Geometry Of Curvature Research Articles

On the Psychophysics of Binocular Space Perception

Binocular vision gives rise to a perceptual space that is endowed with a rich geometrical structure, which according to Luneburg is a non-Euclidean Riemannian geometry of constant Gaussian curvature. This hypothesis, together with certain psychophysical assumptions, provides a qualitative explanation of classical empirical phenomena. The Luneburg theory has been less successful in quantitatively predicting experimental data, a fact that has sometimes been taken as evidence against the presumed geometrical structure. Some of these shortcomings, however, may also be attributed to psychophysical assumptions that are independent of the geometry of visual space, and that were found to be violated systematically. We propose a generalization of Luneburg's psychophysical mapping that can account for these findings. The foundation of a psychologically significant recoordinatization of physical space is formulated within a conjoint measurement approach. It is shown that psychophysical invariances lead to restrictions on the possible form of the coordinate transformation, but its complete specification is left as an open problem. The proposed approach can at least partially explain some empirical observations that are considered to contradict Luneburg's conception of a psychophysical theory of binocular space perception.

Non-planar curvature and branching of arteries and non-planar-type flow

In this study, magnetic resonance imaging techniques have been used to examine the geometry of arterial curvature and branching in casts and in vivo , and to measure the distribution of axial velocity in the associated flow. It is found, contrary to a widely held view, that the geometry is commonly non-planar. Moreover, relatively small values of the parameters which render the geometry non-planar appear significantly to affect the velocity distribution. The findings suggest that non-planarity is an important factor influencing arterial flows, including wall shear. The implications are not restricted to vascular biology, pathology and surgery, but may extend to the design of general piping systems.

On the Psychophysics of Binocular Visual Space

Binocular vision gives rise to a perceptual space that is endowed with a rich geometrical structure, which according to Luneburg [1947 Mathematical Analysis of Binocular Vision (Princeton, NJ: Princeton University Press)] is a non-Euclidean Riemannian geometry of constant curvature. This hypothesis, together with certain psychophysical assumptions, provides a qualitative explanation of classical empirical phenomena. The Luneburg theory has been less successful in quantitatively predicting individual data, a fact that has sometimes been taken as evidence against the presumed geometrical structure. Some of these shortcomings, however, may also be attributed to psychophysical assumptions that do not depend on the geometry of visual space. The locus of perceived egocentric equidistance, for example, has been found to deviate systematically from the Vieth-Müller circle postulated by Luneburg (eg Foley, 1966 Journal of the Optical Society of America56 822 – 827). To account for these findings a generalisation of Luneburg's psychophysical mapping is proposed that actually realises his original intentions. Because of the theoretical basis of its measurement, the adequacy of this general psychophysical frame of reference may be empirically tested. The results of an experimental test of one of the key conditions, which strongly support the presented theory, are reported. It is also shown that psychophysical invariances lead to restrictions on the possible form of the psychophysical mapping, and the approach is discussed from the viewpoint of retinal correspondence.

Effect of anastomosis and geometry of vessel curvature on blood flow velocity and patency in microvessels.

The effect of the geometry of the vessel and the number of anastomoses on the blood flow was studied. Four different shapes of the vessel were constructed by using a 6-cm-long double vein graft model with three anastomoses: (1) an alpha loop, (2) an omega loop, (3) a sigmoid curve, and (4) straight. Blood flow was measured by an ultrasound Doppler flowmeter. The result showed no alternation in blood flow across different geometry and through three patent microanastomoses. However, six out of seven vein grafts were thrombosed at 24 hr postoperative due to vascular kinks. This model demonstrates potential sites of kinking at the dissection end of the femoral artery, the microanastomoses, the side branches of the vein graft, and the adventitial adhesions. This model is recommended to microvascular trainees for the study of kinking and the management of redundant pedicles and vein grafting.

Simplexes in spaces of constant curvature

In hyperbolic, Euclidean and spherical n-space, we determine, for each positive number l, the largest interval of the form λ n (l)≤l ij≤l which guarantees the existence of an n-simplex p 1 p 2 ... p n+1 with edge-lengths p ipj=l ij. (In spherical geometry of curvature 1 the interval is empty unless l≤2 arcsin $$\sqrt {\left( {{{n + 1} \mathord{\left/ {\vphantom {{n + 1} {2n.}}} \right. \kern-\nulldelimiterspace} {2n.}}} \right)}$$ ) The assertion that these intervals are as large as possible is justified because each of them allows certain degenerate simplexes. We determine explicitly all of these critical configurations.

Surfaces with constant exterior geometry of negative curvature

Surfaces with constant exterior geometry of negative curvature

The Differential Geometry of the General Helix as Applied to Mechanical Springs

In order to improve the performance of helical springs, such as increasing the fatigue life and suppressing resonance, variable pitch angle and variable helix radius may be incorporated into the helical spring geometry. Employing the tool of differential geometry, new and complete formulae of curvature, torsion, and spring force are derived. It is shown that these formulae are more general and accurate than Kelvin’s curvature and torsion formulae, than commonly used force formulae (Wahl, 1963). Possible simplifications to the complete formulae and the corresponding errors introduced are both discussed and compared with experimental data.

Certain questions of Lobachevskii geometry, connected with physics

An historical survey of the development of Lobachevskii geometry as a typical representative of a geometry of negative curvature, Friedman cosmology, Lobachevskii geometry, and an interpretation of the velocity space in the special theory of relativity as a Lobachevskii space are presented.

Riemannian geometries of variable curvature in visual space: visual alleys, horopters, and triangles in big open fields.

Luneburg's model for computation of the curvature K of visual two-dimensional space (horizontal visual surface) was tested with equidistant and parallel alleys in large open spaces. Forty-six subjects used stakes to produce 406 experimental alleys of variable sizes (from 5 x 1 to 240 m x 48m). The results show that, contrary to results obtained under laboratory conditions with small alleys and light spots, the individual curvature of visual space does not have negative constant value. K varies in the interval - 1 to + 1 in ninety computed settings: K greater than 0 (N = 38); K less than 0 (N = 52). Therefore the Lobachevskian gemetry currently attributed to visual space ought to be replaced by a Riemannian geometry of variable curvature. Moreover K is an individual function dependent on the size of the alley (distance from the subject), and visual perception would be better understood as scale-dependent. Independently of Luneburg's model we have tested the constancy of the curvature hypothesis in experiments with horopters and visual triangles. The results obtained invalidate Luneburg's hypothesis also.

The Historical and Conceptual Relations between Kant's Metaphysics of Space and Philosophy of Geometry

The Historical and Conceptual Relations between Kant's Metaphysics of Space and Philosophy of Geometry" TED B. HUMPHREY IN THE "TRANSCENDENTAL AESTHETIC" Of the Kritik der reinen Vernun]t, Kant formulates and argues for three related doctrines concerning space and geometry: 9(1) Space is the pure a priori form of outer intuition. (2) Geometrical truth is a priori and synthetic. (3) The metric of humanly intuited space is Euclidean, and therefore the propositions of Euclid's geometry are a priori synthetic truths. The generally accepted view that the third doctrine somehow follows deductively from the first two, indeed, that Kant used it in their initial justification, has been the source of several apparently serious criticisms not only of the Aesthetic but also of Kant's entire epistemology and metaphysics,x In particular, three objections having their source in recent research in mathematics and psychology seem to cast considerable doubt on the defensibility of the second and third doctrines. The two criticisms stemming from mathematics, namely, that geometries other than the Euclidean are formulable with complete logical consistency, a fact many believe Kant could not have been aware of, and that some persons can actually visualize non-Euclidean spatial images, were most powerfully advanced by Hans Reichenbach .2 The second of Reichenbach's criticisms is effectively generalized by the psychological research of R. K. Luneberg and A. A. Blank, the conclusion to be drawn from which, according to Adolf Griinbaum, is that "although the physical space in which sensory depth perception by binocular vision is effective is Euclidean , the binocular r/suM space resulting from psychometric coordination possesses a Lobatchevskian hyperbolic geometry of constant curvature. ''s Because * I wish to thank the Arizona State University Grants Committee for a faculty grant-inaid supporting the research for this paper. TbAnk~ are also due Professor Peter Fuss, University of Missouri, SL Louis, Dr. Frank D. Farmer, Arizona State University, and an unknown reader for this Yournal for extensive, helpful comments. x Much of what I have said thus far about the doctrines concerning space is equally true, mutatis mutandis, of the Aesthetic's doctrines concerning time. Consequently, most of the paper's results regarding the historical and logical relations obtaining among the doctrines pertaining to space will apply mutatis mutandis to time. For the most part, however, time will not be a topic of explicit concern in what follows. 2 Hans Reichenbach, The Philosophy of Space and Time, trans. Maria Reichenbaeh and John Freund, (New York: Dover Publications, Inc., 195"I), pp. 30-58 passim. s Adolf Griinbaum, Philosophical Problems of Space and Time (New York: Alfred A. Knopf, 1963), p. 154. [483] 484 HISTORY OF PHILOSOPHY most persons believe, either implicitly or explicitly, that in the development of the Aesthetic's doctrines, the third is either essential to the defense of the first two or a necessary conclusion from them, 4 they have been dissuaded by these developments from accepting any portion of the position Kant propounds in the Aesthetic . This is a mistake. The Aesthetic contains no argument for the view that humanly intuited space is Euclidean, nor do any of its arguments concerning the origin and metaphysical status of space depend on that view. In this paper I shall argue that critics have misunderstood the historical and conceptual relations obtaining among the Aesthetie's three main doctrines concerning space and therefore have tended to reject the first two on the basis of arguments against which Kant, or at least a Kantian, can defend. The argument will develop in three stages: (1) Kant's theory of mathematical, especially geometrical, truth, particularly with respect to the precise nature of its syntheticity, and his belief that Euclidean geometry is true of humanly intuited space did not serve as premises in the development of his metaphysics of space. (2) The conceptual relations among the three doctrines is different from the one most believe to obtain. (3) Objections from contemporary mathematics to Kant's belief in Euclidean geometry cannot serve as grounds for rejecting his philosophy of mathematics and metaphysics of space. By way of conclusion I will point out some of the ways in which Kant's philosophy of mathematics and beliefs about...

The study of rearrangement reactions by field ionization mass spectrometry: I. Theoretical aspects

The peak shapes of fragment ions caused by rearrangement reactions in a field ionization ( fi) mass spectrometer are derived theoretically. The quasi-equilibrium theory ( qet) is used and different distributions of rate constants are assumed. The peak shapes agree qualitatively with those observed in the fi mass spectra. The shift of the rearrangement peak maxima from the normal fragment peak position is discussed for the case of a single-focusing magnetic sector field mass spectrometer. The peak shifts are studied as a function of the emitter geometry and radius of curvature, surface structure, ion retardation, emitter to cathode distance and emitter temperature. The rate constants as a function of energy are compared for the case of electron impact and fi.

Variation of α and β Values in a Lined Open Channel

A study of the variation in magnitude of the energy coefficient, α, and the momentum coefficient, β, was conducted for subcritical open-channel flow in concrete-lined trapezoidal canals containing horizontal curves. The Manning’s \In\N was 0.013 for reaches without bends; values of α and β were found to be smaller than values proposed in current literature, and there was only small variation of α and β values with stage, channel geometry, position, and radius of curvature. For all cross sections analyzed, the energy and momentum coefficients may be considered constant with values of 1.04 and 1.02, respectively.

Flame Zone Studies. II. Applicability of One-Dimensional Models to Three-Dimensional Laminar Bunsen Flame Fronts

The applicability of one-dimensional models in flame theories has been investigated experimentally in the case of a lean quarter-atmosphere propane air flame. Studies of flow patterns through the flame fronts were made using particle track techniques. From this time-position information, normal gas velocity, stream tube area, and gas density were calculated throughout the flame front. If the density is a single valued measure of the state of the gas, then one-dimensional models can be tested for Bunsen flames. Unidimensionality was verified for this and several other propane-air flames under a number of conditions. In the reaction region preceding the luminous zone it was also found that the velocity and density profiles were invariant with respect to changes in flame geometry, gas flow, and burner diameter. Therefore, it is a good approximation to speak of single reaction path for this flame, which is a function only of the initial state of the gas. The measurements indicate that one-dimensional flame theories should be modified to include the effects of area expansion during the course of the reaction. Continuity, diffusion, and heat conduction equations should also be generalized to include the effect of expansion. It was noted that the hydrocarbon flame reaction consists of two regions: a fast propagating reaction associated with the luminous regions and the major portion of the heat release and a slow secondary reaction associated with the post luminous mantel of the flame. Flow in the primary region is a function only of the initial state of gas and independent of external influences such as gas velocity and flame geometry. Flow in the secondary region could be varied over wide limits with no measureable effect on the propagation of the flame. Burning velocity was found to be independent of gas velocity, flame geometry, and radius of curvature of flame front, and a function only of the initial state of the gas, providing it was defined as the minimum normal gas velocity ahead of the flame front. A discussion is given of the errors introduced by measuring burning velocity by flow measurements in the throat of the burner.

A connectedness principle in the geometry of positive curvature

The main purpose of this paper is to develop a connectedness principle in the geometry of positive curvature. In the form, this is a surprising analog of the classical connectedness principle in complex algebraic geometry. The connectedness principle, when applied to totally geodesic immersions, provides not only a uniform formulation for the classical Synge theorem, the Frankel theorem and a recent theorem of Wilking for totally geodesic submanifolds, but also new connectedness theorems for totally geodesic immersions in the geometry of positive curvature. However, the connectedness principle may apply in certain cases which do not require the existence of totally geodesic immersions.