Multiresolution for Closed Curves and Surfaces Using Wavelets

Multiresolution modeling is becoming a powerful tool for fast display, and geometric data compression and transmission of complex shapes. Most of the existing literatures investigating the multiresolution for B-spline curves and surfaces are concentrated on open ones. In this paper, we focus on the multiresolution representations and editing of closed B-spline curves and surfaces using wavelets. A repetition approach is adopted for the multiresolution analysis of closed B-spline curves and surfaces. Since the closed curve or surface itself is periodic, it can overcome the drawback brought by the repetition method, i.e. introducing the discontinuities at the boundaries. Based on the models at different multiresolution levels, the multiresolution editing methods of closed curves and surfaces are introduced. Users can edit the overall shape of a closed one while preserving its details, or change its details without affecting its overall shape.

On the Mean Absolute Geodesic Curvature of Closed Curves in 2-Dimensional Space Forms

In this paper, we establish some formulas on closed curves in 2-dimensionalspace forms. Mean absolute geodesic curvature is introduced to describe the average curving of aclosed curve. In this sense, a closed curve could be compared with a geodesic circle that is theboundary of a convex geodesic circular disk containing the closed curve. The comparison can be usedto show some properties of space forms only on themselves.

On Equivalence of Simple Closed Curves in Flat Surfaces

By explicit constructions,we give direct proofs of the following results： for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 〉1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX（α） = lX（β）,extX（α） = extX（β） and lq（α） = lq（β）,where lX（·）,extX（·） and lq（·） are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.

COMPLEX RATIONAL TYPE INTERPOLATION AT DISTURBED FEJR POINTS ON A CURVE

The results of accurate order of uniform approximation and simultaneous approximation in the early work ＂Jackson Type Theorems on Complex Curves＂ are improved from Fejer points to disturbed Fejer points in this article. Furthermore, the stability of convergence of Tn,∈（f,z） with disturbed sample values f（z^＊） ＋ Sk are also proved in this article.

Theory of constructing closed parametric curves based on manifolds

A parametric curve based on a manifold is de-signed for constructing an accurate closed curve.A circle was defined as the parametric space and a non-uniform B-splines defined on the unit circle were used as base func-tions.A method to construct knot vectors,control points and corresponding parameters were proposed.A method to de-termine the coordinates for any point on a curve was also proposed.Some non-uniform rational B-splines(NURBS)control techniques,such as curves with an embedded line,a sharp angle,and so on,were used to verify the proposed method’s compatibility with NURBS.Some examples were used to compare the arithmetic with that of NURBS.The results show that the method is not only simple,feasible and reliable but also compatible with a CAD system using NURBS.

Time-Machine Solutions of Einstein's Equations with Electromagnetic Field

In this paper we investigate the time-machine problem in the electromagnetic field. Based on a metric which is a more general form of Ori＇s, we solve the Einstein＇s equations with the energy-momentum tensors for electromagnetic field, and construct the time-machine solutions, which solve the time machine problem in electromagnetic field.

Closed Timelike Curves in Type Ⅱ Non-Vacuum Spacetime

Here we present a cyclicly symmetric non-vacuum spacetime, admitting closed timelike curves(CTCs) which appear after a certain instant of time,i.e.,a time-machine spacetime. The spacetime is asymptotically flat, freefrom curvature singularities and a four-dimensional extension of the Misner space in curved spacetime. The spacetime is of type Ⅱ in the Petrov classification scheme and the matter field pure radiation satisfy the energy condition.

Rotating Squeezed Vacua as Time Machines

Squeezed quantum vacua seems to violate the averaged null energy conditions (ANEC’s), because they have a negative energy density. When treated as a perfect fluid, rapidly rotating Casimir plates will create vorticity in the vacuum bounded by them. The geometry resulting from an arbitrarily extended Casimir plates along their axis of rotation is similar to van Stockum spacetime. We observe closed timelike curves (CTC’s) forming in the exterior of the system resulting from frame dragging. The exterior geometry of this system is similar to Kerr geometry, but because of violation of ANEC, the Cauchy horizon lies outside the system unlike Kerr blackholes, giving more emphasis on whether spacetime is multiply connected at the microscopic level.

A Type D Non-Vacuum Spacetime with Causality Violating Curves, and Its Physical Interpretation

We present a topologically trivial, non-vacuum solution of the Einstein's field equations in four dimensions,which is regular everywhere. The metric admits circular closed timelike curves, which appear beyond the null curve, and these timelike curves are linearly stable under linear perturbations. Additionally, the spacetime admits null geodesics curve, which are not closed, and the metric is of type D in the Petrov classification scheme. The stress-energy tensor anisotropic fluid satisfy the different energy conditions and a generalization of Equation-of-State parameter of perfect fluid p = ωρ. The metric admits a twisting, shearfree, nonexapnding timelike geodesic congruence. Finally, the physical interpretation of this solution, based on the study of the equation of the geodesics deviation, will be presented.

Time-machine in Perfect Fluid Cosmologies

This letter investigates the time-machine problem in perfect fluid cosmologies. It solves the Einstein’s field equations with the energy-momentum tensors for perfect fluid and constructs a class of time-machine solutions, by which the time-machine problem in the perfect fluid cosmologies is solved.