, reduced ON/OFF ratio, and lower maximum current. This usage broadens the term “self-crossing avoidance” beyond spatial geometry to dynamical response curves, but the underlying logic remains one of excluding self-intersection by controlling state evolution.

Taken together, these studies show that self-crossing avoidance is not a single method but a recurrent structural motif. Depending on the domain, it can be enforced geometrically, variationally, probabilistically, combinatorially, or through operating conditions. What remains stable across the settings is that preventing intersections changes long-time behavior, admissible optima, or observable response in a way that is mathematically explicit and empirically testable.