Local Coordinate Frame Research Articles

Free-form design using axial curve-pairs

Deformation of 3D shapes usually requires the use of a deformation tool. The freeform deformation technique requires the use of a lattice of control point for deforming an object. This may require a synchronized movement of the lattice control points in order to obtain the desired effects. The axial deformation technique allows an object to be deformed by manipulating an axial curve. However, unexpected twist of the object may be obtained. This is a result of the lack of control on the local coordinate frame of the curve. This paper presents a technique for deforming objects with a set of axial curve-pairs. The use of a curve-pair allows the local coordinate frame to be controlled intuitively. A curve-pair is composed of a primary and an orientation curve. The orientation curve is an approximate offset of the primary curve. A technique is proposed for maintaining the relation between the primary and the orientation curve when the curve-pair is adjusted. By associating a complex 3D object to a curve-pair, the object can be stretched, bended, and twisted intuitively through manipulating the curve-pair. This deformation technique is particularly suitable for manipulating complex shapes (e.g. decorative components) in industrial and aesthetic design, and is also suitable for modelling characters and animals with flexible bodies. Adjusting the curve-pair according to some motion constraints produces different postures of a character or animal model. This in turn can be used as decorative components for aesthetic design.

A methodology for quantitative evaluation of root canal instrumentation using microcomputed tomography.

The aim of this paper was to present an objective methodology for quantitative evaluation of root canal instrumentation using microcomputer tomography, together with developed software based on a constructed mathematical model. A desktop X-ray micro-CT scanner (SkyScan 1072) was used to provide data sets of an extracted mandibular molar before and after instrumentation (ProFile 0.04 Taper instruments, Dentsply Maillefer, Ballaigues, Switzerland) that were stored for later use by software. A volume visualization package (T3D) was applied to obtain 3D renderings of the molar to illustrate the qualitative visualization capacity. Next, medical image volume fusion software was used to allow alignment of pre and post image volumes. Finally, software was developed to make quantitative measurements and to provide additional qualitative information on the registered image volumes. This procedure implemented a true 3D mathematical model for quantification of instrumentation effects. Using a local co-ordinate frame, perpendicular reslices were made at five different levels within the mesio-buccal canal to evaluate transportation and centring ability. At first accurate and detailed 3D renderings were obtained. The general and local canal shape before and after preparation could visually be examined in 360 degrees rotation. Numerical values were obtained for volume, volume changes and transportation. Centring ability was calculated by centring ratio and centre movement. This methodology is a new and objective way for quantitative evaluation of root canal instrumentation using microcomputer tomography and dedicated software.

Elastic Rod Model of a DNA Loop in theLacOperon

twist [13], using a fast, computationally inexpensive approach that can be used to study DNA conformations in many protein-DNA systems. Model.— The structure of the loop is derived from its elastic properties only, which effectively result from both internal DNA rigidity and the short-range [14] electrostatic self-repulsion of DNA. The long-range electrostatic interactions are screened out under physiological salt conditions and result in negligible forces in the case studied, as shown below. Entropic effects are also neglected since the studied loop has a length of only 26 nm, which is about half of the 50 nm persistence length of DNA [15]. The elastic rod, approximating the DNA loop, is parametrized by its arclength s, according to the classical Kirchhoff approach [16]. The centers of the cross sections of the rod at each point s constitute the centerline rs. The local coordinate frame d1s, d2s, d3s determines the orientation of each cross section. The unit vectors d1 and d2 lie along the principal axes of the DNA cross section [17]; the vector d3 is normal to d1 and d2. The components of these vectors can be expressed in terms of the Euler angles or the Euler parameters [18], which determine the rotation of the local frame relative to the lab coordinate system. The shape of the rod is described in terms of the curvature k of its centerline and the twist V around the centerline [16]. The curvature is decomposed into its components k1 and k2 which account for the bending of the rod about d1 and d2, respectively. The twist is

Local orientational correlations and short time anisotropic motion in molecular liquids: Computer simulations of liquid CO2

Molecular dynamic (MD) simulations of liquid CO2 at 250 MPa pressure and room temperature have been performed using a flexible model potential. A detailed analysis of the data reveal the full three-dimensional local structure of coordination shells that exhibit significant deviations from spherical symmetry with strong angular correlations among the molecules which form the inner coordination shell of the local liquid structure. Structures resembling T-shaped and offset-parallel CO2 dimers similar to those found in molecular beam and low temperature experiments have been identified, the T-shaped dimer having higher probability to be formed than the offset-parallel configuration. Local motion on short time scale is found to be different along three principal directions of a local coordinate frame.

Transformation of Polarization Descriptors Between Coordinate Frames

When accounting for polarization effects in radar and communications systems, it is often necessary to transform descriptions of the polarization state of an antenna or the polarization response of a target between local (or body-centered) and global coordinate frames. Such transformations correspond to rotation of the polarization basis by a prescribed angle which is a function of (1) the coordinate transformation matrix that relates the two coordinate frames and (2) the direction of propagation. Although methods for determining the polarization rotation angle have been presented previously in the literature, they can only be applied in special cases. Here we solve the problem in the general case using methods based on spherical trigonometry and vector algebra, respectively. In order to apply these techniques, the elements of the coordinate transformation matrix must be known. Since conventional methods for determining these elements require information which may be difficult to obtain in practice, we present an alternative method which requires only a pair of directions that have been expressed in terms of both coordinate frames.

Three-dimensional vector field visualization based on tensor decomposition

This paper presents a visualization method called the deformed cube for visualizing 3D velocity vector field. Based on the decomposition of the tensor which describes the changes of the velocity, it provides a technique for visualizing local flow. A deformed cube, a cube transformed by a tensor in a local coordinate frame, shows the local stretch, shear and rigid body rotation of the local flow corresponding to the decomposed component of the tensor. Users can interactively view the local deformation or any component of the changes. The animation of the deformed cube moving along a streamline achieves a more global impression of the flow field. This method is intended as a complement to global visualization methods.

Ray tracing over smooth elastic shells of arbitrary shape

An efficient numerical scheme based on ray theory is developed for the analysis of elastic waves traveling over a fluid-loaded smooth elastic shell of arbitrary shape. The shell’s surface is first discretized into a number of small patches. The local geometry of each patch is then approximated in a parametric form using bi-cubic spline functions. A local curvilinear coordinate frame is defined on each patch. The ray trajectories and ray-tube areas are obtained by solving a set of ordinary differential equations, the ray and transport equations, within each patch. Several numerical tests of the accuracy and efficiency of the scheme were carried out on spherical and ellipsoidal elastic shells. The numerical results for the spherical shell agree well with analytical solutions. The ray trajectories and the ray-tube areas over an ellipsoidal shell with three distinct semiaxes clearly illustrate the inhomogeneous and anisotropic effects due to the variable curvature on the shell’s surface. It is also observed numerically that the magnitude of the ray-tube area along a ray is directly correlated with the stability of its trajectory.

3D Motion recovery via affine Epipolar geometry

Algorithms to perform point-based motion estimation under orthographic and scaled orthographic projection abound in the literature. A key limitation of many existing algorithms is that they operate on the minimum amount of data required, often requiring the selection of a suitable minimal set from the available data to serve as a “local coordinate frame”. Such approaches are extremely sensitive to errors and noise in the minimal set, and forfeit the advantages of using the full data set. Furthermore, attention is seldom paid to the statistical performance of the algorithms.

Localization System of the Hospital Transport Robot FIRST

This work concerns the development of a localization system for the navigation of an autonomous vehicle designed for transportation of heavy loads in an hospital environment. An overview of the physical configuration of the vehicle is presented as well as a description of its locomotion and of its sensing systems. The paper is focused on methods developed for the 2D robot’s localization with respect to a local coordinate frame, and for the topological identification of this local frame. Results on the localization system performances have been evaluated with the real prototype moving in a structured office environment. They indicate that the system can provide data to ensure a reliable, accurate and safe navigation.

Dynamic Model of Excavator

Automation of excavation work calls for a robotic system able to perform the planned digging work that is responsive to interaction forces experienced during excavation. The development of automated excavation control methods requires a dynamic model to describe the evolution of the excavator motion with time. The joint torques of the boom mechanism are generated by hydraulic rams that also affect the torques at other joints. Analyzing each link in succession as a free body and applying the Newton‐Euler formulation in the local coordinate frame, a dynamic model for an excavator can be derived in a straightforward manner. The model presented in this paper is intended for further development of an automated excavation control system for terrestrial, lunar, and planetary excavation.

Electro-absorption of charge transfer states: effect of sample texture

Analytical expressions are derived for the orientationally averaged second derivative electro-absorption signal of CT states in polyacene samples of various texture. It is assumed that the transition moment from the ground state to the CT state of interest and the permanent dipole moment of the CT state, defined in a local coordinate frame, are known. They are transformed to the laboratory coordinate frame in which the external fields are defined. The averaging is done over the orientations of the latter frame, set by sample texture and geometry of the experimental setup. The electro-absorption spectra simulated for a pair of closely spaced states indicate that the shape of the electro-absorption signal may be very sensitive to sample texture. The consequences of this fact for the interpretation of experimental spectra are discussed.

Spiral phase of a doped antiferromagnet.

The stability of the spiral phase is studied in the t-J model for low doping concentrations. A Green's-function formalism is used that is based on a slave-fermion Schwinger-boson representation in a twisted local coordinate frame. The spiral modulation, which is treated within the self-consistent Born approximation, gives rise to a splitting of the band of quasiholes, leading to a stable spiral phase. The wave number of the spiral phase is calculated, and shown to increase with decreasing ratio J/t. We also study the effects of the spiral modulation on spin waves. Spin waves with momenta parallel to the modulation vector couple strongly to electron-hole pairs that have an excitation gap. The resulting hybridized modes are specified, and the strong renormalization of the spin-wave velocity is pointed out.

Spiral phase in doped antiferromagnets

We study the stability of spiral phase in the t-J model for low doping concentrations, using a Green's function formalism based on a slave-fermion Schwinger boson representation in a twisted local coordinate frame. We show that the spiral modulation gives rise to a splitting of the band of quasiholes, leading to a stable spiral phase. We also study the effects of the spiral modulation on spin waves, and show that spin waves with momenta parallel to the modulation vector couple strongly to electron-hole pair excitations, leading to the creation of hybridized modes and a strong renormalization of the spin wave velocity.

Helical repeat and chirality effects on DNA gel electrophoretic mobility.

We determined a value of 10.34 +/- 0.04 base pairs (bp) per turn for the helical repeat of bent DNA sequences of the form A6N4-A6N5 by estimating the sequence repeat required to produce a planar curve, as judged from the maximum in the electrophoretic mobility anomaly of multimers containing different sequence repeats (10.00, 10.33, 10.50, 10.67, and 11.00 bp per turn). This result provides the basis for a method to evaluate the helical repeat of any DNA segment by comparative electrophoresis measurements. The sequence of interest is placed between two A-tract bends and the phasing is varied over an entire helical turn. Knowledge of the number of base pairs between the bends in the cis isomer, which has the lowest electrophoretic mobility, allows calculation of the average helical repeat of the inserted sequence. In the course of these experiments we observed an unexpected dependence of electrophoretic mobility on the shape of DNA molecules: in high-percentage polyacrylamide gels, those bent molecules for which we deduced a right-handed superhelical form are less retarded than their homologous left-handed isomers. To explain this finding we propose that superhelical chirality influences the choice of DNA migration pathway, leading to rotation of the DNA molecule relative to the local coordinate frame in the gel. High-percentage gels have sufficiently close contact with the right-handed DNA helical twist to differentiate the frictional consequences of right- and left-handed twisting motions.

Planning and control of robot orientational path

A novel geometric method for the planning and controlling the orientation of a robot's end-effector as it moves along a prescribed translational path is presented. To eliminate the influence of geometric changes to the path and to be able to predict the orientation, the proposed method plans rotational paths with respect to a natural local coordinate frame formed by three unit vectors, namely, the tangent, normal, and binormal vectors, defined by the geometry of the path. A motion control strategy is developed to maintain the desired rotational speed or acceleration of the end effector. Examples are presented which show the practicality of the method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Ray tracing deformed surfaces

A collection of new methods for ray tracing differentiable surfaces is developed. The methods are general, and extend the set of "ray-traceable" surfaces suitable for use in geometric modeling. We intersect a ray l = at + b , t &amp;gt; 0 with a parametric surface x = f (u, v) , and with implicit surfaces f(x,y,z) = 0. A smooth surface is treated as a deformation of a flat sheet; the intersection problem is converted to a new coordinate system in which the surfaces are flat, and the rays are bent. We develop methods for providing good initial estimates of the parametric intersection values, and a "closeness criterion," to reduce computation. These same criteria help us substitute a set of simpler surfaces for the more complex surface. The parametric method produces the intersection values of u, v , and t . These are suitable for shading calculations and for mapping textures onto the surface; they can also produce the local coordinate frame values, suitable for anisotropic lighting models.

Representation and recognition of the movements of shapes.

The problems posed by the representation and recognition of the movements of 3-D shapes are analysed. A representation is proposed for the movements of shapes that lie within the scope of the Marr & Nishihara (1978) 3-D model representation of static shapes. The basic problem is how to segment a stream of movement into pieces, each of which can be described separately. The representation proposed here is based upon segmenting a movement at moments when a component axis, e.g. an arm, starts to move relative to its local coordinate frame (here the torso). For example, walking is divided into a segment of the stationary states between each swing of the arms and legs, and the actual motions between the stationary points (relative to the torso, not the ground). This representation is called the state-motion-state (SMS) moving shape representation, and several examples of its application are given.

Hierarchical Representation of Three-Dimensional Objects Using Verbal Models

We present a formalism for the computer representation of three-dimensional shapes, that has as its goal to facilitate man-machine communication using verbal, graphic, and visual means. With this method, pieces may be assembled hierarchically using any of severalways of specifying attachment. The primitives of the representation are generalized cylinders, and the creating of assemblies may make use of the axes inherent in the primitives. Generic models may be described that may leave some parameters or dimensions unspecified, so that when a specific instance of the model is described, those parameters may either be explicitly specified or take on default values. The axes of local coordinate frames may be given symbolic names. A set of computer programs translates descriptions of objects into polyhedral models and line drawings.

Long-term evaluation of three-dimensional heliocentric solar sail trajectories with arbitrary fixed sail setting

The solar radiation effects upon the orbital behaviour of an arbitrarily shaped spacecraft (or a solar sail in particular) in a general fixed orientation with respect to the local coordinate frame are investigated. Through introduction of a quasi-angle in the osculating plane, the motion of the orbital plane becomes uncoupled from the in-plane perturbations. Exact solutions in the form of conic sections and logarithmic spirals can readily be formulated for certain specific initial conditions. An effective out-of-plane spiral transfer trajectory is obtained by reversing the force component normal to the orbital plane at specified positions in the orbit. By choosing the appropriate control angles for the sail orientation, any point in space can be reached eventually. In the case of general initial conditions, the long-term orbital behaviour is assessed asymptotically by means of the two-variable expansion procedure. An implicit expression for the eccentricity is derived and explicit results are established by an iteration scheme. The other orbital elements can be expressed in terms of the eccentricity and their asymptotic series for near-circular initial orbits are also obtained. While equations for the higher-order contributions as well as the periodic parts of their solutions can be formulated readily, their secular terms are determined only for a circular initial orbit.

Simplified B-matrix for polymers in irregular conformation

The field of conformational defects in polymers is made accessible to Lifshitz's Green-function method. By introducing local coordinate frames, a fictitious polymer chain is constructed which has the irregular conformation of the real chain but, at the same time, the phonon spectrum of the regular chain, and is described by a dynamical matrix which differs from the real one only in localized regions around the conformational defects. A general and very simple expression for the B-matrix of an isolated polymer skeleton in arbitrary irregular conformation is presented.