A comparison of local parametric C0 Bézier interpolants for triangular meshes

Abstract

Parametric curved shape surface schemes interpolating vertices and normals of a given triangular mesh with arbitrary topology are widely used in computer graphics for gaming and real-time rendering due to their ability to effectively represent any surface of arbitrary genus. In this context, continuous curved shape surface schemes using only the information related to the triangle corresponding to the patch under construction, emerged as attractive solutions responding to the requirements of resource-limited hardware environments. In this paper we provide a unifying comparison of the local parametric C 0 curved shape schemes we are aware of, based on a reformulation of their original constructions in terms of polynomial Bézier triangles. With this reformulation we find a geometric interpretation of all the schemes that allows us to analyse their strengths and shortcomings from a geometrical point of view. Further, we compare the four schemes with respect to their computational costs, their reproduction capabilities of analytic surfaces and their response to different surface interrogation methods on arbitrary triangle meshes with a low triangle count that actually occur in their real-world use.

Blending of mesh objects to parametric surface

“…Fünfzig et al [7] proposed the PNG 1 -Triangle by considering the neighborhood information of each face Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/cag such that each pair of adjacent curved triangles share common tangent plane along the edge. There are other curved triangle methods compared in the work by [8]. Those methods are variations of Bezier patches while the PN-Triangle is the simplest and most efficient scheme and it is therefore the preferred method to represent a smooth mesh in this paper.…”

“…(6) and (7). The error ε and the error bound δ of the eventual surface area from (6) and (7) to (8) and (9) is given as…”

A contact smoothing method for arbitrary surface meshes using Nagata patches

“…However, the quadratic degree of the Nagata interpolation does not allow to generate curves with inflection. Indeed, very sharp patches with inverted orientation can arise for specific arrangements of the nodal normal vectors directions [63]. Therefore, since the surface normal vector orientation defines the sign of the normal gap function (3), the contact status (gap or contact) can be wrongly estimated in some situations.…”

Discretization and fitting of nominal data for autonomous robots

“…In order to obtain an efficient computational method for curvilinear discretization we decided to use a numerical method which was recently proposed by Nagata (2005) and has already been applied successfully to diverse engineering problems (Neto, Oliveira, Alves, & Menezes, 2010;Neto, Oliveira, Menezes, & Alves, 2012;Boschiroli, Fünfzig, Romani, & Albrecht, 2011;Sekine, & Obikawa, 2010). The main idea behind this parametric surface description (subsequently named Nagata patch) is the quadratic discretization of a curved segment from the position and normal vectors at the end points.…”