Construction of periodic adapted orthonormal frames on closed space curves

The construction of continuous adapted orthonormal frames along C1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C1 closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C1 closed–loop PH curves is possible, although this incurs transcendental terms. However, the C1 closed–loop PH quintics admit particularly simple rational periodic adapted frames

Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

Given a polynomial space curve r(ξ) that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves r˜(ξ) with the same rotation–minimizing frame as r(ξ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u(ξ)=r′(ξ)×r″(ξ) and distances from the origin specified in terms of a rational function f(ξ) as f(ξ)/‖u(ξ)‖. An explicit characterization of the rational curves r˜(ξ) generated by a given RRMF curve r(ξ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f(ξ), obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(ξ). Criteria for identifying low–degree instances of the curves r˜(ξ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples

On the use of the Rotation Minimizing Frame for Variational Systems with Euclidean Symmetry

We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimizing frame, also known as the Normal, Parallel or Bishop frame (see [1], [36]). Such systems have previously been studied using the Frenet–Serret frame. However, the Rotation Minimizing frame has many advantages, and can be used to study a wider class of examples. We achieve our results by extending the powerful symbolic invariant cal- culus for Lie group based moving frames, to the Rotation Minimizing frame case. To date, the invariant calculus has been developed for frames defined by algebraic equations. By contrast, the Rotation Minimizing frame is defined by a differential equation. In this paper, we derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants. We then derive the syzygy operator needed to obtain Noether’s conservation laws as well as the Euler–Lagrange equations directly in terms of the invariants, for variational problems with a Euclidean symmetry. We show how to use the six Noether laws to ease the integration problem for the minimizing curve, once the Euler–Lagrange equations have been solved for the generating differential invariants. Our applications include variational problems used in the study of strands of pro- teins, nucleid acids and polymers

Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings

This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S from the tangent space at v. We characterize the `cut locus' (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R^2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo

The Nitsche conjecture

The conjecture in question concerns the existence of a harmonic homeomorphism between circular annuli A(r,R) and A(r*,R*), and is motivated in part by the existence problem for doubly-connected minimal surfaces with prescribed boundary. In 1962 J.C.C. Nitsche observed that the image annulus cannot be too thin, but it can be arbitrarily thick (even a punctured disk). Then he conjectured that for such a mapping to exist we must have the following inequality, now known as the Nitsche bound: R*/r* is greater than or equal to (R/r+r/R)/2. In this paper we give an affirmative answer to his conjecture. As a corollary, we find that among all minimal graphs over given annulus the upper slab of catenoid has the greatest conformal modulus.Comment: 33 pages, 2 figures. Expanded introduction and references; added discussion of doubly-connected minimal surface