Spatial Curvature Research Articles

Constraining dark energy cosmologies with spatial curvature using Supernovae JWST forecasting

Recent cosmological tensions, in particular, to infer the local value of the Hubble constant H 0, have developed new independent techniques to constrain cosmological parameters in several cosmologies. Moreover, even when the concordance Cosmological Constant Cold Dark Matter (ΛCDM) model has been well constrained with local observables, its physics has shown deviations from a flat background. Therefore, to explore a possible deviation from a flat ΛCDM model that could explain the H 0 value in tension with other techniques, in this paper we study new cosmological constraints in spatial curvature dark energy models. Additionally, to standard current Supernovae Type Ia (SNIa) catalogs, we extend the empirical distance ladder method through an SNIa sample using the capabilities of the James Webb Space Telescope (JWST) to forecast SNIa up to z ∼ 6, with information on the star formation rates at high redshift. Furthermore, we found that our constraints provide an improvement in the statistics associated with Ω m when combining SNIa Pantheon and SNIa Pantheon+ catalogs with JW forecasting data.

Clinical study on the treatment of postoperative recurrence of lumbar disc herniation with intervertebral fusion

To explore clinical effect of different intervertebral fusion devices (cage) in treating postoperative recurrent lumbar disc herniation (LDH). One hundred and forty-two LDH patients with recurrence after simple intervertebral disc nucleus pulpoideectomy from January 2019 to January 2021 were retrospectively analyzed. All patients were treated with combined underchannel fixation and interbody fusion and divided into a single anatomical group,two-anatomical group and a single banana group according to types and numbers of implanted cage. There were 51 patients in a single anatomical group,included 29 males and 22 females,aged from 39 to 65 years old with an average of (53.74±5.68) years old;body mass index (BMI) ranged from 18.62 to 28.13 kg·m-2 with an average of (22.08±2.15) kg·m-2;the interval between operation and recurrence ranged from 0.5 to 4.0 years with an average of (2.7±0.8) years;5 patients with L3,4,35 patients with L4,5 and 11 patients with L5S1;a single anatomical cage was implanted. There were 46 patients in two-anatomical group,included 25 males and 21 females,aged from 37 to 66 years old with an average of (54.52±6.02) years old;BMI ranged from 18.25 to 28.44 kg·m-2 with an average of (21.74±1.83) kg·m-2;the interval between operation and recurrence ranged from 0.5 to 5.0 years with an average of (2.7±0.9) years;4 patients with L3,4,32 patients with L4,5 and 10 patients with L5S1;two-anatomical cages were implanted. There were 45 patients in a single banana group,included 22 males and 23 females,aged from 38 to 65 years old with an average of (54.49±6.45) years old;BMI ranged from 18.85 to 28.20 kg·m-2 with an average of (21.63±1.59) kg·m-2;the interval between operation and recurrence ranged from 0.5 to 5.0 years with an average of (2.6±1.0) years;3 patients with L3,4,36 patients with L4,5 and 16 patients with L5S1;a single banana cage was implanted. Operation time,intraoperative blood loss,incision length,postoperative incision drainage volume,hospital stay and complications among 3 groups were observed and compared. The height of intervertebral space before and after operation,curvature of lordosis and the postoperative intervertebral fusion were compared. Visual analogue scale (VAS) and Oswestry disability index (ODI) were used to evaluate degree of lumbar pain and lumbar function before operation,1 and 6 months after operation,respectively. All patients among 3 groups were followed up at least 6 months,and no cases were fell out. There were no significant difference in operation time,intraoperative blood loss,incision length,postoperative incision drainage volume and hospital stay among 3 groups (P>0.05). At 6 months after operation,the height of intervertebral space in two-anatomical group and a single group were [(11.08±1.78) mm,(10.95±1.62) mm],curvature of lordosis were [(12.05±1.86) °,(11.63±1.57) °],which were higher than those in a single dissection group (10.14±1.54) mm,(10.92±1.45) °,and the difference were statistically significant (P<0.05). The interbody fusion rate between two-anatomical and a banana group (95.65%,95.56%) were higher than that in a single anatomical group (78.43%) at 6 months after operation (P<0.05). VAS and ODI of lumbar among 3 groups were decreased at 1 and 6 months after operation (P<0.05). There was no significant difference in complications among 3 groups (P>0.05). The three fusion devices could achieve significant results in treating postoperative recurrence of LDH,but the implantation of two-anatomical cage and a single banana cage are more helpful to maintain the height of intervertebral space and lordosis curvature of patients with postoperative recurrence of LDH,and obtain good intervertebral fusion results.

DistilVPR: Cross-Modal Knowledge Distillation for Visual Place Recognition

The utilization of multi-modal sensor data in visual place recognition (VPR) has demonstrated enhanced performance compared to single-modal counterparts. Nonetheless, integrating additional sensors comes with elevated costs and may not be feasible for systems that demand lightweight operation, thereby impacting the practical deployment of VPR. To address this issue, we resort to knowledge distillation, which empowers single-modal students to learn from cross-modal teachers without introducing additional sensors during inference. Despite the notable advancements achieved by current distillation approaches, the exploration of feature relationships remains an under-explored area. In order to tackle the challenge of cross-modal distillation in VPR, we present DistilVPR, a novel distillation pipeline for VPR. We propose leveraging feature relationships from multiple agents, including self-agents and cross-agents for teacher and student neural networks. Furthermore, we integrate various manifolds, characterized by different space curvatures for exploring feature relationships. This approach enhances the diversity of feature relationships, including Euclidean, spherical, and hyperbolic relationship modules, thereby enhancing the overall representational capacity. The experiments demonstrate that our proposed pipeline achieves state-of-the-art performance compared to other distillation baselines. We also conduct necessary ablation studies to show design effectiveness. The code is released at: https://github.com/sijieaaa/DistilVPR

Spatial quadric calibration method for multi-line laser based on diffractive optical element

The traditional multi-line laser calibration method relies primarily on calibrating the spatial plane equations of multiple laser lines and reconstructing the multi-line laser in three dimensions through the plane equation and camera parameters. In traditional methods, the multi-line laser is projected as a straight line by default, but in reality, the laser line projected by the multi-line laser is not a plane due to the Diffractive Optical Element (DOE). Instead, it is a surface with a certain curvature in space. During the production process, the multi-line laser experiences significant distortion because of the DOE projecting at a large angle. Consequently, if traditional methods are used, accurate calibration of the multi-line laser cannot be achieved. To address this issue, this paper introduces a new calibration algorithm for multi-line laser based on a spatial quadric equation. By calibrating the spatial quadric equation of the multi-line laser, the spatial characteristics of the laser can be accurately captured. The algorithm involves projecting a crossed 7-line laser onto a calibration plate and changing the position of the calibration plate while capturing images. This allows for obtaining the spatial three-dimensional coordinates of each laser line in the camera coordinate system. Next, spatial quadric equations are fitted for each laser line based on the three-dimensional coordinates. Finally, 3D reconstruction is performed using the quadratic surface parameters of the multi-line laser and camera parameters. The results from 3D reconstruction demonstrate that, compared to traditional laser calibration methods based on planes, the proposed algorithm achieves higher calibration accuracy. It also improves the number of successful binocular multi-line laser matches and overall accuracy.

Gravity: Where Quantum Physics and Classical Physics finally merge

In this 3rd paper of a triptych on quantum theory regarding gravity, quantum, and classical physics are merged seamlessly-however with conclusions deviating from classical physics regarding singularities, space-curvature, and graviton. Mathematically, singularities (Schwarzschild, Droste) vanish by proper definition of the gravity source and field descriptions acquiring full validity at the Planck scale. Space(-time) curvature is shown to be geodesic-trajectory curvature by (energy) objects in the field of (a) gravity source(s). The gravity field is found to be a scalar field in which the trajectories are defined only by the principle of least action (LaGrange, Feynman) by compensation of accelerations due to different physical causal sources of acceleration. The graviton is argued to be a massive Higgs-type scalar boson with ubiquitous presence since the creation of quantum and clustered mass in the universe: the influence of a settled gravity field on mass (in) entering this field, therefore, is instantaneous, i.e., unlike a vector boson description. Gravity waves, occurring in mergers/transitions of mass, are defined by changes in time of the gravity field (at max. c m/s) in space and only become observable when substantial mass/energy is involved due to the weakness and spatial decay of gravity fields. All mathematics (e.g. integral transformations, vector space, etc) and related operators in the paper are part of the Abelian group for validity at the Planck scale. Results thus constitute the description of gravity on all scales.‎‎

Emergence of cosmic space and the maximization of horizon entropy

The spatial expansion of the universe can be described as the emergence of space with the progress of cosmic time, through a simple equation, ΔV=ΔtNsurf−Nbulk. This law of emergence suggested by Padmanabhan in the context of general relativity for a flat universe has been generalized by Sheykhi to Gauss Bonnet and Lovelock gravity for a universe with any spatial curvature. We investigate whether this generalized holographic equipartition effectively implies the maximization of horizon entropy. First, we obtain the constraints imposed by the maximization of horizon entropy in Einstein, Gauss Bonnet and Lovelock gravities for a universe with any spatial curvature. We then analyze the consistency of the law of emergence in Ahmad Sheykhi (2013), with these constraints obtained. Interestingly, both the law of emergence and the horizon entropy maximization demands an asymptotically de Sitter universe with ω≥−1. More specifically, when the degrees of freedom in the bulk (Nbulk) becomes equal to the degrees of freedom on the boundary surface (Nsurf), the universe attains a state of maximum horizon entropy. Thus, the law of emergence can be viewed as a tendency for maximizing the horizon entropy, even in a non flat universe. Our results points at the deep connection between the law of emergence and horizon thermodynamics, beyond Einstein gravity irrespective of the spatial curvature.

The interacting vacuum energy models with spatial curvature: A dynamical system perspective with observational constraints

In this paper, we study the cosmic dynamics of varying vacuum models where the dark matter interacts with the vacuum energy. We consider the homogeneous and isotropic spacetime with spatial curvature and apply the dynamical system technique to the varying vacuum models by specifying the form of energy exchange rate [Formula: see text] between the dark energy and dark matter. Further, we utilize the cosmographic parameters and statefinder parameters in the terms of dynamical variables of the cosmological dynamical system to explore the cosmic dynamics of the universe. The Milne universe solution exists as a consequence of spatially curved geometry in the model. The models also yield radiation- matter- and dark energy-dominated phases in order and thus explain the late-time accelerated expansion of the universe. The strength of interaction terms will affect the existence of cosmological phases in the model but there will always be an attractor corresponding to the accelerating universe. We numerically solve the system to illustrate the evolution of cosmological quantities and dynamical variables in the models. The role of curvature is visible during the transition from the decelerated phase into the accelerating phase. The numerical solutions affirm that the cosmological parameters in the models are consistent with their corresponding observational values.

On the moduli space curvature at infinity

We analyse the scalar curvature of the vector multiplet moduli space MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document} of type IIA string theory compactified on a Calabi-Yau manifold X. While the volume of MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document} is known to be finite, cases have been found where the scalar curvature diverges positively along trajectories of infinite distance. We classify the asymptotic behaviour of the scalar curvature for all large volume limits within MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document}, for any choice of X, and provide the source of the divergence both in geometric and physical terms. Geometrically, there are effective divisors whose volumes do not vary along the limit. Physically, the EFT subsector associated to such divisors is decoupled from gravity along the limit, and defines a rigid N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 field theory with a non-vanishing moduli space curvature Rrigid. We propose that the relation between scalar curvature divergences and field theories decoupled from gravity is a common trait of moduli spaces compatible with quantum gravity.

Visualizing high-dimensional loss landscapes with Hessian directions

Analyzing the geometric properties of high-dimensional loss functions, such as local curvature and the existence of other optima around a certain point in loss space, can help provide a better understanding of the interplay between neural-network structure, implementation attributes, and learning performance. In this paper, we combine concepts from high-dimensional probability and differential geometry to study how curvature properties in lower-dimensional loss representations depend on those in the original loss space. We show that saddle points in the original space are rarely correctly identified as such in the expected lower-dimensional representations if random projections are used. The principal curvature in the expected lower-dimensional representation is proportional to the mean curvature in the original loss space. Hence, the mean curvature in the original loss space determines if saddle points appear, on average, as either minima, maxima, or almost flat regions. We use the connection between expected curvature in random projections and mean curvature in the original space (i.e. the normalized Hessian trace) to compute Hutchinson-type trace estimates without calculating Hessian-vector products as in the original Hutchinson method. Because random projections are not suitable for correctly identifying saddle information, we propose to study projections along the dominant Hessian directions that are associated with the largest and smallest principal curvatures. We connect our findings to the ongoing debate on loss landscape flatness and generalizability. Finally, for different common image classifiers and a function approximator, we show and compare random and Hessian projections of loss landscapes with up to approximately 7×106 parameters.

Varying vacuum models with spatial curvature: a dynamical system perspective

Varying vacuum models with spatial curvature: a dynamical system perspective

Crystal Thermal Transport in Altermagnetic RuO_{2}.

We demonstrate the emergence of a pronounced thermal transport in the recently discovered class of magnetic materials-altermagnets. From symmetry arguments and first-principles calculations performed for the showcase altermagnet, RuO_{2}, we uncover that crystal Nernst and crystal thermal Hall effects in this material are very large and strongly anisotropic with respect to the Néel vector. We find the large crystal thermal transport to originate from three sources of Berry's curvature in momentum space: the Weyl fermions due to crossings between well-separated bands, the strong spin-flip pseudonodal surfaces, and the weak spin-flip ladder transitions, defined by transitions among very weakly spin-split states of similar dispersion crossing the Fermi surface. Moreover, we reveal that the anomalous thermal and electrical transport coefficients in RuO_{2} are linked by an extended Wiedemann-Franz law in a temperature range much wider than expected for conventional magnets. Our results suggest that altermagnets may assume a leading role in realizing concepts in spin caloritronics not achievable with ferromagnets or antiferromagnets.

Towards quantum mechanics on the curved cotangent bundle

The minimal-length paradigm is a cornerstone of quantum gravity phenomenology. Recently, it has been demonstrated that minimal-length quantum mechanics can alternatively be described as an undeformed theory set on a nontrivial momentum space. However, there is no fully consistent formulation of these theories beyond Cartesian coordinates in flat space and, in particular, no position representation. This paper is intended to take the first steps in bridging this gap. We find a natural position representation of the position and momentum operators on general curved cotangent bundles. In an expansion akin to Riemann normal coordinates with curvature in both position and momentum space, we apply the formalism perturbatively to the isotropic harmonic oscillator and the hydrogenic atom. Due to the symmetry of the harmonic oscillator under exchange of positions and momenta, we show that it is impossible to distinguish position- from momentum-space curvature with oscillators alone. Thus, we obtain an instantiation of Born reciprocity on the curved cotangent bundle, i.e. in precisely the way Born originally envisioned. It manifests itself as a symmetry mixing UV and IR physics, reminiscent of T-duality in string theory.

N -cutoff regularization for fields on hyperbolic space

We apply a novel background independent regularization scheme, the N-cutoffs, to self-consistently quantize scalar and metric fluctuations on the maximally symmetric but noncompact hyperbolic space. For quantum matter fields on a classical background or full quantum Einstein gravity (regarded here as an effective field theory) treated in the background field formalism, the N-cutoff is an ultraviolet regularization of the fields’ mode content that is independent of the background hyperbolic space metric. For each N&gt;0, the regularized system backreacts on the geometry to dynamically determine the self-consistent background metric. The limit in which the regularization is removed then automatically yields the “physically correct” spacetime on which the resulting quantum field theory lives. When self-consistently quantized with the N-cutoff, we find that without any fine-tuning of parameters, the vacuum fluctuations of scalar and (linearized) graviton fields do not lead to the usual cosmological constant problem of a curvature singularity. Instead, the presence of increasingly many field modes tends to reduce the negative curvature of hyperbolic space, leading to vanishing values in the limit of removing the cutoff. Published by the American Physical Society 2024

From Hubble to snap parameters: a Gaussian process reconstruction

ABSTRACT By using recent H(z) and supernovae Type Ia (SNe Ia) data, we reconstruct the evolution of kinematic parameters H(z), q(z), jerk, and snap, using a model-independent, non-parametric method, namely, the Gaussian processes. Throughout the present analysis, we have allowed for a spatial curvature prior, based on Planck 18 constraints. In the case of SNe Ia, we modify a python package (gapp) in order to obtain the reconstruction of the fourth derivative of a function, thereby allowing us to obtain the snap from comoving distances. Furthermore, using a method of importance sampling, we combine H(z) and SNe Ia reconstructions in order to find joint constraints for the kinematic parameters. We find for the current values of the parameters: H0 = 67.2 ± 6.2 km s−1 Mpc−1, $q_0 = -0.54^{+0.06}_{-0.05}$, $j_0=0.94^{+0.20}_{-0.18}$, and $s_0=-0.62^{+0.26}_{-0.25}$ at 1σ c.l. We find that these reconstructions are compatible with the predictions from flat lambda-cold dark matter model, at least for 2σ confidence intervals.

Gravitational entropy and the flatness, homogeneity and isotropy puzzles

We suggest a new explanation for the observed large scale flatness, homogeneity and isotropy of the universe. The basic ingredients are Einstein's theory of gravity and Hawking's method of computing the gravitational entropy. The new twist is provided by the boundary conditions we recently proposed for “big bang” type singularities dominated by conformal matter, enforcing CPT symmetry and analyticity. Besides allowing us to describe the big bang, these boundary conditions allow new gravitational instantons from which we calculate the total gravitational entropy Sg of cosmologies including radiation, dark energy and space curvature of either sign. We find Sg∼SΛ1/4Sr, where SΛ is the de Sitter entropy and Sr is the total entropy in radiation, which is extensive in the comoving spatial volume. Thus, Sg can be larger than SΛ and we show it is largest for universes which are spatially flat. Extending our analysis to include linearized perturbations, we show that, for a given set of globally conserved quantities, Sg is greatest for universes that are also homogeneous and isotropic on large scales.

Thermodynamic solution of the homogeneity, isotropy and flatness puzzles (and a clue to the cosmological constant)

We obtain the analytic solution of the Friedmann equation for fully realistic cosmologies including radiation, non-relativistic matter, a cosmological constant λ and arbitrary spatial curvature κ. The general solution for the scale factor a(τ), with τ the conformal time, is an elliptic function, meromorphic and doubly periodic in the complex τ-plane, with one period along the real τ-axis, and the other along the imaginary τ-axis. The periodicity in imaginary time allows us to compute the Hawking temperature and the global gravitational entropy of cosmological spacetimes, just as Gibbons, Hawking and Perry did for black holes and de Sitter spacetime. The global gravitational entropy favours universes like our own which are spatially flat, homogeneous, and isotropic, with a small positive cosmological constant.

Simultaneous measurements on cosmic curvature and opacity using latest HII regions and H(z) observations

The different spatial curvatures of the universe affect the measurement of cosmological distances, which may also contribute to explaining the observed dimming of type Ia supernovae. This phenomenon may be caused by the opacity of the universe. Similarly, the opacity of the universe can lead to a bias in our measurements of curvature. Thus, it is necessary to measure cosmic curvature and opacity simultaneously. In this paper, we propose a new model-independent method to simultaneously measure the cosmic curvature and opacity by using the latest observations of HII galaxies acting as standard candles and the latest Hubble parameter observations. The machine learning method-Artificial Neural Network is adopted to reconstruct observed Hubble parameter H(z) observations. Our results support a slightly opaque and flat universe at 1sigma confidence level by using previous 156 HII regions sample. However, the negative curvature is obtained by using the latest 181 HII regions sample in the redshift range zsim 2.5. More importantly, we obtain the simultaneous measurements with precision on the cosmic opacity mathrm Delta tau sim 10^{-2} and curvature mathrm Delta Omega _Ksim 10^{-1}. A strong degeneracy between the cosmic opacity and curvature parameters is also revealed in this analysis.

Viral capsid structural assembly governs the reovirus binding interface to NgR1.

Understanding the mechanisms underlying viral entry is crucial for controlling viral diseases. In this study, we investigated the interactions between reovirus and Nogo-receptor 1 (NgR1), a key mediator of reovirus entry into the host central nervous system. NgR1 exhibits a unique bivalent interaction with the reovirus capsid, specifically binding at the interface between adjacent heterohexamers arranged in a precise structural pattern on the curved virus surface. Using single-molecule techniques, we explored for the first time how the capsid molecular architecture and receptor polymorphism influence virus binding. We compared the binding affinities of human and mouse NgR1 to reovirus μ1/σ3 proteins in their isolated form, self-assembled in 2D capsid patches, and within the native 3D viral topology. Our results underscore the essential role of the concave side of NgR1 and emphasize that the spatial organization and curvature of the virus are critical determinants of the stability of the reovirus-NgR1 complex. This study highlights the importance of characterizing interactions in physiologically relevant spatial configurations, providing precise insights into virus-host interactions and opening new avenues for therapeutic interventions against viral infections.

QUANTUM PARTICLE OF MATTER (CHARGE) AS A FUNDAMENTAL SOURCE OF GRAVITY AND MASS OF MATTER IN NATURE

One of the most intriguing problems of modern physics is the search of particles beyond the Standard Model and the creation of quantum theory of gravity. To these problems, namely to the problem of quantum description of gravitation the closest attention is paid. The importance of applying the Standard Model to describe three types of force physical interactions involving elementary particles is beyond doubt. These particles are active participants of physical processes of electromagnetism, strong and weak interactions. However, there is still an open question about the meaning and role in nature of gravitation. Moreover, it is not clear how gravity interacts with matter of any mass and what is the essence of these differences. Also, there is no answer to the question about the meaning in the nature of the discreteness factor and the possibility of real quantisation of gravitation. Besides, it is unknown whether in the surrounding world there exists a quantum of gravitational energy – graviton and what is its actual energy. To explain the significance of gravity on cosmic scales, the General Theory of Relativity of A. Einstein was applied, which describes the curvature of space around stars and planets. However, this theory affects only the interaction of large masses of matter and does not apply to quantum objects in the form of atoms and elementary particles. As for the objective application in the scientific literature of other known theories of gravity, including its quantum forms, they all use a mathematical apparatus where the criterion of quantum discreteness – Planck’s constant – is missing. The apparent incompleteness of research in the field of development and creation of an objective theory of gravitation and its quantum forms requires practical application of the existing apparatus of quantum mechanics for realisation of this scientific direction. Taking into account such a view, it is necessary to consider the possibility of applying the thermodynamic nature of the movement of gravity in nature, where it is necessary to raise the issue of the existence and manifestation of the quantum form of gravitational energy in nature, the source of which may be a quantum particle of matter (charge) outside the Standard Model. Due to the unique coincidence of physical properties of quantum particles of matter (charge) and physical principles inherent in the operation of any collider, it is possible to use the Large Hadron Collider to study the properties of quantum particles of matter (charge) as a source of gravity in nature. Keywords: gravity, quantum particle of matter (charge), energy and Planck’s constant, Schrödinger equation, Heisenberg uncertainty, graviton, oscillator of gravity of matter (charge), de Broglie wave, laws of gravity, converging effect of gravitational fields, standard model, collider

A Novel Radial Basis Function Description of a Smooth Implicit Surface for Musculoskeletal Modelling

As musculoskeletal illnesses continue to increase, practical computerised muscle modelling is crucial. This paper addresses this concern by proposing a mathematical model for a dynamic 3D geometrical surface representation of muscles using a Radial Basis Function (RBF) approximation technique. The objective is to obtain a smoother surface while minimising data use, contrasting it from classical polygonal (e.g. triangular) surface mesh models or volumetric (e.g. tetrahedral) mesh models. The paper uses RBF implicit surface description to describe static surface generation and dynamic surface deformations based on its spatial curvature preservation during the deformation. The novel method is tested on multiple data sets, and the experiments show promising results according to the introduced metrics.