Topic
Conical coordinates
About: Conical coordinates is a research topic. Over the lifetime, 196 publications have been published within this topic receiving 3715 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: In this article, a general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived, defined as the forces acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero.
Abstract: A new, general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calculating the free energy changes along these coordinates. This method is tested in two examples - rotation around the C-C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calculated free energies are compared with those obtained by two commonly used methods. One of them relies on determining the probability density function of finding the system at different values of the selected coordinate and the other requires calculating the average force at discrete locations along this coordinate in a series of constrained simulations. The free energies calculated by these three methods are in excellent agreement. The relative advantages of each method are discussed.
1,143 citations
[...]
TL;DR: Green coordinates as discussed by the authors are motivated by Green's third integral identity and respect both the vertices position and faces orientation of the cage, leading to space deformations with a shape-preserving property.
Abstract: We introduce Green Coordinates for closed polyhedral cages. The coordinates are motivated by Green's third integral identity and respect both the vertices position and faces orientation of the cage. We show that Green Coordinates lead to space deformations with a shape-preserving property. In particular, in 2D they induce conformal mappings, and extend naturally to quasi-conformal mappings in 3D. In both cases we derive closed-form expressions for the coordinates, yielding a simple and fast algorithm for cage-based space deformation. We compare the performance of Green Coordinates with those of Mean Value Coordinates and Harmonic Coordinates and show that the advantage of the shape-preserving property is not achieved at the expense of speed or simplicity. We also show that the new coordinates extend the mapping in a natural analytic manner to the exterior of the cage, allowing the employment of partial cages.
383 citations
Book•
17 Oct 2016
TL;DR: Equations with three space variables in Cylindrical Coordinates as discussed by the authors have been shown to be equivalent to Cartesian Equations with two Space Variables in Cartesian Coordinates and Equations in Polar Coordinates.
Abstract: 1.Introduction.- 2.Mathematical Preliminaries.- 3.Physical Backgrounds.- 4.Equations with one Space Variable in Cartesian Coordinates.- 5.Equations with one Space Variable in Polar Coordinates.- 6.Equations with one Space Variable in Spherical Coordinates.- 7.Equations with two Space Variables in Cartesian Coordinates.- 8.Equations in Polar Coordinates.- 9.Axisymmetric equations in Cylindrical Coordinates.- 10.Equations with three Space Variables in Cartesian Coordinates.- 11.Equations with three space Variables in Cylindrical Coordinates.- 12.Equations with three space Variables in Spherical Coordinates.- Conclusions.- Appendix: Integrals.- References.
258 citations
208 citations
159 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2020 | 1 |
| 2018 | 1 |
| 2017 | 6 |
| 2016 | 6 |
| 2015 | 4 |