Cusp Curves Research Articles

Topology driven approximation to rational surface-surface intersection via interval algebraic topology analysis

Computing the intersection between two parametric surfaces (SSI) is one of the most fundamental problems in geometric and solid modeling. Maintaining the SSI topology is critical to its computation robustness. We propose a topology-driven hybrid symbolic-numeric framework to approximate rational parametric surface-surface intersection (SSI) based on a concept of interval algebraic topology analysis (IATA) , which configures within a 4D interval box the SSI topology. We map the SSI topology to an algebraic system's solutions within the framework, classify and enumerate all topological cases as a mixture of four fundamental cases (or their specific sub-cases). Various complicated topological situations are covered, such as cusp points or curves, tangent points (isolated or not) or curves, tiny loops, self-intersections, or their mixtures. The theoretical formulation is also implemented numerically using advanced real solution isolation techniques, and computed within a topology-driven framework which maximally utilizes the advantages of the topology maintenance of algebraic analysis, the robustness of iterative subdivision, and the efficiency of forward marching. The approach demonstrates improved robustness under benchmark topological cases when compared with available open-source and commercial solutions, including IRIT, SISL, and Parasolid.

Singularity Computation for Rational Parametric Surfaces Using Moving Planes

Singularity computation is a fundamental problem in Computer Graphics and Computer Aided Geometric Design, since it is closely related to topology determination, intersection, mesh generation, rendering, simulation, and modeling of curves and surfaces. In this article, we present an efficient and robust algorithm for computing all the singularities (including their orders) of rational parametric surfaces using the technique of moving planes. The main approach is first to construct a representation matrix whose columns correspond to moving planes following the parametric surface. Then, by substituting the parametric equation of the rational surface into this representation matrix, one can extract the singularity information from the corresponding matrix and return all the singular loci including self-intersection curves, cusp curves, and isolated singular points of the rational surface, together with the order of each singular locus. We present some examples to compare our algorithm with state-of-the-art methods from different perspectives including robustness, efficiency, order computation, and numerical stability, and the experimental results show that our method outperforms existing methods in all these aspects. Furthermore, applications of our algorithm in surface rendering, mesh generation and surface/surface intersections are provided to demonstrate that correctly computing the self-intersection curves of a surface is essential to generate high quality results for these applications.

Bifurcation Analysis of a Heat-Sensitive Epigenetic Regulatory Network.

An epigenetic regulatory network that influences transgenerational inheritance of a heat-altered phenotype was recently discovered in Arabidopsis. Our analysis shows that transgenerational inheritance of the heat-altered phenotype operates in a switch-like manner and can be turned on or off as a function of heat. We also show that trans-acting small interfering RNAs act as an "inverse amplifier" of HTT5, the protein that controls the heat-altered phenotype by a currently unknown mechanism. Our analysis uses the resultant to find an analytic expression for a cusp curve in parameter space and to find a parameter bound on switch-like behavior.

Duality of the Evolute-Involute Pair and Its Application

We studied the type of cusps on the caustic in the optical model that the coplanar point light source irradiates a singular planar curvilinear mirror in our last paper. In this paper, we give a oneto-one correspondence between the wavefront (orthotomic) and the caustic. According to this one-toone correspondence, we can derive the property of one of them from that of the other one. To prove it, we formulate the involutes of (n, m)-cusp curves and study their singular properties. Furthermore, we give a duality theorem for evolute-involute pairs. This theorem plays a crucial role in the one-to-one correspondence mentioned earlier.

Evolutes of (n, m)-cusp curves and application in optical system

The focal properties of generic curvilinear planar mirrors illuminated by a coplanar light source are very clear. However, the case is vague when the mirror has singularities. In this paper, we define the evolutes of (n, m)-cusp curves in Euclidean 2-space. The (n, m)-cusp curve is a generalization of regular curves which might have singularities. Furthermore, we study the behavior and the singular property of the evolute at the point corresponding to the (n, m)-cusp of the base curve. As an application, we study the caustic of the curvilinear planar mirror which is illuminated by a coplanar light source when the mirror is an (n, m)-cusp curve.

IMAGE-PLANE ANALYSIS OFn-POINT-MASS LENS CRITICAL CURVES AND CAUSTICS

The interpretation of gravitational microlensing events caused by planetary systems or multiple stars is based on the n-point-mass lens model. The first planets detected by microlensing were well described by the two-point-mass model of a star with one planet. By the end of 2014, four events involving three-point-mass lenses had been announced. Two of the lenses were stars with two planetary companions each; two were binary stars with a planet orbiting one component. While the two-point-mass model is well understood, the same cannot be said for lenses with three or more components. Even the range of possible critical-curve topologies and caustic geometries of the three-point-mass lens remains unknown. In this paper we provide new tools for mapping the critical-curve topology and caustic cusp number in the parameter space of n-point-mass lenses. We perform our analysis in the image plane of the lens. We show that all contours of the Jacobian are critical curves of re-scaled versions of the lens configuration. Utilizing this property further, we introduce the cusp curve to identify cusp-image positions on all contours simultaneously. In order to track cusp-number changes in caustic metamorphoses, we define the morph curve, which pinpoints the positions of metamorphosis-point images along the cusp curve. We demonstrate the usage of both curves on simple two- and three-point-mass lens examples. For the three simplest caustic metamorphoses we illustrate the local structure of the image and source planes.

A NOTE ON THE HILBERT SCHEME OF POINTS ON A CUSP CURVE

We study the Hilbert scheme of points on a cusp curve, mostly the punctual Hilbert scheme [Formula: see text] which parameterizes m points supported at a cusp. We show that the reduced punctual scheme [Formula: see text] is isomorphic to ℙ1 and the Hilbert scheme has one singularity along [Formula: see text].

HITCHIN SYSTEMS ON SINGULAR CURVES II: GLUING SUBSCHEMES

In this paper we continue our studies of Hitchin systems on singular curves (started in [1]). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec {f ∈ ℂ[z] : f(A(∊)) = f(B(∊)); ∊N = 0}, where A(∊), B(∊) are arbitrary polynomials). The most simple examples are the generalized cusp curves which are projectivizations of Spec {f ∈ ℂ[z] : f′(0) = f″(0) = ⋯ fN-1(0) = 0}. We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials A(∊), B(∊) defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop r-matrix formalism for the functions on the truncated loop group GLn(ℂ[z]), zN = 0. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space.

Plane curves, wavefronts and Legendrian knots

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Goryunov V. V. 2001Plane curves, wavefronts and Legendrian knotsPhil. Trans. R. Soc. A.3591497–1510http://doi.org/10.1098/rsta.2001.0837SectionRestricted accessPlane curves, wavefronts and Legendrian knots V. V. Goryunov V. V. Goryunov Department of Mathematical Sciences, Divison of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, UK () Google Scholar Find this author on PubMed Search for more papers by this author V. V. Goryunov V. V. Goryunov Department of Mathematical Sciences, Divison of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, UK () Google Scholar Find this author on PubMed Search for more papers by this author Published:15 July 2001https://doi.org/10.1098/rsta.2001.0837AbstractWe survey some of the recent results on Legendrian knots and links in the standard contact 3–space and solid torus. These include the description of finite–order invariants and estimates of the self–linking number coming from the classical polynomial link invariants. We also describe the combinatorial invariant introduced by Chekanov and Pushkar, which allowed them to prove Arnold's conjecture on the necessity of four–cusp curves in generic eversions of a circular front in the plane. Previous Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by SAYGILI K (2012) TOPOLOGICALLY MASSIVE ABELIAN GAUGE THEORY, International Journal of Modern Physics A, 10.1142/S0217751X08039840, 23:13, (2015-2035), Online publication date: 20-May-2008. Thornby J, MacKay R and Williams M (2015) Mathematical principles for the design of isostatic mount systems for dynamic structures, IMA Journal of Applied Mathematics, 10.1093/imamat/hxv020, (hxv020) This Issue15 July 2001Volume 359Issue 1784Discussion Meeting Issue ‘Topological methods in the physical sciences’ organized by V. I. Arnold, J. W. Bruce, H. K. Moffatt and R. B. Pelz Article InformationDOI:https://doi.org/10.1098/rsta.2001.0837Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/07/2001Published in print15/07/2001 License: Citations and impact KeywordsLegendrian linksChekanov–Pushkar invariantpolynomial knot invariantsBennequin numberarnold–s conjecturefinite–type invariants

Change Curve Estimation via Wavelets

The recently developed theory of wavelets has a remarkable ability to “zoom in” on very short-lived frequency phenomena, such as transients in signals and singularities in functions, and hence provides an ideal tool to study localized changes. This article proposes a wavelet method for estimating jump and sharp cusp curves of a function in the plane. The method involves first computing wavelet transformation of data and then estimating jump and sharp cusp curves by wavelet transformation across fine scales. Asymptotic theory is established, and simulations are carried out to lend some credence to the asymptotic theory. The wavelet estimate is nearly optimal and can be computed by fast algorithms. The method is applied to a real image.

Wave Fronts for Hamilton-Jacobi Equations:¶The General Theory for Riemann Solutions in $\calR^n$

The Hamilton-Jacobi equation describes the dynamics of a hypersurface in \(\). This equation is a nonlinear conservation law and thus has discontinuous solutions. The dependent variable is a surface gradient and the discontinuity is a surface cusp. Here we investigate the intersection of cusp hypersurfaces. These intersections define (n-1)-dimensional Riemann problems for the Hamilton-Jacobi equation. We propose the class of Hamilton-Jacobi equations as a natural higher-dimensional generalization of scalar equations which allow a satisfactory theory of higher-dimensional Riemann problems. The fist main result of this paper is a general framwork for the study of higher-dimensional Riemann problems for Hamilton-Jacobi equations. The purpose of the framwork ist to unterstand the structure of Hamilton-Jacobi wave interactions in an explicit and constructive manner. Specialized to two-dimensional Riemann problems (i.e., the intersection of cusp curves on surfaces embedded in \(\)), this framework provides explicit solutions to a number of cases of interest. We are specifically interested in models of deposition and etching, important processes for the manufacture of semiconductor chips.

Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops: errata

Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [ P. L. Marston E. H. Trinh , Nature London312, 529– 531 ( 1984)]. The principal curvatures of the generic local wave front that produces the far-field transverse cusp are examined. This wave front is shown to generate a transverse cusp curve (U − Uc)3 = − d∞V2, where U and V are horizontal and vertical scattering angles and Uc is the cusp point direction. The far-field opening rate d∞ is calculated for the transverse cusp. It is shown that d∞ has a simple dependence on the parameters of the generic wave front. We define the aspect ratio of the drop q = D/H, where H is the height and D is the equatorial width for the scattering drop. The method of generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wavefront and hence to d∞. Measurements of d∞. for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q4 ≈ 1.31, the critical value for generation of a hyperbolic–umbilic focal section, the predicted d∞ goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d∞ vanishes as q approaches critical values for lips and transition events.

Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops

Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [P. L. Marston and E. H. Trinh, Nature London 312, 529-531 (1984)]. The principal curvatures of the generic local wave front that produces the far-field transverse cusp are examined. This wave front is shown to generate a transverse cusp curve (U - U(c))(3) = - d(infinity)V(2), where U and V are horizontal and vertical scattering angles and U(c) is the cusp point direction. The far-field opening rate d(infinity) is calculated for the transverse cusp. It is shown that d(infinity) has a simple dependence on the parameters of the generic wave front. We define the aspect ratio of the drop q = D/H, where H is the height and D is the equatorial width for the scattering drop. The method of generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wave front and hence to d(infinity). Measurements of d(infinity) for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q(4) approximately 1.31, the critical value for generation of a hyperbolic-umbilic focal section, the predicted d(infinity) goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d(infinity) vanishes as q approaches critical values for lips and transition events.

Observations of transverse cusp diffraction catastrophes produced by reflecting long and short ultrasonic bursts from a curved metal surface in water

Transverse cusp caustics are predicted to be produced when sound or light from a point source is reflected by a smooth curved surface having the general shape h(x,y) = h1x2 + h2xy2 + h3y2 + h4x + h5y (with h2≠0) [P. L. Marston, in Acoustical Imaging (Plenum, New York, 1988), Vol. 16, p. 579]. The transverse cusp curve partitions space into regions where either one or three rays contribute to the wave field. The diffraction pattern exhibited by a cusp caustic is proportional to the Pearcey function. Experiments were conducted that produced a transverse cusp using a focused source of 1‐MHz sound and a smooth curved metal surface as a reflector. The shape of the reflector was measured to determine the parameters in h(x,y) for that particular surface. Long tone bursts were used to simulate a steady‐state signal in order to image the acoustical wave field in a plane distant from the reflector. Experimental and calculated diffraction patterns are in fair agreement when the measured values of h1 and h2 were used in the calculation. Single‐cycle bursts were used to show the transition from the three‐ray to the one‐ray region across the cusp curve. Temporal records manifest the transition expected for slices through the imaged wave field. An optical source and receiver replaced the acoustical source and receiver to image the corresponding optical wave field. A distinct optical cusp curve was visible as expected for the short wavelength limit. [Work supported by ONR.]

Lips caustics in light backscattered from an acoustically levitated spheroidal water drop

Cusp diffraction catastrophes that open up roughly transverse to the propagation direction are known to exist for both light and sound [P. L. Marston, J. Acoust. Soc. Am. 81, 226–232 (1987)]. They may be produced by reflecting high-frequency sound from curved surfaces. In the present research, a closely related caustic in which the cusp curves join to form a pair of lips is studied. This caustic was predicted to exist [J. F. Nye, Nature 312, 531–532 (1984)] in light backscattered from horizontally illuminated oblate drops of water, provided the axis ratio q = D/H was within certain ranges. The associated rays are diffracted at the drop's surface and have only one internal reflection. As q is increased above the critical value q4≈1.31 associated with a hyperbolic umbilic focal section, the cusp points at the ends of the lips caustic were predicted to merge in the backward direction at a “lips event” when q = qL1 ≈ 1.42. For qL1 < q < qL2 ≈ 1.58, no caustics for this class of ray are expected while a second lips event occurs at qL2. Observations of farfield scattering from levitated drops support Nye's analysis and illustrate a mechanism for producing lips caustics. The observed backscattering is weak for q between qL1 and qL2. [Work supported by ONR.]

Weak foci in three-dimensional linear shock waves and the stability of real shock waves

A first approximation for identifying foci in weak shocks is to consider the focusing process for a linear discontinuity. This is sometimes known as the “acoustic shock” approximation. A case studied previously is the longitudinal cusp of “arête” located where three rays merge at the focus of a converging cylindrical wave; for finite amplitudes, the wave can become unstable with respect to the formation of a pair of shock-shocks in the focal region [B. Sturtevant and V. Kulkarny, J. Fluid Mech. 73, 651–671 (1976)]. In the present research, catastrophe theory is used to identify wave-front shapes W(x,y) that propagate to produce novel acoustic foci with caustics spread out roughly transverse to the direction of propagation [P. L. Marston, J. Acoust. Soc. Am. 81, 226–232 (1987) and Proceedings of the APS 1987 Topical Conference on Shock Waves in Condensed Matter (in press)]. The initial displacement of the wave front from the xy plane is given by W(x,y). Linear waves with W = − (a1x2 + a2y2x + a3y2), where the shape parameter a2≠0, propagate to produce transverse cusps described by a cubic cusp curve. A stronger focus, the hyperbolic umbilic, is produced if W = − [(αx3 + 3γy2x)/6 + a1(x2 + y2)] with a1⩽0. A focal section is produced at a distance z = − (2a1)−1 from the xy plane. In this section, caustic lines form a V with an apex angle of 2 arctan (γ/α)1/2 Issues related to shock stability are noted. [Work supported by ONR.]