Arc Length Parameter Research Articles

Fleshing: Spine‐driven Bending with Local Volume Preservation

AbstractSeveral design and animation techniques use a one‐dimensional proxy C (a spine curve in 3D) to control the deformation or behavior of a digital model of a 3D shape S. We propose a modification of these “skinning” techniques that ensures local volume preservation, which is important for the physical plausibility of digital simulations. In the proposed “fleshing” techniques, as input, we consider a smooth spine C0, a model S0 of a solid that lies “sufficiently close” to C0, and a deformed version C1 of C0 that is “not overly bent”. (We provide a precise characterization of these restrictions.) As output, we produce a bijective mapping M, that maps any point X of S onto a point M(X) of M(S). M satisfies two properties: (1) The closest projection of X on C0 and of M(X) on C1 have the same arc length parameter. (2) U and M(U) have the same volume, where U is any subset of S. We provide three different closed form expressions for radial, normal and binormal fleshing and discuss the details of their practical real‐time implementation.

The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at singledead-endpoint, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

An adaptive feedrate scheduling method of dual NURBS curve interpolator for precision five-axis CNC machining

Non-uniform rational b-spline (NURBS) tool path is becoming more and more important due to the increasing requirement for machining geometrically complex parts. However, NURBS interpolators, particularly related to five-axis machining, are quite limited and still keep challenging. In this paper, an adaptive feedrate scheduling method of dual NURBS curve interpolator with geometric and kinematic constraints is proposed for precision five-axis machining. A surface expressed by dual NURBS curves, which can continuously and accurately describe cutter tip position and cutter axis orientation, is first used to define five-axis tool path. For the given machine configuration, the calculation formulas of angular feedrate and geometric error aroused by interpolation are given, and then, the adaptive feedrate along the tool path is scheduled with confined nonlinear geometric error and angular feedrate. Combined with the analytical relations of feed acceleration with respect to the arc length parameter and feedrate, the feed profiles of linear and angular feed acceleration sensitive regions are readjusted with corresponding formulas and bi-directional scan algorithm, respectively. Simulations are performed to validate the feasibility of the proposed feed scheduling method of dual NURBS curve interpolator. It shows that the proposed method is able to ensure the geometric accuracy and good machining performances in five-axis machining especially in flank machining.

Why and how to objectively evaluate medical gestures?

Junior physicians mainly learn during their observation in the operative room. The senior physicians evaluate them based on the same kind of observation. Knowledge transfer evaluation is thus done without quantitative methods and it mainly lies on a subjective assessment. In this paper, we present some recent techniques used to objectively evaluate medical gestures. The classical techniques used are Hidden Markov Models (HMM) or Dynamic Time Warping (DTW). Both techniques lies on the temporal analysis of the gestures. We proposed here a technique based on the arc length parametrization in order to analyze the gestures in space which is more appropriate because it gives information about the shape of the gestures independently of the chosen coordinate system.

NURBS Curve Fitting Based on Arc Centripetal Parameterization for Tool Path Generation

NURBS curve fitting is used for tool path planning for sculptured surface NC machining. The cutter location data is parameterized by equal chord arc length parameterization, and Centripetal Parameterization is improved. It is not only more approach to curves nature equation in theory, but also closer to the interpolated curves in actual fitting. It is directly to reflect the curvature of curves of cutter location in NC machining.

Correlation of anterior segment optical coherence tomography measurements with graft trephine diameter following descemet stripping automated endothelial keratoplasty

BackgroundTo assess repeatability of the Zhongshan Assessment Program (ZAP) software measurement of Anterior Segment Optical Coherence Tomography (ASOCT) images and correlate with graft trephine diameter following Descemet Stripping Automated Endothelial Keratoplasty (DSAEK)MethodsRetrospectively evaluated interventional case series. 121 consecutive eyes undergoing DSAEK over a 26 month period underwent ASOCT imaging 1month after their surgery. ASOCT images were processed using ZAP software which measured the graft and cornea parameters including anterior and posterior graft arc length and cord length, posterior cornea arc length (PCAL) and anterior chamber width.ResultsThe graft measurements showed good repeatability on ASOCT using ZAP with high intra class coefficient and small variation in the coefficient of variation. On ASOCT, the mean recipient PCAL was 12.99+/−0.69mm and the anterior chamber width was 11.16+/−0.57mm. The mean Graft anterior arc length was 9.69+/−0.66mm and the mean Graft anterior cord length was 8.92+/−2.94mm. The mean graft posterior arc length was 9.24+/−0.75mm and the mean graft posterior cord length was 8.15+/−0.57mm. Graft posterior arc length (rho=0.788, p< 0.001) correlated best with intra-operative graft trephine diameter. The mean ratio of posterior graft arc length to PCAL was 0.712 +/− 0.056.ConclusionsWe have validated the repeatability of the ZAP software for DSAEK graft measurements from ASOCT images and shown that the graft arc length parameters calculated from the ASOCT images correlate well with the intra-operative graft trephine diameter. This software may help surgeons determine the optimal DSAEK graft size based on pre-operative ASOCT measurements of the recipient eye.

Piecewise B-Spline Tool Paths With the Arc-Length Parameter and Their Application on High Feed, Accurate CNC Milling of Free-Form Profiles

To conduct B-spline curve machining, first, B-spline tool paths with feed rates are planned; and second, the B-spline interpolator generates tool trajectories in real-time based on the paths fed into the computer numerically controlled (CNC) controller. Currently, the paths are often planned geometrically with a nonarc-length parameter. Literally, the interpolator can process B-spline paths with the arc-length parameter well, while it sometimes is challenged to work with the nonarc-length parameterized B-spline paths. As a consequence, it is difficult to ensure high accuracy of the tool trajectories in B-spline machining in terms of their corresponding paths; especially, if the feed is very high, smooth tool kinematics cannot be well maintained. To root out these problems, a new type of tool path—piecewise B-spline tool paths with the arc-length parameter—is first proposed in this work. Given a B-spline path with a nonarc-length parameter, it is accurately converted into a B-spline path with an arc-length parameter before sending it into the CNC controller. Furthermore, if the prescribed feed rate is very high and the arc-length parameterized B-spline path is disqualified, it is split into pieces represented with distinct arc-length parameterized B-spline paths in different feed rates. The main advantage of these piecewise paths is that they can eliminate the problems encountered by the existing B-spline interpolator with input of nonarc-length parameterized B-spline paths. Therefore, the piecewise arc-length parameterized B-spline paths are a genuine solution to high feed-and-accuracy B-spline machining.

The behavior of curvature functions at cusps and inflection points

At a 3/2-cusp of a given plane curve γ(t), both of the Euclidean curvature κg and the affine curvature κA diverge. In this paper, we show that each of |sg|κg and (sA)2κA (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a C∞-function of the variable t, where sg (resp. sA) is the Euclidean (resp. affine) arclength parameter of the curve corresponding to the 3/2-cusp sg=0 (resp. sA=0). Moreover, we give a characterization of the behavior of the curvature functions κg and κA at 3/2-cusps. On the other hand, inflection points are also singular points of curves in affine geometry. We give a similar characterization of affine curvature functions near generic inflection points. As an application, new affine invariants of 3/2-cusps and generic inflection points are given.

Arc-Length Parameterization of Linear Features for Photogrammetric Tasks

Arc-Length Parameterization of Linear Features for Photogrammetric Tasks

TANGENT-POINT SELF-AVOIDANCE ENERGIES FOR CURVES

We study a two-point self-avoidance energy [Formula: see text] which is defined for all rectifiable curves in ℝn as the double integral along the curve of 1/rq. Here r stands for the radius of the (smallest) circle that the is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of [Formula: see text] for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle 𝕊1 or to a closed interval I. For q &gt; 2 the energy [Formula: see text] evaluated on curves in ℝ3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in ℝ3 with finite [Formula: see text]-energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q crit = 2, the arclength parametrization of γ is of class C1, 1-2/q, with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 - 2/q is optimal.

One-dimensional gratings evolving through high-temperature annealing: sine-generated solutions

Sine-generated curves (i.e. curves in which the curvature is a sine function of the arc-length parameter) have been used in the past to describe river meanders. Here we show how these curves spontaneously appear during the decay of high-aspect-ratio surfaces mediated by surface diffusion. We obtained analytical results for the kinetic evolution of such processes relevant to a wide class of initial geometries. Our theoretical results were satisfactorily compared with numerical simulations and with results from previous approaches to the same problem, and they can be useful for interpreting and designing experiments related to the technologically important process of high-temperature annealing on nano/micro-structured samples.

A completely integrable flow of star-shaped curves on the light cone in Lorentzian

In this paper, we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian —invariants under the centro-affine action of the Lorentzian group—is Poisson equivalent to the space of conformal differential invariants for curves in the Möbius sphere. We use this relation to find realizations of solutions of a complexly coupled system of KdV equations as flows of curves in the cone.In memory of Ron Peters

EXTERNAL FIELD INFLUENCE ON SEMIFLEXIBLE MACROMOLECULES: GEOMETRIC COUPLING

We suggested a geometric approach to address the external field influence on the DNA molecules, described by the WLC model via geometric coupling. It consists in the introduction of the effective metrics depending on the potential of the external field, with further re-definition of the arc-length parameter and of the extrinsic curvatures of the DNA molecules. It yields the nontrivial impact of the external field in the internal energy of macromolecules. We give the Hamiltonian formulation of this model and perform its preliminary analysis in the redefinition of the initial energy density.

On meromorphic parameterizations of real algebraic curves

A singular flat geometry may be canonically assigned to a real algebraic curve Γ; namely, via analytic continuation of unit speed parameterization of the real locus \({\Gamma_\mathbb{R}}\) . Globally, the metric \({\rho=|Q|=|q(z)|dzd\bar{z}}\) is given by the meromorphic quadratic differential Q on Γ induced by the standard complex form dx2 + dy2 on \({\mathbb{C}^2=\{(x,y)\}}\) . By considering basic properties of Q, we show that the condition for local arc length parameterization along \({\Gamma_\mathbb{R}}\) to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.

Constant Speed Parametrization Mapping of Curved Boundary Surfaces in Higher-Order Moment-Method Electromagnetic Modeling

A constant speed parametrization (CSP) mapping of boundary surfaces is proposed for moment-method analysis of antennas and scatterers, along with its approximation using large higher-order Lagrange-type curved quadrilateral patches. The importance of the proper placement of interpolation nodes that ensures minimum mapped parametric space distortion (arc-length parametrization) is explained and demonstrated on simple examples. The CSP mapping results in on average five times lower radar cross section (RCS) error for a spherical scatterer than with the ray casting (central projection) parametrization mapping. The extension of the CSP concept to arbitrary surfaces is illustrated in modeling of the double-ogive target.

Le problème de Helmoltz pour des obstacles peu réguliers

The classical Helmholtz model for wakes behind an obstacle in perfect fluid mechanics boils down in complex analysis into a Riemann problem with free boundary, the free boundary being the wake and being such that the Riemann map parametrizes it as arclength parametrization. We generalize a method due to Coifman and Meyer [CM] to solve this problem for a class of non-smooth obstacles.

A Mathematical and Numerically Integrable Modeling of 3D Object Grasping under Rolling Contacts between Smooth Surfaces

A computable model of grasping and manipulation of a 3D rigid object with arbitrary smooth surfaces by multiple robot fingers with smooth fingertip surfaces is derived under rolling contact constraints between surfaces. Geometrical conditions of pure rolling contacts are described through the moving‐frame coordinates at each rolling contact point under the postulates: (1) two surfaces share a common single contact point without any mutual penetration and a common tangent plane at the contact point and (2) each path length of running of the contact point on the robot fingertip surface and the object surface is equal. It is shown that a set of Euler‐Lagrange equations of motion of the fingers‐object system can be derived by introducing Lagrange multipliers corresponding to geometric conditions of contacts. A set of 1st‐order differential equations governing rotational motions of each fingertip and the object and updating arc‐length parameters should be accompanied with the Euler‐Lagrange equations. Further more, nonholonomic constraints arising from twisting between the two normal axes to each tangent plane are rewritten into a set of Frenet‐Serre equations with a geometrically given normal curvature and a motion‐induced geodesic curvature.

Estimates of gradient and of Jacobian of harmonic mappings defined in the unit disk

Let $F:\mathbb T\to \gamma$ be a bounded measurable function of the unit circle $\mathbb T$ onto a rectifiable Jordan curve $\gamma$ with the length $|\gamma |$, and let $w=P[F]$ be its harmonic extension to the unit disk $\mathbb U$. By using the arc length parametrization of $\gamma$ we obtain the following results: (i) If $F$ is a quasi-homeomorphism and $1\le p<2$, the $L^p$-norm of the Hilbert-Schmidt norm of the gradient of $w$ is bounded as follows: $\|D(w)\|_{p}\le \frac {|\gamma |}{4\sqrt 2}(\frac {16}{\pi (2-p)})^{1/p}$. (ii) If $F$ is $p$-Lipschitz continuous and $\gamma$ is Dini smooth, then the Jacobian of $w$ is bounded in $\mathbb U$ by a constant $C(p,\gamma )$. The first result is an extension of a recent result of Verchota and Iwaniec, and Martin and Sbordone, while the second result is an extension of a classical result of Martio where $\gamma =\mathbb T$.

An Improved Bubble Packing Method for Unstructured Grid Generation with Application to Computational Fluid Dynamics

An improved bubble packing method (BPM) is proposed to generate high-quality unstructured grids for prediction of the flow field in a domain with complex geometry. For a curved-boundary domain, bubble departure from the curved boundaries during the dynamic movement of bubbles can be avoided by using the mapping and the arc-length parameterization methods. Furthermore, the grid density of the whole region can be controlled effectively. Local mesh refinement is achieved by adding bubbles with different sizes to the real and artificial vertices of the domain, and vertex information is transferred to the inner nodes of the domain using the Shepard interpolation method. In order to validate the proposed BPM, a finite-volume solver on an unstructured collocated grid is developed to simulate both square and polar lid-driven cavity flows. The numerical simulation results agree well with the experimental data under different Reynolds numbers.

Curve interpolation based on the canonical arc length parametrization

We use the canonical equations (CE) of differential geometry, a local Taylor series representation of any smooth curve with parameter the arc length, as a unifying framework for the development of new CNC algorithms, capable of interpolating 2D and 3D curves, represented parametrically, implicitly or as surface intersections, with accurate feedrate control. We use a truncated form of the CE to compute a preliminary point, at an arc distance from the last interpolation point selected to achieve a desired feedrate profile. The next interpolation point is derived by projecting the preliminary point on the curve. The coefficients in the CE involve the curve’s curvature, torsion and their arc length derivatives. We provide computing procedures for them for common Cartesian representations, demonstrating the generality of the proposed method. In addition, our algorithms admit corrections, which render them more accurate in terms of the programmed feedrate, compared to existing parametric algorithms of the same order.