Handwriting production is viewed as a constrained modulation of an underlying oscillatory process. Coupled oscillations in horizontal and vertical directions produce letter forms, and when superimposed on a rightward constant velocity horizontal sweep result in spatially separated letters. Modulation of the vertical oscillation is responsible for control of letter height. Modulation of the horizontal oscillation is responsible for control of corner shape through altering phase or amplitude. The vertical velocity zero crossing in the velocity space diagram is important from the standpoint of control. Changing the horizontal velocity value at this zero crossing controls corner shape. Changing the slope at this zero crossing controls writing slant. The corner shape and slant constraints completely determine the amplitude and phase relations between the two oscillations. This theory applies generally to a number of acceleration oscillation patterns such as sinusoidal rectangular and trapezoidal oscillations. The oscillation theory also provides an explanation for how handwriting might degenerate with speed. An implementation of the theory in the context of the spring muscle model is developed. Here sinusoidal oscillations arise from a purely mechanical source; orthogonal antagonistic spring pairs generate particular cycloids depending on the initial conditions. Modulating between cycloids can be achieved by changing the spring zero settings at the appropriate times. Frequency can be modulated either by shifting between coactivation and alternating activation of the antagonistic springs or by presuming variable spring constant springs. An acceleration and position measuring apparatus was developed for measurements of human handwriting. Measurements of human writing are consistent with the oscillation theory.

http://diabeto.enseeiht.fr/download/handwriting/Hollerbach_1981.pdf

An oscillation theory of handwriting

The hdograph of a plane parametric curve r(t) = (x(t), y(t)f is the locus described by the first parametric derivative r' (t) = (x ' (t), y ' (t)) of that curve. A polynomial parametric curve is said to have a Pythagorean hodograph if there exists a polynomial a(t) such that x t 2 (t) + y " (t) " Pythagorean triple. " Although Pythagorean-hodograph curves have fewer degrees of freedom than general polynomial curves of the same degree, they exhibit remarkably attractive properties for practical use. For example, their arc length is expressible as a polynomial function of the parameter, and their offsets are rational curves. We present a sufficient-and-necessary algebraic characterization of the Pythagorean-hodograph property, analyze its geometric implications in terms of Bernstein-Bezier forms, and survey the useful attributes it entails in various applications. ~ ' (t) , i.e., (x' (t), y ' (t) , a (t)) form a 1. Introduction The representation of curves and surfaces in a form amenable to efficient, systematic computation is a basic issue in computer-aided design. Those representations Topyright 1990 by International Business Machines Corporation. Copying in printed form for private use is permitted without payment of royalty provided that (1) each reproduction is done without alteration and (2) the Joz~rnal reference and IBM copyright notice are included on the first page. The title and abstract, but no other portions, of this paper may be copied or distributed royalty free without further permission by computer-based and other information-service systems. Permission to republish any other 736 portion of this paper must be obtained from the Editor. that have won widespread acceptance in practical use are almost exclusively parametric formulations, based on (piecewise) polynomial functions (see [ 11 and references therein). Plane curve segments, for example, are usually defined in a form equivalent to " x(t) = aktk, k=O n y (t) = 1 bktk for t E [0, 11. k=O Such segments may be pieced together with various orders of continuity to form spline curves for smooth data interpolation; they are easily rendered by uniformly incrementing t and evaluating the polynomials (1); and algorithmic procedures are available for computing their intersections (see [ 2 ]). An immediate shortcoming of the form (1)-its inability to accommodate conic loci other than the parabola [3]-may be remedied by allowing the rational form r(t) = (X(t)/W(t), Y(t)/ W (t)) , where X(t), Y(t), and W(t) are polynomials. …

Pythagorean hodographs

Short-wave fields can be well approximated by families of trajectories. These families are dominated by their singularities, i.e. by caustics, where the density of trajectories is infinite. Thom's theorem on singularities of mappings can be rigorously applied and shows that structurally stable caustics—that is those whose topology is unaltered by ‘generic’ perturbation—can be classified as ‘elementary catastrophes’. Accurate asymptotic approximations to wave functions can be built up using the catastrophes as skeletons: to each catastrophe there corresponds a canonical diffraction function. Structurally unstable caustics can be produced by special symmetries, and the detailed form of the caustic that results from symmetry-breaking can often be determined by identifying the structurally unstable caustic as the special section of a higher-dimensional catastrophe. Sometimes it is clear that the unstable caustic is a special section of a catastrophe of infinite co-dimension; these fall outside the sc...

Waves and Thom's theorem

The consequences of impact on the solid bodies of the solar system are manifest and legion. Although the visible effects on planetary surfaces, such as the Moon's, are the most obvious testimony to the spatial and temporal importance of impacts, less dramatic chemical and petrographic characteristics of materials affected by shock abound. Both the morphologic and petrologic aspects of impact cratering are important in deciphering lunar history, and, ideally, each should complement the other. In practice, however, a gap has persisted in relating large-scale cratering processes to petrologic and geochemical data obtained from lunar samples. While this is due in no small part to the fact that no Apollo mission unambiguously sampled deposits of a large crater, it can also be attributed to the general state of our knowledge of cratering phenomena, particularly those accompanying large events. The most common shock-metamorphosed lunar samples are breccias, but a substantial number are impact-melt rocks. Indeed, numerous workers have called attention to the importance of impact-melt rocks spanning a wide range of ages in the lunar sample collection. Photogeologic studies also have demonstrated the widespread occurrence of impact-melt lithologies in and around lunar craters. Thus, it is clear that impact melting has been a fundamental process operating throughout lunar history, at scales ranging from pits formed on individual regolith grains to the largest impact basins. This contribution examines the potential relationship between impact melting on the Moon and the interior morphologies of large craters and peaking basins. It then examines some of the implications of impact melting at such large scales for lunar-sample provenance and evolution of the lunar crust.

http://www.planetary.brown.edu/planetary/geo2810/papers/Cintala_Grieve_1998.pdf

Scaling impact melting and crater dimensions: Implications for the lunar cratering record

SANISAND is the name used for a family of simple anisotropic sand constitutive models developed over the past few years within the framework of critical state soil mechanics and bounding surface plasticity. The existing SANISAND models use a narrow open cone-type yield surface with apex at the origin obeying rotational hardening, which implies that only changes of the stress ratio can cause plastic deformations, while constant stress-ratio loading induces only elastic response. In order to circumvent this limitation, the present member of the SANISAND family introduces a modified eight-curve equation as the analytical description of a narrow but closed cone-type yield surface that obeys rotational and isotropic hardening. This modification enables the prediction of plastic strains during any type of constant stress-ratio loading, a feature lacking from the previous SANISAND models, without losing their well-established predictive capability for all other loading conditions including the cyclic. In the process the plausible assumption is made that the plastic strain rate decomposes in two parts, one due to the change of stress ratio and a second due to loading under constant stress ratio, with isotropic hardening depending on the volumetric component of the latter part only. The model formulation is presented firstly in the triaxial stress space and subsequently its multiaxial generalization is developed following systematically the steps of the triaxial one. A detailed calibration procedure for the model constants is presented, while successful simulation of both drained and undrained behavior of sands under constant and variable stress-ratio loadings at various densities and confining pressures is obtained by the model.

/pdf/sanisand-simple-anisotropic-sand-plasticity-model-4h2ohbjwyx.pdf

J. Dennis Lawrence

Papers

A Catalog of Special Plane Curves