Straightest Path Research Articles

General relativity as a unified fluid and field theory

Einstein’s dream for a unified field theory of Nature is attained with a classical fluid theory founded on space, time and spin, rather than on Einstein’s spacetime. An invariant quantum theory for the primordial fluid obeys the homogeneous Klein-Gordon equation, which is the same three-dimensional classical wave equation (CWE) initially tried by Schrödinger to formulate quantum mechanics (QM), but abandoned by linear superposition considerations. Primordial fluid pervades universe, obeys energy and momentum conservation, and is formed by sagions: energy-like, discrete, extended Planck-size objects of finite size, carriers of linear momentum and spin, moving in absolute 3D-curved space with speed C along straightest path. We briefly describe our novel non-harmonic and inherently quantized solutions for CWE in spherical coordinates, discovered in 1995. Solutions include a steady-state background field (possibly related to the CMB, to non-locality, and to action-at-a-distance), quantized helices, and inherently quantized functions exhibiting stable dynamic equilibrium, and isomorphism under many transformations, including the classical Doppler case, and the relativistic Lorentz, Poincaré, and Einstein transformations. Isomorphism pre-empts ab initio a few interpretative issues regarding relativistic and classical transformations. Mathematical fields represent the realistic physical temporal evolution of the primordial fluid in curved 3D-space.

The Time Scale Calculus Approach to the Geodesic Problem in 3D Dynamic Data Sets

Geodesics have a fundamental role in the geometry of curved surfaces, as well as in discrete geometry. We present the time scale analogy of the dynamic data sets parameterized by a tensor product of two times scales. The goal of our study is the find the shortest and straightest path between two points on a point cloud like data sets which also involves continuous data.

Approximate straightest path computation and its application in parameterization

This paper proposes an approaching method to compute the straightest path between two vertices on meshes. An initial cutting plane is first constructed using the normal information of the source and destination vertices. Then an optimal cutting plane is iteratively created by comparing with previous path distance. Our study shows that the final straightest path based on this optimal cutting plane is more accurate and insensitive to the mesh boundary. Furthermore, we apply the straightest path result to compute the measured boundary in the parameter domain for mesh parameterization, and we obtain a new computing formula for vertex stretch in the planar parameterization. Experimental results show that our parameterization method can effectively reduce distortions.

Fast Computing for the Approximate Straightest Path on Triangular Mesh Surfaces

Fast Computing for the Approximate Straightest Path on Triangular Mesh Surfaces

Least action principles and their application to constrained and task-level problems in robotics and biomechanics

Least action principles provide an insightful starting point from which problems involving constraints and task-level objectives can be addressed. In this paper, the principle of least action is first treated with regard to holonomic constraints in multibody systems. A variant of this, the principle of least curvature or straightest path, is then investigated in the context of geodesic paths on constrained motion manifolds. Subsequently, task space descriptions are addressed and the operational space approach is interpreted in terms of least action. Task-level control is then applied to the problem of cost minimization. Finally, task- level optimization is formulated with respect to extremizing an objective criterion, where the criterion is interpreted as the action of the system. Examples are presented which illustrate these approaches.

The choice of arboreal escape paths and its consequences for the locomotor behaviour of four species of Anolis lizards

The direction and speed of escape locomotion can affect the ability of an animal to evade a predator, and variation in habitat structure often affects speed. Consequently, the escape paths chosen by animals may affect their performance and subsequent survival. Arboreal locomotion is well suited for gaining insight into the choice of escape routes because of the discrete paths formed by branches. Decreased branch diameter and increased angles between branches can significantly decrease locomotor speeds, but no previous study has determined whether arboreal lizards selectively choose alternative paths. We quantified choice of escape paths and locomotor performance of four syntopic species of arboreal Anolis lizards in their natural habitat and in the laboratory. In the field, species with shorter limbs occurred more commonly on narrow perches than did long-limbed species, but all species favoured escape paths with larger-diameter perches and straighter interperch angles. Thus, short-limbed species used narrower perches than long-limbed species merely as a result of what they encountered, rather than as a result of a biased choice at branching points. In natural vegetation, choosing branches with the largest diameter often results in the straightest path. However, in the laboratory, most lizards preferred large-diameter perches with a sharp turn to continuing a straight path onto a small-diameter perch. Although an overriding preference for larger perch diameter may optimize escape speed within a single perch, a maladaptive side-effect could be a compromise of the overall rate of gaining distance from starting points in paths with turns.

Islamic Democracy and Moderate Muslims

Islamic Democracy and Moderate Muslims

Islamic Democracy and Moderate Muslims

Islamic Democracy and Moderate Muslims

Gravitation in Hertz Mechanics

In a previous paper, one of us presented an axiomatic framework for non-relativistic classical particle mechanics where the concept of force is not assumed as a primitive notion. Such a formulation was inspired by Hertz's mechanics, which considers only three primitive concepts: time, space, and mass. It is also emphasized that there is only one Fundamental Law: Every free system persists in its state of rest or of iniform motion in a straightest path. In the present paper we formulate a theory of gravitation which seems to be compatible with Hertz's mechanics, in the sense that it makes no explicit reference to the concepts of force or actions-at-a-distance. We show how to derive Kepler's Laws in our axiomatic framework.

The Aerialist, and: Midway, and: Studies for a Portrait, and: Tarantula

THE AERIALIST/Gabriel Spera (Philippe Petit, August 7, 1974, New York) Here goes nothing, he thinks, taking his first smaU resolute step, here goes nothing. Through the soles of his tight shoes, the iron wire goes effortlessly to the crease in the baUs of his feet, as he starts, effortlessly, walking as if on air, his posture stiff as a young debutante's walking holding books on her head. The long slight frown of pole he's holding amplifieseveryleastmuscletremor,eachpause,shift,regroup,ampUfies how little room there is for error, how much hangs in the balance, how most men could never follow. The straightest path is always the most difficulttostickto—everyoneknowsthat—thoughwhat'smostdifficult points us most directly to heaven. The taut cable, from all viewpoints but his, cuts the world neatly in half, a bold slice, a dividing Une, but he doesn'tbuy those outdated dualities, leftand right, east and west, he refuses to choose, refuses to swerve, insists there's another way, refuses even to lean. He's reached an accord between two ends that are even after all. How high is he? Ahundred stories? As if that mattered—after so many floors, the stakes are the same. But as for the crowd, so many hundreds offeetbelow, theyhaveno real idea. There mustbehundreds, complete strangers, gathered Uke brothers, bound by their complete wonderand tension. Dotheywanthimtosucceed,hecan'thelpwonder, hoping to take part in his literal transcendence? Or are they all hoping secretly he'll fail, so they can go back to their pedestrian lives, secretly convinced thatnone fall farther and harder than those who've convinced themselvesthey'reaboveitall?Maybeboth. Orneither.Theythemselves surely don't know, and he doesn't need to take sides. Not now, surely. At last the edge of Tower Two appears a few feet away, and he's at last aware of the crowd, police, photographers, heUcopters. He's not aware how long it took to get this far—minutes? days? hours?—or how long until it's all over. Hold steady, he thinks, the prize is never lost until practicaUyinhand.Everyoneknowsthat.Takeiteasy—you'vepracticaUy assuredyourplaceinhistory—thecitywon'tforget.Youcanrestassured. This is yourmoment, notthenext. What'snext? Fill yourlungswith this rarest of air, slightly redolent of the river, fill your eyes with this rarest pictureofdowntown,hearthedicksofcamerassnappingendlesspictures, forceyourarmstobearupa Utilelonger. Savorthisinstant,whichnoforce on earth can take away. Soon, you'll be firmly, humanly, back on earth— there'snogettingaround it,no lookingback. Onestep, and you're there. 24 · The Missouri Review MiOWAY/Gabriel Spera The same arcs repeat themselves everywhere my eye goes—the same measured progression, same Ulusion of narrative. Take now: I am cranking one end of a jump rope tethered to a fence of my own making while my daughter skips, singing a song with no meaning, no purpose outside the simple repetition of sound. And as it rises and curls, faUs and scoops, the rope traces through the air a heUx too subtle to be perceived— I see only a circle radiating from my wrist; she sees a simple repetition of lines like the frames of a broken film reel, or the crossbars of a Ferris wheel. She gravitates naturaUy to the midpoint, where time and space are on her side, touching down and lifting off before her feet can grow steady on the ground. And just as surely as her rhythm is a cycle, the cycle a wave, the wave a song with no purpose but to sing itself, she steps in time with the dervishing planets, the snoring tides, the faucets leaking aU over creation. I used to see the eye as a prism, projecting the future on its upside-down screen, but now I see it as the narrows of an hourglass, itself a victim of time, the future above, aU past below, converging and retreating from this bifocal moment that divides or at least separates the two. The rope skips Uke a broken metaphor whose meaning degrades in the noise of settling sand as even my own double-heUcal signal will decay, grow indistinguishable The Missouri Review · 25 in time, an arc of ciphers spliced into the length of a patchwork spiral. Already I have reached the point where aU that I remember exceeds all...