There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim ...at improving the interoperability between finite element method and computer aided design (CAD) software. However, these approaches have difficulty when the domain has singularities since the solution at the singularity may be multivalued. This paper develops a novel numerical approach to solve elliptic PDEs on real, closed, connected, orientable, and almost smooth algebraic curves and surfaces. Our method integrates numerical algebraic geometry, differential geometry, and a finite difference scheme which is demonstrated on several examples.
IR-806 is a near-infrared cyanine dye that forms assemblies in aqueous solutions which in turn orientationally order into a liquid crystal phase at concentrations as low as 0.5 wt %. Unlike many ...chromonic liquid crystals, the absorption spectrum of IR-806 changes dramatically with concentration, showing an isodesmic assembly process at lower concentrations followed by a second process at higher concentrations that is not isodesmic. The lower concentration assembly process is characterized by a free energy change per molecule of about 9 kBT, not unlike other chromonic systems. However, X-ray scattering measurements suggest that the assemblies that form during the higher concentration process are much larger than what is observed for many chromonic liquid crystals. Although there is a transitional region between the liquid crystal and isotropic phases of 10–15 °C, unlike most chromonic liquid crystals, no biphasic region is observed using polarizing microscopy.
We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and (ii) ...three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Grobner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver, which dramatically speeds up previous HC solving by specializing HC methods to generic cases of our problems. We show in simulated experiments that our solvers are numerically robust and stable under image noise. We show in real experiment that (i) SIFT features provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches where the state of the art structure from motion initialization fails.
Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are ...homogeneous, we investigate options for performing computations on an adaptively chosen affine coordinate patch. Second, for parameterized systems which are overdetermined, we investigate options for adaptively selecting a well-constrained subsystem to restore numerical stability. Finally, since one is typically interested in only computing real solutions for parameterized problems which arise from applications, we investigate a scheme for heuristically identifying solution paths which appear to be ending at nonreal solutions and truncating them. We demonstrate these three aspects on two problems arising in computer vision.
Chromonic liquid crystals form when certain molecules are dissolved in a solvent, usually water, and the molecules spontaneously assemble into anisotropic structures. If the density of these ...structures is high enough and the temperature is low enough, they organize into a liquid crystal phase with orientational and sometimes with positional order also. Chromonic liquid crystals have been studied for more than half a century, yet theoretical, computational, and experimental investigations in the last decade have revealed many more details about them. The molecular structures that form chromonic liquid crystals are quite varied, and as a result the assemblies that these molecules form vary significantly also. Recent research has begun to shed light on these assembly processes, revealing that these too can be quite different from one system to another.
Real monodromy action Hauenstein, Jonathan D.; Regan, Margaret H.
Applied mathematics and computation,
05/2020, Volume:
373
Journal Article
Peer reviewed
Open access
The monodromy group is an invariant for parameterized systems of polynomial equations that encodes structure of the solutions over the parameter space. Since the structure of real solutions over real ...parameter spaces are of interest in many applications, real monodromy action is investigated here. A naive extension of monodromy action from the complex numbers to the real numbers is shown to be very restrictive. Therefore, we introduce a real monodromy structure which need not be a group but contains tiered characteristics about the real solutions over the parameter space. An algorithm is provided to compute the real monodromy structure. In addition, this real monodromy structure is applied to an example in kinematics which summarizes all the ways performing loops parameterized by leg lengths can cause a mechanism to change poses.
Kinematic geometry of spatial RSSR mechanisms Rotzoll, Mirja; Regan, Margaret H.; Husty, Manfred L. ...
Mechanism and machine theory,
July 2023, 2023-07-00, Volume:
185
Journal Article
Peer reviewed
Two different novel methods to derive the input–output (IO) equation of arbitrary RSSR linkages are described. Both methods share some common elements, i.e., they use the standard Denavit–Hartenberg ...notation to first describe the linkage as an open kinematic chain, and Study’s kinematic mapping to describe the displacement of the coordinate frame attached to the end-effector of the chain with respect to the relatively non-moving base frame. The kinematic closure equation is obtained in the seven-dimensional projective kinematic mapping image space by equating the eight Study soma coordinates to the identity array. Then two methods are successfully applied to eliminate the intermediate joint angle parameters leading to the degree four biquadratic implicit algebraic IO equation: (a) the linear implicitisation algorithm, which can be applied after rearranging the closure equation such that the linkage can be viewed as two serial RS chains, and (b) numerical elimination theory using pseudowitness sets. Both approaches lead to the same IO equation. The utility of this algebraic form of the IO equation is illustrated with three detailed application examples.
•The algebraic input–output equation for spatial RSSR mechanisms is derived.•Algebraic geometry and numerical elimination methods are used.•Both methods project the motion onto S62.•The algebraic method uses linear implicitasion.•The numerical elimination method uses pseudowitness sets.
PDF
