VOLUMETRIC COORDINATE SYSTEM (VCS)
5DVNS Technologies
Offline volumetric positioning and decision support systems for Sub-Sea, Sub-Surface and Space Applications maintaining navigation integrity during total GNSS loss.
Our technology reduces drift-induced instability and strategic risk in physical and non-physical environments where traditional analytical tools fail.
Cartesian to Volumetric VPS Reversible Converter Tool
About Us
5DVNS Technologies is a European spatial intelligence company currently in formal incorporation.
Our solutions convert geographic and operational information into volumetric decision maps that perform independently of GPS, online services, or cloud analytics.
The Volumetric Coordinate System provides deterministic state integrity modeling for system operating within bounded geometric constraints under degraded reference conditions. Across subsea, subsurface, and orbital domains, it preserves admissible state evolution and derives guidance, stability, and risk indicators from volumetric boundary proximity rather than coordinate correction.
Determinism is achieved by giving bounded uncertainty sets and admissibility constraints. State validity evolves predictably without probabilistic correction layers.
Modern operations produce enormous data but the systems designed to interpret it remain linear, reactive, and sensitive to fragmentation.
We do the opposite.
We use offline volumetric modeling to identify:
This produces actionable clarity without exposing proprietary logic or internal customer data.
Every industrial organization has models that work until they don't.
VPS stabilizes those models before collapse occurs.
In geometry and applied systems theory, a Volumetric Coordinate System (VCS) is a coordinate framework in which the position of an entity is defined as a state within a bounded admissible volume rather than as a point in an unbounded Cartesian space.
Unlike a Cartesian coordinate system, which specifies a point by signed distances from mutually perpendicular axes, a volumetric coordinate system specifies a state relative to:
A volumetric state at time t is defined as:
S(t) = [x, y, d | V]
Where:
The lateral coordinates specify the position of a state within the cross-sectional geometry of the admissible volume. These coordinates do not necessarily correspond to global Euclidean axes and may be defined relative to:
The parameter d represents progression through the admissible volume. Unlike the z-axis in Cartesian systems, d does not represent elevation but advancement along a permitted corridor or structured region.
It may encode:
The vector V encodes properties of the bounded volume. Depending on application, it may include:
The context vector defines the admissible region within which valid states may exist.
In a volumetric coordinate system, valid positions are restricted to those that satisfy the constraints defined by V. A state transition:
S(t₁) → S(t₂)
is admissible only if volumetric consistency is preserved. This introduces a constraint-based formulation of position integrity rather than an absolute point-based definition.
Volumetric coordinates are used to describe positions as admissible states within bounded geometric regions rather than as isolated points in unbounded Euclidean space. This formulation allows problems of navigation, control, and constrained motion to be expressed in terms of state consistency under geometric and boundary constraints.
Unlike Cartesian coordinates, which are most commonly used in analytic geometry and computational graphics to represent points in open Euclidean space, volumetric coordinates are particularly suited to representing motion and state integrity within bounded regions subject to geometric and physical constraints.
A volumetric coordinate system may be extended to n-dimensional constrained manifolds. In such cases, the state becomes:
S(t) = [u₁, u₂, ..., uₖ | V]
where the admissible region is defined by constraint functions:
Cᵢ(S) ≤ 0
for all i in the constraint set.
Using a Volumetric Coordinate System, structured regions (such as corridors, bounded domains, or constrained manifolds) may be described as the set of all admissible states satisfying a collection of constraint relations involving the state variables. For example, a cylindrical corridor of radius r, centered along a reference progression axis, may be described as the set of all states whose lateral embedding coordinates satisfy a bounded inequality of the form:
x² + y² ≤ r²
together with admissibility conditions on the progression parameter d. Stability properties, transition boundaries, and constraint-preserving trajectories may be derived from these relations using methods from differential geometry, control theory, and constrained optimization.
Volumetric coordinate systems are applicable in contexts where:
Examples include:
Volumetric coordinate formulations are applicable in domains where position must be maintained relative to structural boundaries rather than absolute global axes. Such domains include robotics within confined workspaces, navigation in constrained environments, corridor-based motion planning, state-constrained control systems, and structured spatial modeling.
Volumetric coordinates provide a framework for representing state evolution in bounded environments and offer geometric interpretations of admissibility, constraint satisfaction, and progression under structural limits. The system generalizes naturally to higher-dimensional constrained manifolds and may be formulated using constraint functions or admissibility operators.
A Deterministic Framework for State-Based Positioning and Navigation
Prepared by: Hamdy Samy
Founder & IP Holder — 5DVNS
Date: February 2026
Positioning and navigation systems have historically been constructed around a single assumption: that location is best represented as a point within an absolute reference frame. This assumption underlies global navigation satellite systems (GNSS), inertial navigation systems (INS), simultaneous localization and mapping (SLAM), and most contemporary autonomous navigation architectures.
This paper introduces a different premise.
Rather than treating position as a point in space, the Volumetric Positioning State (VPS / 5DVNS) models position as a deterministic state embedded within a bounded volume, defined by geometric admissibility and contextual integrity rather than absolute coordinates. Determinism is achieved by giving bounded uncertainty sets and admissibility constraints. State validity evolves predictably without probabilistic correction layers.
In this framework, position is not continuously corrected toward an external reference. Instead, positional coherence is maintained through volumetric consistency under constraints. Guidance, stability, and navigation outputs emerge from the persistence of admissible states over time.
The framework does not depend on privileged global reference frames, it does not require a persistent global reference frame for state validity, it is robust to signal loss, vertical ambiguity, and map unavailability. It reframes navigation from coordinate accuracy to state integrity, enabling coherent operation across surface, subsurface, indoor, underwater, orbital, and non-physical domains.
This document formalizes the VPS state model, its invariance properties, and its interpretive shift, without prescribing implementation details or domain-specific realizations.
Classical positioning systems represent location as a point:
P(t) = (x, y, z)
This formulation assumes:
While effective in open and well-instrumented environments, point-based positioning exhibits structural fragility when:
In such cases, systems compensate by layering probabilistic corrections, sensor fusion, or increasingly complex maps—without addressing the underlying representational assumption.
The core limitation is not sensor quality or computation.
It is the choice of point as the primitive of position.
The VPS framework replaces point-based localization with state-based embedding.
Position is not asked as “Where am I?”
It is evaluated as “Is my current state admissible within the volume I inhabit?”
Formally, position is expressed as:
S(t) = [x, y, d | V]
Where:
The state is deterministic under bounded uncertainty.
Position emerges from volumetric consistency, not absolute reference alignment.
In this model:
Position becomes a property of state survivability under constraints over time.
The VPS formulation exhibits the following invariances:
These properties arise from the geometry of the state itself, not from redundancy or correction frequency.
The framework introduces two fundamental transformations:
Navigation guidance is not prescribed.
It emerges from the admissible evolution of states within constraints.
This shifts navigation from optimization toward targets to maintenance of coherence.
This document describes an operational-level construct.
Implementations may vary by domain, resolution, sensing modality, and risk profile. Certain structural aspects of the framework are subject to prior art disclosures and active patent filings and are intentionally described here at a conceptual level.
Within the VPS framework, navigation is the evolution of a state rather than the traversal of coordinates.
State evolution is governed by admissibility:
A state is valid if it remains geometrically and contextually consistent with the constraints encoded in V.
Formally, state transition does not seek an optimal target. It evaluates whether the next state remains inside the admissible volume defined by:
This introduces a critical distinction:
Trajectory is not planned in advance
Path viability is evaluated continuously
As long as the evolving state remains admissible, navigation proceeds. When admissibility degrades, guidance emerges as a corrective signal—not toward a coordinate, but toward restored coherence.
In this sense, navigation is not command-driven.
It is constraint-driven.
The volumetric context vector V is not metadata.
It is a first-class component of the positional state. Unlike traditional systems where uncertainty, constraints, or environmental factors are treated as error terms or auxiliary layers, VPS embeds them directly into the state definition.
The vector V may encode, depending on domain and implementation:
Crucially:
V does not describe the environment
V defines the conditions under which a state remains valid
This allows the system to reject impossible states deterministically, rather than correcting them probabilistically after failure.
All operational outputs in VPS are derived, not prescribed.
Typical emergent outputs include:
These outputs are consequences of state evolution under constraints.
No map, waypoint, or route is required for these outputs to exist.
Guidance emerges when the system detects approaching loss of admissibility.
In this sense:
Guidance is a symptom
Stability is the objective
A defining architectural principle of VPS is the separation between:
This separation ensures that:
This makes the framework suitable as a foundational positioning layer, rather than a domain-specific navigation tool.
The VPS formalism does not require physical space.
Any system that satisfies the following conditions may be modelled:
This allows the same formalism to apply to:
In such cases, x, y, d need not correspond to physical dimensions.
They represent embedding, progression, and phase inside a constrained manifold.
This extension is not metaphorical.
It is a direct consequence of the state definition.
| Classical Navigation | VPS / 5DVNS |
|---|---|
| Classical Navigation | VPS / 5DVNS |
| Point-based | State-based |
| Coordinate-driven | Constraint-driven |
| Map-dependent | Map-agnostic |
| Correction-centric | Integrity-centric |
| Accuracy-focused | Confidence-focused |
The VPS framework introduces a structural inversion: position corresponds to a state that remains within the admissible set over time, not what is measured.
The VPS framework does not compete with existing navigation systems.
It reframes the layer at which positioning is defined.
It treats navigation not as the pursuit of coordinates, but as the maintenance of admissible state evolution.
This makes it a domain-invariant admissibility layer for state-based positioning, autonomy, and reasoning across domains.
Let the system evolve over continuous time t ∈ ℝ⁺.
Define a bounded manifold ℳ ⊆ ℝⁿ representing the admissible embedding domain.
The Volumetric Positioning State is defined as:
S(t) = (p(t), d(t), V(t))
where
and 𝒞 is the constraint space.
Define the admissible set:
𝒜 ⊆ ℳ × ℝ × 𝒞
A state is valid if:
S(t) ∈ 𝒜
Unlike classical positioning, validity is not defined by proximity to a reference coordinate but by membership in the admissible set.
Let system evolution be governed by transition operator:
S(t+Δt) = Φ(S(t), u(t), η(t))
where
The VPS condition requires:
Φ(S(t), u(t), η(t)) ∈ 𝒜 ∀ t
Navigation is therefore the maintenance of invariance of 𝒜.
Instead of probabilistic error models, VPS uses bounded uncertainty sets:
η(t) ∈ ℰ(t)
with
ℰ(t) ⊆ ℝᵏ (compact)
Deterministic coherence exists if:
∀ η(t) ∈ ℰ(t), S(t) ∈ 𝒜
Thus position remains defined even under unknown disturbances.
Let R(t) denote an external reference measurement.
Classical navigation requires:
lim ‖p̂(t) − p_R(t)‖ → 0 as t → ∞
VPS removes this requirement.
Instead, coherence requires only:
S(t) ∈ 𝒜 ∀ t
Therefore state validity is independent of the existence of R(t).
Define admissibility margin:
γ(t) = dist(S(t), ∂𝒜)
Guidance is generated when:
γ(t) → 0
Control actions seek to increase admissibility margin rather than minimize coordinate error.
A VPS state is observable if measurements constrain admissible set to a bounded subset:
|𝒜_t| < ∞
Position is therefore defined by constraint convergence rather than coordinate convergence.
Classical navigation:
Position = estimated coordinate
VPS:
Position = persistent membership in admissible set
This single section does three critical things:
Setup. Let the VPS state evolve as
S(t+Δt) = Φ(S(t), u(t), η(t))
with bounded disturbance η(t) ∈ ℰ(t), where ℰ(t) is compact for all t. Let 𝒜 ⊆ ℳ × ℝ × 𝒞 denote the admissible set.
Assumptions.
Statement. If S(t₀) ∈ 𝒜, then there exists a control policy u(·) such that
S(t) ∈ 𝒜 ∀ t ≥ t₀
i.e., admissibility is a robust forward-invariant property and coherence is preserved without reference-frame correction.
Interpretation. VPS coherence is the persistence of state membership inside 𝒜 under bounded disturbance.
Setup. Let R(t) denote any external reference signal (GNSS, map anchor, absolute heading, etc.). Define VPS validity solely by admissible membership: S(t) ∈ 𝒜.
Assumptions.
Statement. The truth value of “state is valid” is invariant to the presence or absence of external reference:
(S(t) ∈ 𝒜) is well-defined even if R(t) ≡ ∅
Therefore, loss of external reference does not imply loss of state definability, only a potential change in admissibility margin.
Corollary 2.1 (Graceful degradation). When R(t) is removed, coherence can persist as long as the bounded disturbance set remains compatible with 𝒜.
Setup. Define an admissibility margin:
γ(t) = dist(S(t), ∂𝒜)
where ∂𝒜 is the boundary of admissibility.
Assumptions.
Statement. Any control policy that maintains or increases γ(t) produces navigation guidance as a byproduct of preserving admissibility:
γ(t) ↓ 0 ⇒ emergent correction signal
and the corrective action is directed toward restoring admissibility (increasing γ), not minimizing coordinate residuals.
Corollary 3.1 (Collision/risk as admissibility loss). Collision risk, instability risk, or corridor failure correspond to trajectories that drive γ(t) toward zero.
Author: Hamdy Samy
Date of Public Disclosure: January 18, 2026
Jurisdiction: Global (public disclosure)
This notice establishes prior art for the conceptual and theoretical foundations of the Volumetric Positioning State (VPS), also referred to as the 5DVNS framework.
This prior art disclosure intentionally covers conceptual structure and formalism, not implementation.
The following elements are publicly disclosed and asserted as prior art:
Position is defined as a state inside a bounded volume, not as a point in Cartesian space, formalized as:
S(t) = [x, y, d | V]
where positional meaning arises from volumetric consistency rather than absolute coordinates.
Navigation, localization, and reasoning are governed by volumetric state integrity and admissibility under bounded uncertainty, rather than continuous external correction.
The framework does not require a privileged global reference frame for state validity and is applicable across physical and non-physical environments where agents operate under constraints.
Detection, anomaly awareness, or risk indication is an emergent consequence of violations or stress within volumetric state consistency, not a standalone detection method.
Navigation guidance, collision risk, and decision confidence are derived outputs, not primary objectives or hard-coded rules.
The framework shifts navigation and reasoning from:
This disclosure does not reveal:
These remain protected under active and future intellectual property filings.
This notice serves to:
The author asserts that systems materially embodying the structural elements described above may fall within the conceptual scope disclosed hereunder,
are operating within the conceptual space established by this prior art.
This disclosure is publicly timestamped and intended to serve as prior art under applicable patent law, including but not limited to the Paris Convention and relevant national statutes.
Patent and Registration
PATENTS:
STATUS:
Defence cleared for PCT routing.
Legal and Safety
Our solutions operate offline, without telemetry, and without integration into third-party systems.
We do not collect or store operational data.
All engagements are governed by confidentiality and restricted use.
5DVNS Technologies develops spatial intelligence tools for industrial and civilian applications.
Our intellectual property is protected under Belgian and European frameworks. All VPS algorithms, methodologies, and system architectures are proprietary and confidential.
We do not collect or store operational data.
Our offline-first architecture means: no telemetry or data transmission, no cloud storage or third-party integrations, no tracking or analytics. All processing occurs locally in client environments.
5DVNS Technologies develops spatial intelligence tools for industrial and civilian applications.
We do not provide predictive, military, or surveillance services.
Our systems are designed to enhance operational resilience and protect critical environments.
What We Do
Offline / deterministic / autonomous
VPS converts classical coordinates (latitude/longitude, decimal, or grid) into volumetric spatial positions.
It allows resilient reference, redundancy, and indexing in:
VPS is offline and self-contained. It does not rely on satellite signals or external datasets.
Interpretive enterprise analysis
Most forecasting and planning models fail when confronted with:
VPS-SPARK stabilizes these environments by mapping how distortion accumulates spatially and how it propagates through corridors and operational nodes.
We do not predict. We do not replace ERP/AI. We correct the environments where those tools break down.
Confidential partnership model
We work with tier-one industrial partners in controlled, confidential formats:
Our engagements are designed to demonstrate value without exposing proprietary algorithms or internal client infrastructure.
The Volumetric Coordinate can be adapted to systems that represent position as bounded volumetric state membership and derive navigation or detection signals from admissibility evaluation and embody structural elements disclosed herein.
Traditional navigation systems represent position as a point in Cartesian space, defined by coordinates that must be continuously corrected against external references.
The VPS framework replaces this assumption.
In VPS, position is modeled as a state within a bounded volume, not as an absolute coordinate. A position is considered valid if it remains coherent within geometric, environmental, and operational constraints.
This means positional integrity is maintained through volumetric consistency, rather than coordinate accuracy. The system evaluates whether an object remains in a valid spatial state, even when absolute references are unavailable, degraded, or conflicting.
VPS represents position as a structured spatial state composed of:
This structure defines where an entity exists in relation to its environment, rather than where it exists on a global map.
The purpose of this structure is not measurement precision, but state validity and continuity under constrained and uncertain conditions.
VPS is deterministic not because it eliminates uncertainty, but because it operates within bounded uncertainty.
Determinism is achieved through geometric and constraint coherence, rather than continuous correction from external signals or reference frames.
As long as the system remains within a valid volumetric corridor, positional continuity is preserved. Signal loss, ambiguity, or degradation do not invalidate the state.
This allows stable operation in environments where traditional navigation systems become unstable or fail entirely.
VPS does not produce predefined or hard-coded outputs.
Navigation guidance, corridor stability, collision risk, trajectory confidence, and decision signals are derived properties of the volumetric state.
These outputs emerge from the integrity of the spatial state and its constraints, rather than from prediction models, rule engines, or optimization targets.
VPS does not forecast outcomes. It maintains valid spatial context so downstream systems and operators can act with confidence.
VPS is not a GIS system. It does not depend on maps or spatial databases.
VPS is not SLAM. It does not build or require persistent environmental models.
VPS is not a predictive AI system. It does not forecast behavior or optimize outcomes.
VPS is not a cloud analytics platform. It operates fully offline without telemetry.
VPS is not a surveillance or tracking system. It does not collect, transmit, or store operational data.
VPS is a volumetric positioning framework focused on maintaining spatial state integrity in constrained environments.
VPS stands for Volumetric Positioning State — a volumetric positioning framework designed to preserve spatial state integrity in constrained environments. A deterministic admissibility engine for bounded state evolution in degraded-reference complex environments.
Traditional navigation systems represent position as a point in Cartesian space, defined by coordinates that must be continuously corrected against external references.
The VPS framework replaces this assumption.
In VPS, position is modeled as a state within a bounded volume, not as an absolute coordinate. A position is considered valid if it remains coherent within geometric, environmental, and operational constraints.
Positional integrity is therefore maintained through volumetric consistency, rather than coordinate accuracy.
VPS represents position as a structured spatial state composed of:
This structure defines where an entity exists in relation to its environment, rather than where it exists on a global map.
The objective is state validity and continuity under constrained and uncertain conditions.
VPS is deterministic not because it eliminates uncertainty, but because it operates within bounded uncertainty.
Determinism is achieved through geometric and constraint coherence, not through continuous correction from external signals or fixed reference frames.
As long as the spatial state remains valid within its volumetric corridor, positional continuity is preserved even during signal loss or ambiguity.
VPS does not produce predefined or hard-coded outputs.
Navigation guidance, stability indicators, collision risk, trajectory confidence, and decision signals are derived properties of the volumetric state.
These outputs emerge from state integrity and constraint satisfaction, not from prediction engines or optimization logic.
VPS does not forecast outcomes. It maintains valid spatial context.
VPS is not a GIS system and does not depend on maps or spatial databases.
VPS is not SLAM and does not build persistent environmental models.
VPS is not a predictive AI system and does not optimize or forecast behavior.
VPS is not a cloud analytics platform and operates fully offline.
VPS is not a surveillance or tracking system and does not collect or transmit operational data.
VPS is a volumetric positioning framework designed to preserve spatial state integrity in constrained environments.
Volumetric Positioning State (VPS / 5DVNS) introduces a foundational shift in how position and navigation are defined.
Rather than representing position as a point within an absolute reference frame, VPS models position as a deterministic state embedded within a bounded volume, defined by geometric admissibility and contextual integrity.
In this framework, position is not continuously corrected toward an external reference. Positional coherence is preserved through volumetric consistency under constraints, allowing navigation, guidance, and stability signals to emerge naturally from valid state evolution.
Classical navigation systems treat position as a point in space, assuming:
These assumptions break down in environments where references degrade, vertical ambiguity dominates, or maps are incomplete or unavailable.
The limitation is not sensor quality or computation.
It is the choice of point-based representation as the primitive of position.
VPS replaces point-based localization with state-based embedding.
Position is no longer asked as "Where am I?"
It is evaluated as "Is my current state admissible within the volume I inhabit?"
Coordinates become observations rather than authorities.
Sensors inform state evolution, but do not define position.
Drift is not eliminated, but made explicit and bounded.
Loss of external reference does not imply loss of coherence.
Position becomes a property of state survivability under constraints over time.
The VPS framework exhibits the following invariance:
These properties arise from the geometry of the state itself, not from redundancy or correction frequency.
The framework introduces two fundamental transformations:
Navigation guidance is not prescribed in advance.
It emerges from the admissible evolution of states within constraints.
This shifts navigation from optimization toward targets to maintenance of coherence.
All operational outputs in VPS are derived, not prescribed.
Typical emergent outputs include:
Guidance emerges when the system detects approaching loss of admissibility.
Stability is the objective.
VPS enforces a strict separation between:
This ensures that visualization and human interpretation remain downstream of machine coherence, preserving domain invariance and system integrity.
The VPS formalism does not require physical space.
Any system where states exist inside bounded constraints and transitions occur over time may be modeled using the same framework.
This includes abstract decision spaces, multi-agent coordination, and non-spatial state systems where position represents coherence rather than location.
This extension is not metaphorical.
It is a direct consequence of the state definition.
VPS introduces a structural inversion of classical navigation:
Position corresponds to a state that remains within the admissible set over time, not what is measured.