Curves Of Width Research Articles

Rasterizing curves of constant width

This paper gives a fast, linear-time algorithm for generating high-quality pixel representations of curved lines. The results are similar to what is achieved by selecting a circle whose diameter is the desired line width, and turning on all pixels covered by the circle as it moves along the desired curve. However, the circle is replaced by a carefully chosen polygon whose deviations from the circle represent subpixel corrections designed to improve the aesthetic qualities of the rasterized curve. For nonsquare pixels, equally good results are obtained when an ellipse is used in place of the circle. The class of polygons involved is introduced, an algorithm for generating them is given, and how to construct the set of pixels covered when such a polygon moves along a curve is shown. The results are analyzed in terms of a mathematical model for the uniformity and accuracy of line width in the rasterized image.

Some global properties and constructions for closed curves in the plane

This is an extended comment on the paper [1] by W. Cieślak and W. Mozgawa. It is shown that several theorems presented there for rosettes can be extended to a larger class of closed curves by using different methods of proof. Furthermore, the notion of rosettes of constant width is discussed in more detail and some classical constructions for convex curves of constant width are generalized to this class of curves.

Curves of Constant Width from a Linear Viewpoint

Curves of Constant Width from a Linear Viewpoint

Study of the Kα emission spectrum of copper

AbstractA simple model of the x‐ray Kα emission spectrum of copper metal has been proved to fit the experimental distribution curve of diffracted intensity fairly well, after correcting for profile distortion due to the spectrometer geometry and the diffraction process. Each Kα line can be described by a doublet of overlapping dispersion curves of different height and width. Although the physical relevance of the model is not clear at present, the simple analytical expression of the spectrum may be useful for some problems in x‐ray diffractometry.

Convex Curves with Periodic Billiard Polygons

Let C be a piecewise-smooth closed-convex curve in the plane R 2. A billiard ball is a point interior to C which moves with unit velocity until it hits a point p E C. If C has a tangent at p the ball bounces in the direction determined by reflection off of the unique tangent line at p, again with unit velocity. The billiard path is then the well-determined orbit traced by the point in the course of time, where we shall neglect orbits which encounter any corner point of C. A billiard path is said to be periodic if the orbit is a closed polygon with finitely many vertices on C. The polygon need not be simple; for example, nonsimple star figures occur as billiard orbits in circles. It is of interest to determine if a class of convex curves (or, more generally, a class of convex figures in Rd) can be characterized by a corresponding property of the billiard paths within the figure. Indeed there are a number of results of this type, both recent and classical. The billiard characterizations of the conic sections have long been known. For an example of a more recent result, a closed convex polygon P is regular if and only if P contains a periodic billiard path P' which is similar to P (see [1]). Two billiard characterizations of curves of constant width are known, the more recent an elegant result found in [3]. The other characterization motivates the work of this paper and is obtained by simply restating the well-known double normal property: a smooth closed convex curve C has constant width if and only if there is a periodic billiard 2-gon (i. e., double segment) at each point of C. This characterization of curves of constant width raises a rather intriguing question:

Smooth Curves of Constant width and Transnormality

Smooth Curves of Constant width and Transnormality

Cooling rates of seven hexahedrites

Cooling rates for seven hexahedrites, (Uwet, Coahuila, Walker County, Lombard, Quillagua, Hex River Mountains and Tocopilla) have been determined using a ternary diffusion controlled phase growth analysis developed by the authors. The model is applied to the exsolution and growth of plate phosphides in the kamacite phase of hexahedrites during cooling of the meteorite in its parent body. The effects of cooling rate, bulk composition, nucleation temperature and diffusion field length are considered. A unique cooling rate was determined by comparing the Ni content and width of several phosphides in a given hexahedrite to computer-generated curves of Ni content vs phosphide width. Six hexahedrites have cooling rates of approximately 2°C/10 6yr. One hexahedrite, Coahuila, has a somewhat higher cooling rate of 10°C/10 6 yr. These cooling rates fall within the range calculated by an independent method for octahedrites. The cooling rate analysis indicates that the hexahedrites, except for one possible exception, were formed in or close to the core of a parent body ~ 150 km in radius.

A Characterisation of Plane Curves of Constant Width

Journal of the London Mathematical SocietyVolume s2-16, Issue 3 p. 536-538 Notes and papers A Characterisation of Plane Curves of Constant Width K. J. Falconer, K. J. Falconer Corpus Christi College, CambridgeSearch for more papers by this author K. J. Falconer, K. J. Falconer Corpus Christi College, CambridgeSearch for more papers by this author First published: December 1977 https://doi.org/10.1112/jlms/s2-16.3.536Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes2-16, Issue3December 1977Pages 536-538 RelatedInformation

Scattering in the atmosphere of Venus: III. Line profiles and phase curves for Rayleigh scattering

Spectral line profiles, curves of growth, and curves for the equivalent width of a line as a function of Venus phase angle have been computed for a Rayleigh scattering cloud and compared with those for a cloud of isotropic scatterers. The results are very similar for the two kinds of scattering, with the exception of the curves of equivalent width as a function of Venus phase angle. These latter curves exhibit the “inverse phase effect” and rule out the possibility that the scale height of the clouds can be much less than half the scale height of the gas. The optical depth of the clouds, τc, is approximately 100.

Smooth curves of constant width

The construction of curves of constant width using circular arcs is well known; the procedure may be found, for example, in [1]. This article describes a different method for constructing a family of ‘smooth’ curves of constant width. Basic properties of such curves may be found in [1]. Let C be a regular, smooth, and convex curve in the euclidean plane. Regularity implies that each point of C lies on only one support line and each support line contains only one point of C, smoothness implies the existence of derivatives at each point of C, and convexity implies the curve is a simple closed curve whose interior points form a convex set. Select a point O on C as origin, use the support line to C at O as the x-axis, and give the curve a counter-clockwise orientation (see Fig. 1).

An electro-optical curve follower with analogue control

A four-quadrant photodiode detector was used in an automatic system for tracking convoluted bright lines produced on a screen by back-projection. Error signals from the detector amplifier were processed by a small analogue computer which controlled the detector movement by the action of stepping motors. The detector did not need to be rotated while following the line. Tracking accuracies of the order +or-0.3 mm have been achieved while recording digital data by this method from test curves of width 1.5 mm.

Intersections of circles and curves of constant width

Intersections of circles and curves of constant width

A Characterization of Curves of Constant Width

This contradicts the assumption k = 27r, since the quantity on the left of (11) is no larger that k2/4 = 7r2 (square formula 5). Hence equalities (6), (7), and (8) hold. These equalities are true of any selection xo, x1, *., x,, (as described). We conclude from (8) that the diameters of C have the same midpoint M; from (7) that they have the same length (which must be 2); hence the points of C are all on the sphere of radius 1 about M. The only path on this sphere, from xo to its antipode A(xo), having length ir, is half of a great circle; so formula (6) shows C must consist of two great semi-circles, which in fact form a single great circle because the diameters all pass through M. This verifies the second conjecture.

A Characterization of Curves of Constant Width

A Characterization of Curves of Constant Width

Intersection properties of curves of constant width

Two curves (boundaries of $2$-dimensional disks) of constant width in the plane, whose interiors intersect, must meet in an even or infinite number of omponents. Some new constant width curves are constructed. C0rollariesand partial converses suggest severalopen questions.

Equivalent Capacitances of Microstrip Open Circuits

The integral equations that describe the charge distribution near an open-circuited microstrip end are formulated and subsequently solved by a projective method. The solution hinges on development of computationally efficient techniques for dealing with the singularities that occur by special quadrature formulas. The necessary formulas are described and tabulated. These techniques are then used to find open-circuit capacitance values for microstrip. It is found that the curves of excess capacitance versus width are easily describable by empirical equations; such equations are presented, along with the curves themselves. For strip widths between 0.1 and 10 times the substrate thickness and for dielectric constants in the range 1-51, the given data are believed accurate to within a few percent.

Do Self-Intersections Characterize Curves of Constant Width?

(1972). Do Self-Intersections Characterize Curves of Constant Width? The American Mathematical Monthly: Vol. 79, No. 5, pp. 505-506.

Do Self-Intersections Characterize Curves of Constant Width?

Do Self-Intersections Characterize Curves of Constant Width?

Variation of Stripe-Domain Spacing with Applied Field

Theory has predicted that because of the exchange and demagnetizing energies, the stripe-domain spacing is a function of the amplitude of the (periodic) component of the magnetization normal to the film plane. Experimentally the stripe width was reported not to vary when the normal component of the magnetization was changed by applying a field parallel to the stripes. This disagreement, which was attributed to the coercive force of the stripes, has prevented a close comparison between theory and experiment. It has been found that the application of an oscillating field above some critical value in the plane of the film and normal to the stripes overcomes the coercivity associated with the stripes and allows the domain spacing to relax to that of a lower energy state. A change of spacing up to 50% has been observed in some Ni–Fe films. Experimental curves of the stripe width as a function of amplitude of the normal component of magnetization in the stripes are in rough agreement with the theoretical curve of Saito et al., except that his model and other one-dimensional models do not predict the change in stripe spacing to be as large as that observed. The dependence of the change in stripe spacing on the amplitude of the oscillating field and film parameters is discussed.

Friction stress and measurements made on hysteresis loops observed in the microstrain region

A number of experimental observations indicate that the model of microstrain in which dislocation motion is opposed by a uniform friction stress is inadequate, and that the model should be refined to allow for an orientation-dependent friction stress. This cannot be done using the line tension approximation, but a computer method has now been developed to allow the effect of a non-uniform stress on the bowing of a dislocation segment to be taken into account. Stress-strain curves for dislocations bowing against the Peierls-Nabarro friction stress are presented, and the ways in which this stress effects the dependence of the irreversible work done against friction on the plastic strain amplitude for dislocation sources of different orientations are described. The results are used to show how curves of mechanical hysteresis-loop area vs. loop width should vary with dislocation substructure, and the predictions are found to be in good agreement with experimental observations in a number of materials. Finally, the anelastic limit is shown to be associated with the stress at which the most-flexible Frank-Read source becomes unstable.