Virtual Point Formulation

• Virtual point formulation is a strategy that introduces synthetic or auxiliary points to recast complex geometric, algebraic, or numerical problems into more tractable forms across various fields.
• It preserves key solver structures by enhancing data representations (e.g., densifying 3D detection or creating virtual stencils in CFD) without altering the underlying operator frameworks.
• Empirical results demonstrate significant performance gains in applications such as multimodal 3D detection and generalized pose estimation, though limitations vary with domain-specific challenges.

Searching arXiv for the specified papers to ground the article and citations. Search query: (Yin et al., 2021) Multimodal Virtual Point 3D Detection

In the literature considered here, virtual point formulation denotes a class of constructions in which physically absent, implicit, auxiliary, or reparameterized points are introduced so that a target problem can be represented in a more tractable geometric, algebraic, or numerical form. The term is used explicitly in multimodal 3D detection, meshfree viscous-flow simulation, and generalized pose estimation, and it also provides a precise interpretive lens for virtual element methods and generalized treatments of the principle of virtual work. Across these settings, the “virtual” object is not uniform: it may be a synthetic 3D sample lifted from image evidence, an auxiliary interpolation location in a virtual staggered stencil, an implicit nodal degree of freedom on a polygonal element, a redefined world point that absorbs camera offsets, or a phase-space point equipped with a virtual variation (Yin et al., 2021, Park et al., 2014, Artioli et al., 2017, Mishra et al., 18 Feb 2026, Li et al., 8 Jun 2026, Delphenich, 2022).

1. Terminological scope and recurring structure

The phrase does not name a single standardized formalism. In "Multimodal Virtual Point 3D Detection" (Yin et al., 2021), a virtual point is a synthetic 3D point generated from 2D detections and LiDAR depth. In "Virtual Interpolation Point Method for Viscous Flows in Complex Geometries" (Park et al., 2014), virtual interpolation points are auxiliary locations that emulate a staggered arrangement on a single collocated node set. In "Virtual-point-based Solutions to Handle Generalized Absolute Pose Problem" (Li et al., 8 Jun 2026), a virtual point is a reparameterized 3D world point that converts the generalized absolute pose problem into a standard PnP form. In the virtual element literature, the phrase is not always used explicitly, but the detailed formulations in elasticity and in the Stokes–Poisson–Boltzmann system fit a virtual-point interpretation because the discrete unknowns are attached to vertices, edge nodes, and internal moments while the interior basis functions remain implicit (Artioli et al., 2017, Mishra et al., 18 Feb 2026). Delphenich’s generalized Hamiltonian treatment is similarly interpretable through virtual displacements and Pfaffian pairings rather than through explicit pointwise trajectories (Delphenich, 2022).

Context	Virtual point object	Operative role
Multimodal 3D detection	$v_i=(x_i,y_i,z_i,e_i)$	Augments sparse LiDAR with dense camera-derived 3D samples
Meshfree viscous flow	$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$	Creates a virtual staggered structure on one node set
Polygonal VEM	Vertex/edge DOFs and internal moments	Represents unknown fields through implicit basis functions and projections
Generalized pose	$Q'_i = Q_i - R^{-1}V_i$	Converts GAP constraints to standard PnP form
Generalized virtual work	Phase-space point plus virtual variation	Pairs Pfaffian 1-forms with virtual displacements

This suggests that the expression functions less as a single algorithmic label than as a cross-domain design pattern: difficult geometry, sparsity, coupling, or noncentrality is transferred into auxiliary pointwise objects while the main solver retains a familiar operator structure.

2. Synthetic 3D points in multimodal 3D detection

In multimodal autonomous-driving perception, the virtual point formulation addresses the mismatch between sparse LiDAR returns and dense RGB evidence. A virtual point is a synthetic 3D point

$v_i = (x_i, y_i, z_i, e_i),$

where $(x_i,y_i,z_i)$ is a 3D location in the LiDAR coordinate frame and $e_i \in \mathbb{R}^D$ is a semantic feature derived from RGB detections, specifically a one-hot class vector concatenated with an objectness score. These points are generated only near detected 2D objects, use LiDAR depth as a reference to lift image pixels into 3D, are much denser than the original LiDAR for small or far objects, and are fed into the 3D detector together with real LiDAR points after voxelization (Yin et al., 2021).

The geometric pipeline begins with standard LiDAR-to-camera projection under a rigid transform and perspective camera model. For a LiDAR point in homogeneous coordinates, the transform

$$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$

maps LiDAR coordinates to the camera frame while accounting for ego-motion. Projected LiDAR points that fall inside an instance mask $m_j$ form an object frustum

$$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$

The method then samples $\tau$ pixels uniformly from the mask,

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$0

and assigns each sampled pixel the depth of its nearest projected LiDAR neighbor,

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$1

Each sampled pixel is unprojected back into 3D and paired with the semantic feature

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$2

The resulting construction is explicitly object-centric: all virtual points generated for one 2D instance share the same semantic descriptor.

Integration into LiDAR detection is deliberately minimal. The combined point set is

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$3

Virtual points are injected before voxelization, and the method uses split voxelization because real and virtual point features have different semantics and dimensions. For each voxel, the average of real-point features and the average of virtual-point features are computed separately and then concatenated. Both point types traverse the same geometric pipeline thereafter, and the distinction lies only in feature channels and separate averaging; no explicit dynamic weighting or flags are introduced. The framework was instantiated with CenterPoint using VoxelNet and PointPillars backbones, and virtual points were also plugged into CenterPoint’s optional second stage without introducing new loss terms.

The default nuScenes setting uses $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$4 virtual points per 2D instance. No raw RGB values or CNN feature maps are attached to the points; semantics come only from 2D detector outputs. The gains reported are substantial. On the main nuScenes leaderboard, a CenterPoint baseline of mAP 58.0 and NDS 65.5 becomes mAP 66.4 and NDS 70.5 with MVP, corresponding to +8.4 mAP and +5.0 NDS. The abstract additionally reports that the framework improves a strong CenterPoint baseline by 6.6 mAP. The improvements are especially strong for small objects, including +20.6 mAP for bicycle, +16.3 mAP for motorcycle, +5.7 mAP for pedestrian, and +8.3 mAP for traffic cone. On the validation setup with VoxelNet, adding virtual points, split voxelization, and two-stage refinement yields 67.1 mAP, 70.8 NDS, and on PointPillars the gain is +10.4 mAP and +4.8 NDS. A range-based breakdown reports gains of +1.9 mAP at $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$5 m, +7.4 mAP at $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$6 m, and +10.1 mAP at $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$7 m.

The formulation differs from PointPainting, MVX-Net, and PointAugmenting because it creates additional 3D points rather than only appending image features to existing LiDAR returns. It differs from BEV fusion and frustum-specialized pipelines because it remains LiDAR-centric and can be inserted into a standard voxel-based 3D detector. Its main limitations are equally explicit: depth is approximated by nearest-neighbor interpolation in image space; there is no explicit occlusion reasoning; performance depends on 2D detection quality and on calibration and synchronization; and the semantic feature $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$8 is low-dimensional. The reported Chamfer distance between completed depth and held-out LiDAR points is approximately 0.33 m on nuScenes.

3. Auxiliary interpolation points and virtual staggered structures in meshfree flow solvers

In incompressible-flow computation, the Virtual Interpolation Point method uses a fundamentally different notion of virtual point. All unknowns, velocity $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$9 and pressure $Q'_i = Q_i - R^{-1}V_i$0, are stored at the same physical node locations $Q'_i = Q_i - R^{-1}V_i$1. For each node $Q'_i = Q_i - R^{-1}V_i$2, the method introduces four auxiliary points $Q'_i = Q_i - R^{-1}V_i$3 arranged in east, west, north, and south directions. These VIPs are not nodes of the unknowns; they are locations where variables are interpolated from surrounding nodes by moving least squares. Together with the associated local stencil, they form a virtual staggered structure that mimics the pressure–velocity coupling of a staggered grid while keeping a single collocated node set (Park et al., 2014).

The governing equations are the non-dimensional incompressible Navier–Stokes system,

$Q'_i = Q_i - R^{-1}V_i$4

advanced by a second-order fractional-step method. The intermediate velocity solve uses implicit diffusion and explicit convection; the pressure correction solves

$Q'_i = Q_i - R^{-1}V_i$5

followed by

$Q'_i = Q_i - R^{-1}V_i$6

The spatial machinery combines MLS derivative operators with finite-difference-like evaluations between VIPs. Values at VIPs are interpolated by MLS shape functions, whereas gradients and divergences are formed as central differences between VIP locations.

The meshfree character of the method is central. MLS approximates a field locally by a quadratic polynomial in scaled coordinates and yields derivative shape functions $Q'_i = Q_i - R^{-1}V_i$7 so that function values and derivatives at $Q'_i = Q_i - R^{-1}V_i$8 are represented as weighted sums of surrounding nodal values. The Laplacian is obtained directly from $Q'_i = Q_i - R^{-1}V_i$9, but the convective term, divergence of the intermediate velocity, and pressure gradient are all evaluated through VIP differences. This is precisely why the method is described as staggered-like without an actual staggered grid.

The role of the virtual points is therefore stabilizing rather than reconstructive. The authors note that a pure MLS meshfree collocation of a projection method is unstable and prone to checkerboard pressure. By introducing VIPs and computing divergence and gradients via differences of MLS-interpolated VIP values, the discrete operators behave as if they were defined on a staggered layout. The method directly discretizes the strong forms of the incompressible Navier–Stokes equations and requires no numerical integration.

The reported numerical evidence is broad. For Taylor decaying vortices on $v_i = (x_i, y_i, z_i, e_i),$0, the method achieves $v_i = (x_i, y_i, z_i, e_i),$1 convergence in space and $v_i = (x_i, y_i, z_i, e_i),$2 in time. For the lid-driven cavity at $v_i = (x_i, y_i, z_i, e_i),$3, velocity profiles agree well with Ghia et al.’s benchmark data across node sets ranging from 7,897 to 22,925 nodes. In the triangular cavity, the computed Moffatt-eddy ratios satisfy $v_i = (x_i, y_i, z_i, e_i),$4 around 1.96–2.01 and $v_i = (x_i, y_i, z_i, e_i),$5 around 380–406, close to the analytical values 2.01 and 407. For flow past a circular cylinder, the method reports $v_i = (x_i, y_i, z_i, e_i),$6 at $v_i = (x_i, y_i, z_i, e_i),$7, and at $v_i = (x_i, y_i, z_i, e_i),$8 it captures periodic vortex shedding with time-averaged drag $v_i = (x_i, y_i, z_i, e_i),$9, lift amplitude $(x_i,y_i,z_i)$0, and Strouhal number $(x_i,y_i,z_i)$1. For bumpy cylinders at $(x_i,y_i,z_i)$2, node counts reach 236,604, and for $(x_i,y_i,z_i)$3 the drag reduction exceeds 7.9% relative to the smooth cylinder.

The limitations are equally specific. Boundary-region VIP construction is technically nontrivial because a symmetric virtual stencil may leave the fluid region. The paper notes these technical problems and states that a second-order accurate VIP scheme is designed to remain stable irrespective of the relative position between the virtual local stencil and the domain boundary, but it does not fully spell out the geometric algorithm. MLS conditioning and Reynolds-number scalability also remain practical concerns.

4. Implicit nodes, projections, and stabilization in virtual element formulations

A third major meaning of virtual point formulation appears in virtual element methods on polygonal meshes. In "Arbitrary order 2D virtual elements for polygonal meshes: Part I, elastic problem" (Artioli et al., 2017), the phrase is not used in the original title, but the detailed formulation can be understood in precisely that sense because the method is organized around degrees of freedom at geometrically defined points and internal moments, while the basis functions inside the element are never constructed explicitly. In "A higher order pressure-stabilized virtual element formulation for the Stokes-Poisson-Boltzmann equations" (Mishra et al., 18 Feb 2026), the same perspective is extended to a coupled multiphysics system on general polygonal meshes, including meshes with hanging nodes.

For 2D linear elasticity, the local polygonal displacement space on an element $(x_i,y_i,z_i)$4 contains all polynomial displacements in $(x_i,y_i,z_i)$5 but also a virtual complement controlled only through DOFs and projections. The local space has

$(x_i,y_i,z_i)$6

degrees of freedom on a polygon with $(x_i,y_i,z_i)$7 edges and approximation order $(x_i,y_i,z_i)$8. These are the two displacement components at each vertex, the two displacement components at $(x_i,y_i,z_i)$9 internal Gauss–Lobatto points per edge, and $e_i \in \mathbb{R}^D$0 internal moment DOFs based on a basis of $e_i \in \mathbb{R}^D$1. Vertex and edge DOFs are literal point values, whereas internal DOFs are moments

$e_i \in \mathbb{R}^D$2

The discrete bilinear form is split into a consistent term and a stabilization term,

$e_i \in \mathbb{R}^D$3

with a computable strain projection $e_i \in \mathbb{R}^D$4 onto symmetric polynomial tensors. Projection is made possible by integration by parts: boundary traces are known through point DOFs, and internal polynomial moments are known through the internal DOFs. The stabilization acts only on the non-polynomial component and is built entirely in DOF space, ensuring exactness on polynomials and coercivity on the complementary modes.

This formulation is inherently polygonal. It accommodates any number of edges, nonconvex elements, and highly distorted polygons, requires no mapping to a reference element, and uses Gauss–Lobatto nodes on each physical edge as both DOF locations and quadrature points. Patch tests reproduce constant stress states up to machine precision, $e_i \in \mathbb{R}^D$5 show the expected convergence rates, and sensitivity to the stabilization parameter is reported to be very mild over $e_i \in \mathbb{R}^D$6 with $e_i \in \mathbb{R}^D$7. In an energy-type norm $e_i \in \mathbb{R}^D$8, standard FEM slightly outperforms VEM; in an $e_i \in \mathbb{R}^D$9-type edge-based norm $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$0, VEM often outperforms FEM on distorted meshes.

The Stokes–Poisson–Boltzmann formulation retains the same virtual-element logic but applies it to a stationary electrokinetic model with unknown velocity $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$1, pressure $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$2, and electrostatic potential $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$3. The key obstacle is the Laplacian drag term $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$4 in the momentum equation. The formulation removes this second-order term by inserting the nonlinear Poisson–Boltzmann equation,

$$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$5

so that

$$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$6

The momentum equation can then be written with a first-order weighted advection term and a lower-order load, which fits naturally into $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$7-based virtual spaces.

All three fields are approximated in equal-order VEM spaces of degree $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$8, with local scalar space $$T_{\mathrm{rgb}\leftarrow \mathrm{lidar}^{t_1\leftarrow t_2}} = T_{(\mathrm{rgb}\leftarrow \mathrm{car})} \,T_{(t_1\leftarrow t_2)} \,T_{(\mathrm{car}\leftarrow \mathrm{lidar})}$$9 characterized by boundary polynomial traces, a polynomial Laplacian, and a constraint tying the non-polynomial part to its polynomial projection. The degrees of freedom are nodal values at vertices, edge values at $m_j$0 internal Gauss–Lobatto points, and internal moments up to order $m_j$1. Discrete operators are computed from the $m_j$2- and $m_j$3-projections $m_j$4 and $m_j$5, together with stabilization terms acting on the difference between a virtual function and its projection. The pressure stabilization is residual-based and PSPG-like; a grad-div term is also added. Because everything is expressed through projections, explicit basis functions are never needed inside the elements.

The formulation is designed for arbitrary polygonal meshes satisfying a star-shapedness and edge-length condition and explicitly supports distorted elements, non-convex polygons, Voronoi tessellations, and meshes with hanging nodes. Well-posedness is established for the continuous problem by Banach fixed-point arguments and for the discrete problem by Brouwer fixed-point arguments under small-data conditions. The a priori error estimate gives convergence of order $m_j$6 in the energy norm. Numerical experiments on convex and non-convex domains, mixed meshes with hanging nodes, and L-shaped Voronoi meshes confirm first-order convergence for $m_j$7 and second-order convergence for $m_j$8. In nanopore-flow simulations with T-shaped and curved obstacles, the method captures pressure drops, recirculation zones, and potential gradients on Voronoi meshes with 18,000 elements. Relative to Taylor–Hood FEM, the equal-order VEM uses up to approximately 43% fewer DOFs for $m_j$9 and about 15% fewer DOFs for $$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$0, depending on mesh topology.

5. Reparameterized 3D points in generalized absolute pose estimation

In multi-camera pose estimation, the virtual point formulation is a geometric reparameterization. The generalized absolute pose problem uses a ray origin $$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$1 and a unit-bearing vector $$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$2 for each observation, so the projection equation is

$$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$3

The central idea is to define a virtual point

$$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$4

which transforms the generalized model into

$$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$5

This equation is identical in form to the standard single-camera PnP model. The paper therefore treats the generalized camera as a monocular equivalent camera at the origin of the rig frame observing virtual points $$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$6, and the mapping $$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$7 is presented as an equivalence between GAP and PnP (Li et al., 8 Jun 2026).

The main consequence is algorithmic reuse. Instead of designing a generalized solver from scratch, the formulation shows how to transform an existing PnP solver into a generalized solver by replacing $$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$8 with the virtual point structure and adjusting the translation-elimination step. Three concrete solvers are derived in this way. VGPc is based on DLSU and uses a Cayley parameterization, leading to a quartic cost in the Cayley variables and three third-degree polynomial stationarity equations. VGPq is based on OPnP and uses a non-unit quaternion parameterization, yielding a fourth-degree polynomial cost and four cubic optimality equations while maintaining the global optimality guarantee of OPnP. VGPr is based on SQPnP and keeps the rotation-matrix parameterization, translation elimination, and QCQP structure, with refinement by sequential quadratic programming. The paper also extends the formulation to a generalized pose-with-scale model

$$F_j = \{(\mathbf{p}_i,d_i)\mid \mathbf{p}_i\in m_j\}.$$9

by augmenting the translation vector with a scale component.

The reported properties are differentiated rather than uniform. VGPc is the most accurate under heteroscedastic noise because it inherits the uncertainty-aware structure of DLSU. VGPq maintains global optimality. VGPr is the most efficient and is explicitly positioned for real-time multi-camera pose estimation. The formulation claims exact equivalence, inheritance of optimality properties from the underlying PnP solvers, and no additional local minima because the offsets $\tau$0 are absorbed into $\tau$1 without adding new nonlinear degrees of freedom in the rotation space.

The empirical results are correspondingly strong. In synthetic experiments with $\tau$2 points and heteroscedastic noise, VGPc achieves rotation error of about 0.29° and translation error 0.16% at $\tau$3, improving UPnP by about 33% in rotation and about 43% in translation. VGPr runs at about 0.08 ms at $\tau$4, approximately 5× faster than gDLS+++ and 12× faster than UPnP. Under increasing image noise, VGPc reports zero failures even at $\tau$5 px, whereas gDLS+++ and others fail frequently. On ETH3D Many-view and Oxford RobotCar, VGPc attains the lowest rotation and translation errors, with average error reductions relative to UPnP of about 23% in rotation and about 57% in translation at $\tau$6. On a real UAV-mounted five-camera system with non-overlapping fields of view, multi-camera configurations improve accuracy for all methods; under large field of view and almost isotropic noise, UPnP, VGPc, VGPq, and VGPr reach very similar centimeter-level accuracy, while VGPr remains the most efficient.

A notable limitation is conceptual rather than geometric: the virtual point

$\tau$7

cannot be precomputed directly because it depends on the unknown rotation $\tau$8. The solver therefore introduces symbolic coefficient matrices such as modified $\tau$9, $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$00, and derived matrices $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$01, $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$02, and $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$03, so that the effect of $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$04 is absorbed into algebraic coefficients while the core PnP machinery remains intact.

6. Virtual work, generalized Hamiltonian structure, and abstract phase-space interpretations

Delphenich’s generalized Hamiltonian formulation extends the principle of virtual work rather than introducing synthetic geometric samples. Its relevance to virtual point formulation is interpretive: a “virtual point” may be understood as a point in extended configuration space or phase space together with an infinitesimal virtual variation, and the primary operation is the pairing of a Pfaffian 1-form with that variation (Delphenich, 2022).

The starting point is a general 1-form on the first jet bundle,

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$05

where $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$06 is power, $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$07 are generalized forces, and $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$08 are generalized momenta. Given a curve $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$09 and a virtual displacement

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$10

the virtual work density is

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$11

Restricting to fixed parameterization and integrable variations $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$12, the total virtual work functional becomes

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$13

After integration by parts, the equations of motion follow from vanishing of total virtual work for all admissible variations: $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$14 When $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$15, this reduces to the Euler–Lagrange equations.

The generalized Hamiltonian structure is built on phase space with canonical symplectic form

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$16

A generalized Hamiltonian 1-form is introduced as

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$17

With a Pfaff normal form

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$18

the symplectic-dual vector field yields generalized Hamilton equations

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$19

The usual Poisson bracket

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$20

then appears in the generalized energy balance and in the decomposition of Lie brackets for arbitrary vector fields when their symplectic dual 1-forms are written in Pfaff normal form.

The generalized Hamilton–Jacobi theory likewise shifts emphasis from a single generating function to a contact field $$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$21. Pulling back the generalized 2-form yields the nonlinear first-order PDE system

$$\mathbf{x}_e,\mathbf{x}_w,\mathbf{x}_n,\mathbf{x}_s$$22

When the Pfaff correction vanishes, the standard Hamilton–Jacobi equation is recovered. The paper is mainly structural and does not work through detailed explicit mechanical examples, but it provides a rigorous geometric extension of virtual work to nonconservative and nonholonomic settings.

7. Comparative interpretation and domain-specific limitations

Taken together, these literatures suggest that a virtual point formulation is a recurrent methodological move rather than a single formalism. In multimodal perception, virtual points densify sparse geometry by lifting image evidence into LiDAR coordinates. In meshfree CFD, they create virtual staggered stencils on a single node set. In polygonal VEM, they localize unknowns at vertices, edge nodes, and moment functionals while replacing interior basis functions by projections and stabilization. In generalized pose estimation, they absorb camera offsets into reparameterized 3D points so that a noncentral camera becomes algebraically equivalent to a standard PnP system. In generalized virtual work, they correspond to points in extended state space endowed with admissible virtual variations (Yin et al., 2021, Park et al., 2014, Artioli et al., 2017, Mishra et al., 18 Feb 2026, Li et al., 8 Jun 2026, Delphenich, 2022).

The limitations are correspondingly domain-specific. The detection formulation depends on accurate calibration and synchronization, assumes nearest-neighbor depth completion in image space, has no explicit occlusion reasoning, and uses a low-dimensional semantic feature. The VIP method requires careful boundary-region stencil construction and is motivated by the instability of pure MLS collocation. The virtual-element formulations rely on projection operators and stabilization terms because the interior basis functions are never explicit; in the coupled Stokes–Poisson–Boltzmann setting, well-posedness and optimal error estimates are established only under sufficiently small data assumptions. The generalized Hamiltonian construction is local in character, with Pfaff normal forms and most formulas expressed in local coordinates. The generalized pose formulation cannot literally precompute its virtual points because they depend on the unknown rotation.

A plausible implication is that the chief value of virtual points lies in operator preservation. Each formulation keeps a desirable algebraic or numerical skeleton intact—voxelized LiDAR detection, staggered pressure–velocity coupling, polynomial consistency on polygonal elements, standard PnP least-squares structure, or Hamiltonian symplectic duality—while shifting geometric difficulty into auxiliary points, projections, or symbolic eliminations. Under that interpretation, virtual point formulation names a broad and technically productive strategy for reconciling rich geometry with solver structures that were originally designed for simpler settings.