https://u.math.biu.ac.il/~katzmik/goldman05.pdf

Curvature formulas for implicit curves and surfaces

A distance field is a representation where, at each point within the field, we know the distance from that point to the closest point on any object within the domain. In addition to distance, other properties may be derived from the distance field, such as the direction to the surface, and when the distance field is signed, we may also determine if the point is internal or external to objects within the domain. The distance field has been found to be a useful construction within the areas of computer vision, physics, and computer graphics. This paper serves as an exposition of methods for the production of distance fields, and a review of alternative representations and applications of distance fields. In the course of this paper, we present various methods from all three of the above areas, and we answer pertinent questions such as How accurate are these methods compared to each other? How simple are they to implement?, and What is the complexity and runtime of such methods?.

/pdf/3d-distance-fields-a-survey-of-techniques-and-applications-2qj0vd6xsa.pdf

3D distance fields: a survey of techniques and applications

Regularised marching tetrahedra: improved iso-surface extraction

Remeshing is a key component of many geometric algorithms, including modeling, editing, animation and simulation. As such, the rapidly developing field of geometry processing has produced a profusion of new remeshing techniques over the past few years. In this paper we survey recent developments in remeshing of surfaces, focusing mainly on graphics applications. We classify the techniques into five categories based on their end goal: structured, compatible, high quality, feature and error-driven remeshing. We limit our description to the main ideas and intuition behind each technique, and a brief comparison between some of the techniques. We also list some open questions and directions for future research.

/pdf/recent-advances-in-remeshing-of-surfaces-27z7ve434c.pdf

Recent Advances in Remeshing of Surfaces

The precision of photometric and spectroscopic observations has been systematically improved in the last decade, mostly thanks to space-borne photometric missions and ground-based spectrographs dedicated to finding exoplanets. The field of eclipsing binary stars strongly benefited from this development. Eclipsing binaries serve as critical tools for determining fundamental stellar properties (masses, radii, temperatures, and luminosities), yet the models are not capable of reproducing observed data well, either because of the missing physics or because of insufficient precision. This led to a predicament where radiative and dynamical effects, insofar buried in noise, started showing up routinely in the data, but were not accounted for in the models. PHOEBE (PHysics Of Eclipsing BinariEs; http://phoebe-project.org) is an open source modeling code for computing theoretical light and radial velocity curves that addresses both problems by incorporating missing physics and by increasing the computational fidelity. In particular, we discuss triangulation as a superior surface discretization algorithm, meshing of rotating single stars, light travel time effects, advanced phase computation, volume conservation in eccentric orbits, and improved computation of local intensity across the stellar surfaces that includes the photon-weighted mode, the enhanced limb darkening treatment, the better reflection treatment, and Doppler boosting. Here we present the concepts on which PHOEBE is built and proofs of concept that demonstrate the increased model fidelity.

https://iopscience.iop.org/article/10.3847/1538-4365/227/2/29/pdf

Physics of eclipsing binaries. ii. toward the increased model fidelity

Surface triangulations are necessary in applying finite element methods for solving mechanical problems and for displaying surfaces by ray tracing or other hidden line algorithms. A parametric surface can be a triangulated by triangulating its (plane) area of definition. However, the images of these triangles in object space may vary unacceptably for the application. Thus we need suitable methods of triangulation even for parametric surfaces. Triangulation algorithms for implicit surfaces are available in the literature. [ALGN'91; BL'88; LO'87; SC'93; WY'86]. All these methods divide the space into suitable polyhedrons (cubes, tetrahedrons) and determine the section of the given implicit surface with the edges of these polyhedrons. The intention of this paper is to introduce a marching method to build a mesh of triangles successively by starting with a point or a prescribed polygon. The triangulation is terminated by several bounding polygons (on the given surface) or a global bounding box. (A similar idea is used in the recently published paper [BAXU'97] on algebraic surfaces.) The method will be established for implicit surfaces. With the idea of numerical implicitization introduced in [HA'97], the triangulation is applicable to any surface for which foot points (i.e., points of minimal distance to the surface) can be determined.The main advantages of the triangulation presented in this paper are:

/pdf/a-marching-method-for-the-triangulation-of-surfaces-16545kd6e5.pdf

A marching method for the triangulation of surfaces

http://www.cgeo.ulg.ac.be/CAO/hartmann_normalforms.pdf

On the curvature of curves and surfaces defined by normalforms

G n-1 -functional splines for interpolation and approximation of curves, surfaces and solids

We introduce a method for curvature-continuous (G2) interpolation of an arbitrary sequence of points on a surface (implicit or parametric) with prescribed tangent and geodesic curvature at every point. The method can also be used forG2 blending of curves on surfaces. The interpolation/blending curve is the intersection curve of the given surface with a functional spline (implicit) surface. For the construction of blending curves, we derive the necessary formulas for the curvature of the surfaces. The intermediate results areG2 interpolation/blending methods in IR2.

G2 interpolation and blending on surfaces

Numerical implicitization for intersection and G n -continuous blending of surfaces

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