Virtual Geometries Overview

• Virtual Geometries are non-standard spatial constructions that extend beyond traditional Euclidean frameworks to redefine immersive VR, gravitational thermodynamics, synthetic structures, and numerical discretizations.
• Key applications include immersive VR simulations in hyperbolic, product, and Thurston geometries, enabling tangible experiences of curvature, holonomy, and anisotropy.
• They also inform abstract frameworks in gravitational, synthetic, and computational contexts, leveraging off-shell configurations, virtual invariants, and adaptive discretization methods.

Across the literature surveyed here, “virtual geometries” names several distinct constructions in which geometry is not treated as an ordinary fixed Euclidean backdrop. In immersive visualization, it denotes virtual-reality systems whose ambient space is itself non-Euclidean, such as $\mathbb H^3$, $\mathbb H^2\times E$, $\mathsf{Nil}$, or $\mathsf{Sol}$ (Hart et al., 2017, Hart et al., 2017, Coulon et al., 2020, Coulon et al., 2020). In gravitational thermodynamics, it denotes off-shell black-hole-like geometries that preserve asymptotics and a horizon while not necessarily satisfying the field equations (Dumitru et al., 24 Jul 2025). In synthetic and algebro-geometric settings, it denotes virtual or motivic classes attached to incidence geometries and Quot schemes (Thas, 2022, Arbesfeld et al., 2020). In numerical analysis, it denotes discretization frameworks in which geometric flexibility is obtained through virtual element spaces, virtual local stencils, or virtual interpolation points rather than explicit classical shape functions or staggered meshes (Huyssteen et al., 2022, Park et al., 2014).

1. Principal meanings of the term

The term is used in several technically distinct ways.

Research area	Meaning of “virtual geometries”	Representative papers
Non-Euclidean VR	Intrinsic inhabitation of non-Euclidean spaces in VR	(Hart et al., 2017, Coulon et al., 2020, Skrodzki et al., 2024)
Black-hole thermodynamics	Off-shell geometries with fixed asymptotics and a horizon	(Dumitru et al., 24 Jul 2025)
Synthetic/motivic geometry	Grothendieck-ring or virtual-invariant treatments of geometry	(Thas, 2022, Arbesfeld et al., 2020)
Numerical discretization	Geometry-flexible methods using virtual spaces or virtual stencils	(Veiga et al., 2019, Huyssteen et al., 2022, Park et al., 2014)

The earliest strand in this collection is the “Non-Euclidean Virtual Reality” program, which treats VR as a medium for intrinsic exploration of homogeneous geometries rather than for external depiction of Euclidean models of them (Hart et al., 2017, Hart et al., 2017). A later survey explicitly frames this as a broader program extending toward the eight Thurston geometries and emphasizes embodiment, stereoscopy, and head-tracked motion as the decisive difference from screen-based illustration (Skrodzki, 2020). Other works broaden the same idea toward hyperbolic interfaces for higher-dimensional grids and 2D engines that permit runtime variation of curvature (Kopczyński et al., 2021, Osudin et al., 2019).

A different usage appears in gravity, where “virtual geometries” are deformations of black-hole spacetimes that preserve asymptotic boundary conditions and the existence of a horizon but are not required to satisfy Einstein’s equations (Dumitru et al., 24 Jul 2025). Another appears in synthetic geometry, where a Grothendieck ring $K_0(Q_\ell)$ is introduced for generalized quadrangles and related incidence geometries, with virtual classes defined by scissor relations and graph-theoretic products (Thas, 2022). In the moduli theory of Quot schemes, virtual $K$-theoretic and cobordism invariants supply yet another virtual-geometric layer (Arbesfeld et al., 2020). Taken together, these literatures suggest a common theme: geometry is retained, extended, or operationalized through structures that are intrinsic but not reducible to ordinary Euclidean realization.

2. Intrinsic virtual reality in constant-curvature and product geometries

In the initial VR formulation, the simulated space is not Euclidean $E^3$ but three-dimensional hyperbolic space $\mathbb H^3$, represented computationally by the hyperboloid model in Minkowski space $E^{3,1}$ (Hart et al., 2017). In $d$ dimensions the model is

$\mathbb H^2\times E$0

with the induced metric from Minkowski space. For rendering, the implementation relies primarily on the hyperboloid model for transformations and the Klein model for display, since geodesics become straight Euclidean segments there and inverse hyperbolic trigonometric evaluations can often be avoided (Hart et al., 2017).

The motion update is expressed as a global isometric action rather than an ordinary Euclidean camera motion. The user is kept at the origin $\mathbb H^2\times E$1, and the world is updated by an element of $\mathbb H^2\times E$2 determined by the headset displacement $\mathbb H^2\times E$3. The infinitesimal generator is

$\mathbb H^2\times E$4

with finite update

$\mathbb H^2\times E$5

This is not merely a numerical device; it realizes locomotion as a Lie-group action by hyperbolic isometries (Hart et al., 2017).

To provide landmarks, $\mathbb H^2\times E$6 is decorated with the regular honeycomb $\mathbb H^2\times E$7, in which six cubes meet around each edge rather than the Euclidean four of $\mathbb H^2\times E$8. The paper further studies truncations whose exposed triangular faces lie on horospheres, allowing users to encounter Euclidean triangular tilings as induced geometry inside hyperbolic space (Hart et al., 2017). This supports one of the central experiential claims of the work: tracked VR makes curvature, geodesic divergence, and parallel transport bodily perceptible.

The companion system for $\mathbb H^2\times E$9 retains hyperbolic horizontal motion while preserving a Euclidean vertical direction (Hart et al., 2017). The model sits in $\mathsf{Nil}$0 as

$\mathsf{Nil}$1

with parametrization

$\mathsf{Nil}$2

Here the implementation cannot rely on the Klein-model shortcut used in $\mathsf{Nil}$3; instead it computes the inverse exponential map explicitly so that sight rays in tangent space agree with actual geodesics of the product geometry (Hart et al., 2017). This is the paper’s main technical distinction. It also exposes anisotropy directly: height in the Euclidean direction scales linearly, while width in the hyperbolic directions scales exponentially, so object aspect ratios change as one approaches them (Hart et al., 2017).

3. Thurston geometries, perceptual effects, and navigable non-Euclidean worlds

Subsequent work extends the same intrinsic VR program to the anisotropic Thurston geometries $\mathsf{Nil}$4 and $\mathsf{Nil}$5, where inverse-exponential rendering is no longer practical and geodesic ray marching becomes necessary (Coulon et al., 2020, Coulon et al., 2020). In $\mathsf{Nil}$6, modeled as the Heisenberg group with metric

$\mathsf{Nil}$7

geodesics can spiral, conjugate points occur along the vertical axis, and distant objects can generate ring-shaped multiple images (Coulon et al., 2020). In $\mathsf{Nil}$8, modeled as $\mathsf{Nil}$9 with group law

$\mathsf{Sol}$0

and metric

$\mathsf{Sol}$1

rays must follow Sol geodesics, some of which make visual “u-turns,” so horizontal planes can appear rolled into tubes and rays can strike the same plane twice (Coulon et al., 2020).

A recurring misconception in this literature is that such effects are rendering bugs. The papers state the opposite. In $\mathsf{Sol}$2, the apparent dropping of the floor under forward motion is a manifestation of geodesic divergence, and the apparent rotation of the world after a loop is an instance of parallel transport or holonomy (Hart et al., 2017). The 2020 survey makes the same point across $\mathsf{Sol}$3 and $\mathsf{Sol}$4: immersive VR exposes geodesic divergence, holonomy, isotropy, and anisotropy as sensorimotor facts rather than as diagrammatic abstractions (Skrodzki, 2020).

The HOLONOMY system pushes this further by using the order-5 square tiling of the hyperbolic plane together with a finite $\mathsf{Sol}$5 meter walk area subdivided into nine $\mathsf{Sol}$6 meter cells (Skrodzki et al., 2024). Because five squares meet at a vertex in the virtual tiling, a physical four-turn loop does not close virtually; this holonomy effect allows a small Euclidean room to access an infinite hyperbolic world without teleportation (Skrodzki et al., 2024). The implementation uses a discrete graph of the tiling as its primary state space, a Poincaré disk mini-map for display, and Minkowski hyperboloid coordinates for direction indication. Shortest-path planning is performed on a lazily generated state graph by A$\mathsf{Sol}$7, with the state including virtual location, move-area position, and rotation (Skrodzki et al., 2024).

Related work uses hyperbolic tessellations not to simulate hyperbolic space itself but to navigate higher-dimensional grids. The construction labels cells of the tessellation $\mathsf{Sol}$8 by points of $\mathsf{Sol}$9 so that screen adjacency coincides with grid adjacency, and extends to hyperbolic honeycombs in $K_0(Q_\ell)$0 for immersive display (Kopczyński et al., 2021). A separate 2D engine for spherical and hyperbolic worlds uses polar coordinates together with azimuthal equidistant projection precisely because the same projection rule can be used for $K_0(Q_\ell)$1, $K_0(Q_\ell)$2, and $K_0(Q_\ell)$3, making runtime curvature changes possible without rebuilding the world representation (Osudin et al., 2019).

4. Off-shell virtual geometries in black-hole thermodynamics

In black-hole thermodynamics, “virtual geometries” denotes a family of off-shell deformations that preserve asymptotic boundary conditions and the existence of a horizon, but do not necessarily satisfy the Einstein equations and may have arbitrary temperature (Dumitru et al., 24 Jul 2025). The framework begins from the static spherically symmetric ansatz

$K_0(Q_\ell)$4

with $K_0(Q_\ell)$5 defined by $K_0(Q_\ell)$6. After Wick rotation, regularity of the Euclidean section fixes

$K_0(Q_\ell)$7

and this assignment of temperature does not require the field equations (Dumitru et al., 24 Jul 2025).

The central object is the virtual thermodynamic potential

$K_0(Q_\ell)$8

obtained by evaluating the Euclidean action on the virtual geometry in the grand canonical ensemble, with horizon radius $K_0(Q_\ell)$9 treated as an independent order parameter (Dumitru et al., 24 Jul 2025). Its differential yields

$K$0

so the modified first law becomes

$K$1

The paper identifies the extra term with the “virtual work” term and proves that

$K$2

is proportional to the horizon Einstein equation. Physical equilibria are therefore selected by the off-shell extremality condition

$K$3

The same framework is then used for criticality. The paper proposes the simultaneous conditions

$K$4

as the black-hole analogue of Landau criteria for a critical endpoint, and applies them to a generalized Kaluza–Klein hairy black hole with dilaton potential (Dumitru et al., 24 Jul 2025). In that example the formalism yields a grand-canonical critical point and an inverted swallowtail structure in the free energy. In this usage, virtual geometry is not immersive or visual but thermodynamic: it is an off-shell configuration space on which a Landau–Ginzburg-type potential becomes well defined.

5. Synthetic, motivic, and virtual-invariant geometries

A synthetic use of the term appears in the construction of a Grothendieck ring $K$5 for generalized quadrangles and related incidence geometries (Thas, 2022). The paper defines $K$6 by starting from thick generalized quadrangles with $K$7 points per line, closing under finite products and disjoint unions, and then adjoining point-line subgeometries of such objects. A synthetic Zariski topology is introduced via prime geometries and closed sets, and the Grothendieck ring is defined as the free abelian group on isomorphism classes modulo scissor relations

$K$8

with multiplication

$K$9

where $E^3$0 comes from the Cartesian product of collinearity graphs (Thas, 2022). The paper also defines a synthetic Krull dimension through chains of prime geometries and proves, for example, that thick generalized quadrangles have dimension at least $E^3$1. Here virtual geometry means that incidence geometries are studied through motivic classes rather than through direct coordinatization.

A related but distinct algebro-geometric virtualization appears in the virtual $E^3$2-theory of Quot schemes of surfaces (Arbesfeld et al., 2020). For $E^3$3, the paper uses the canonical $E^3$4-term perfect obstruction theory with virtual tangent class

$E^3$5

and studies generating series of virtual Euler characteristics, virtual Segre classes, virtual Verlinde numbers, and virtual cobordism classes (Arbesfeld et al., 2020). The main rationality conjecture states that the generating series of virtual $E^3$6-theoretic invariants are rational functions of $E^3$7, and the paper proves this in several cases, including punctual quotients on all smooth projective surfaces and dimension-$E^3$8 quotients on surfaces with $E^3$9 (Arbesfeld et al., 2020).

The same work also shows that rationality is not universal across all virtual invariants: the generating series of virtual cobordism classes can be irrational (Arbesfeld et al., 2020). It further establishes a virtual Segre/Verlinde correspondence in three settings, including punctual Quot schemes, and proves a new symmetry exchanging the rank $\mathbb H^3$0 of a $\mathbb H^3$1-theory class with the ambient rank $\mathbb H^3$2 for punctual Quot schemes of the trivial sheaf (Arbesfeld et al., 2020). These results place virtual geometry in a fully algebro-geometric setting: geometry is encoded by virtual fundamental classes, virtual structure sheaves, and universal generating functions rather than by spatial models.

6. Virtual geometries in numerical discretization

In numerical analysis, virtual geometry is realized through methods that permit general polygonal or polyhedral cells while avoiding explicit interior basis functions. For the Virtual Element Method, a first-order conforming local space on a polygon $\mathbb H^3$3 is written as

$\mathbb H^3$4

and the practical computation is driven by projections and stabilization rather than by explicit interior shape functions (Huyssteen et al., 2022). This is the basis for adaptive refinement on arbitrary polygonal and Voronoi meshes, where local remeshing is performed by Voronoi submeshing inside a marked polygon and followed by edge-node optimization to improve compatibility and suppress short edges (Huyssteen et al., 2022). The same geometric flexibility motivates a complementary coarsening strategy in which patches of elements are merged into a single coarse polygon, with eligibility tests preserving non-convex corners and holes and with mean value coordinates used to relocate trapped nodes (Huyssteen et al., 2023).

Curved geometry is treated exactly in several VEM extensions. One construction for polygons with one curved edge defines the trace on the curved edge as the restriction of a two-dimensional polynomial rather than as a polynomial in the edge parameter, thereby preserving

$\mathbb H^3$5

and hence the patch test of order $\mathbb H^3$6 (Veiga et al., 2019). A solid-mechanics variant replaces the direct vector extension of the scalar curved space with a new edge space

$\mathbb H^3$7

where $\mathbb H^3$8 contains rigid body motions and $\mathbb H^3$9 supplies the higher-order mapped-polynomial part (Artioli et al., 2019). Another development extends VEM to axisymmetric elasticity and plasticity by augmenting the projected meridional strain with the hoop strain $E^{3,1}$0, using mean value coordinates to evaluate the needed centroidal shape-function values on arbitrary polygons (Yaw, 2023). Mixed VEM for Darcy flow with curved interfaces likewise retains the exact curved geometry of internal and boundary interfaces in two and three dimensions to avoid geometry error in flux-dominated applications (Dassi et al., 2020).

A parallel, meshfree use of virtual geometry appears in the Virtual Interpolation Point method for incompressible Navier–Stokes flow (Park et al., 2014). There the “virtual staggered structure” is an imaginary geometric arrangement of virtual interpolation points and a virtual local stencil around each physical node. All variables are stored at one set of physical nodes, but directional differences for convection, divergence, and pressure gradients are computed at virtual east, west, north, and south points reconstructed by moving least squares (Park et al., 2014). This replaces explicit staggered or body-fitted mesh geometry with a computational surrogate. The broader implication is that, in discretization theory, virtual geometries provide a way to separate physical geometry from computational geometry while still retaining exact or near-exact geometric information where it matters most.