Method Of Moving Frames Research Articles

Geometrical characteristics of some typical spatial constraints

The invariants of some typical constraint ruled surfaces are extensively studied by means of the moving frame method of differential geometry. Constraint ruled surfaces correspond to combinations of certain kinematic pairs, such as the R– C, H– C, C′– C and C′– P– C. In this paper, the sufficient and necessary conditions are given for a ruled surface to be one of the above constraint ruled surfaces. Then, the contact order between two ruled surfaces along a common ruling is discussed, and contact conditions are derived for a ruled surface to contact a constraint ruled surface along their common ruling to a given order. As a result, some equations are derived for the lines of a moving body to lie on one of the above constraint ruled surfaces in several infinitesimally separated positions.

Developable surfaces in Euclidean space

AbstractThe classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.

Brachiopod outline and episodic growth

Accretionary outlines of many brachiopods are closely matched by one or more episodes of logarithmic spiral growth where each episode has a constant growth gradient. An episodic model supports the observations of abrupt change, but not that of gradual change in accretionary growth. A reliable guide to the number of growth episodes is given by the pattern of residuals after fitting a single spiral to an outline. An episodic model exposes subtle variations in growth that are not easily described by visual examination. A three-episode model clarifies a shape difference between two species of the endemic New Zealand genus Neothyris. Direct observation, along with linear measurements and their ratios, was unable to pinpoint that the well-documented dorsoventral compression in one species arises from a growth episode in the ventral valve. Logarithmic spirals about a single coiling axis are not suited for large, disjunct changes in outline for which the moving-frame methods or aperture-deposition models are applicable.

Ordering in automated theorem proving of differential geometry

A new ordering method is proposed for automated theorem proving of differential geometry, by which Cartan's moving frame method can be combined with Wu's elimination principle.

Cartan connections and Kawaguchi geometry of spaces obtained by the moving Frame method

Cartan connections and Kawaguchi geometry of spaces obtained by the moving Frame method

Complex points of minimal surfaces in almost Kähler manifolds

If (N, ο, J,g) is an almost Kahler manifold and M is a branched minimal immersion which is not a $J$-holomorphic curve, we show that the complex tangents are isolated and that each has a negative index, which extends the results in the Kahler case by S. S. Chern and J. Wolfson [2] and S. Webster [7] to almost Kahler manifolds. As an application, we get lower estimates for the genus of embedded minimal surfaces in almost Kahler manifolds. The proofs of these results are based on the well-known Cartan's moving frame methods as in [2, 7]. In our case, we must compute the torsion of the almost complex structures and find a useful representation of torsion. Finally, we prove that the minimal surfaces in complex projective plane with any almost complex structure is a J-holomorphic curve if it is homologous to the complex line.

The Poisson-Lie structure of nonlinear O(N) sigma -model by using the moving-frame method

We discuss the Poisson-Lie structure of the integrable nonlinear O(N) sigma -model with the moving-frame method. The corresponding r- and s-matrices are given explicitly. We also perform the gauge transformation for the Lax potential and the r and s matrices. Furthermore, we discover that the field-dependent terms in our r- and s-matrices only depend on the Riemannian connection of the target manifold.

Time-domain-finite-wave analysis of the engine exhaust system by means of the stationary-frame method of characteristics. Part I. Theory

Time-domain-finite-wave analysis of the engine exhaust system is usually done using the method of characteristics. This makes use of either the moving frame method, or the stationary frame method. The stationary frame method is more convenient than its counterpart inasmuch as it avoids the tedium of graphical computations. In this paper (part I), the stationary-frame computational scheme along with the boundary conditions has been implemented. The analysis of a uniform tube, cavity-pipe junction including the engine and the radiation ends, and also the simple area discontinuities has been presented. The analysis has been done accounting for wall friction and heat-transfer for a one-dimensional unsteady flow. In the process, a few inconsistencies in the formulations reported in the literature have been pointed out and corrected. In the accompanying paper (part II) results obtained from the simulation are shown to be in good agreement with the experimental observations.

Finslerian structures: The Cartan–Clifton method of the moving frame

A theory of Finslerian structures is presented using an unpublished method of Clifton. It consists of using Cartan’s moving frame method with fields of special bases, and the Cartan calculus of tensor-valued forms with nonredundant coordinates on sphere bundles (which are the base spaces of the Finsler bundles). The formulation only requires a modicum of new concepts and does not involve Lagrangians at any fundamental level, including the definitions of metrics and connections. Finslerian connections on Riemannian (and even flat) metrics are discussed. The invariant forms of the sphere bundle of Minkowski space are obtained. The possibility of describing equations of motion through autoparallels, rather than extremals, is illustrated.

On the equivalence of variational problems, I

We deal with the absolute equivalence of one-dimensional variational problems. The term “absolute” means that the underlying space is not given in advance: the transformations may be quite general and need not preserve the order of derivatives. Then the use of jets of infinite order is advisable and an appropriately modified E. Cartan's moving frame method proves to be very effective.

Biological aspects of theoretical shell morphology

Savazzi, E. 1990 04 15: Biological aspects of theoretical shell morphology. Lethaia. Vol. 23, pp. 195–212. Oslo. ISSN 0024–1164 Among the available methods in theoretical shell morphology. moving-frame method, produce the broadest range of shell shapes and appear to emulate most closely the biological processes involved in shell morphogenesis and growth. In addition, moving-frame methods can easily be enhnneed in both respects by adding operations that are similar in nature to the original procedures. Improvements beyond this point. however. require a qualitative change in approach. The focus of attention must he transferred from the shell lo the soft parts involved in shell construction. This may be done by (1) regarding the soft parts a5 pneus. and (2) controlling their shape and metabolism through morphogenetic programme based on biochemical reactions. *Theoretical morphology. functional morphology, modelling. computer graphics, Mollusca. Gastropoda. Bivalvia. Brachiopoda.

Riemannian metrics on tangent bundles

Some «natural» metrics on the tangent and on the sphere tangent bundle of Riemannian manifold are constructed and studied via the moving frame method.