Dupin indicatrix

About: Dupin indicatrix is a research topic. Over the lifetime, 17 publications have been published within this topic receiving 178 citations.

Geometric Continuity: A Parametrization Independent Measure of Continuity for Computer Aided Geometric Design

Abstract: : Parametric spline curves and surfaces are typically constructed so that some number of derivatives match where the curve segments or surface patches abut. If derivatives up to order n are continuous, the segments or patches are said to meet with Cn, or nth order parametric continuity. It has been shown previously that parametric continuity is sufficient, but not necessary, for geometric smoothness. The geometric measures of unit tangent and curvature vectors for curves (objects of parametric dimension one), and tangent plane and Dupin indicatrix for surfaces (objects of parametric dimension two), have been used to define first and second order geometric continuity. These measures are intrinsic in that they are independent of the parametrizations used to describe the curve or surface. In this work, the notion of geometric continuity as a parametrization independent measure is extended for arbitrary order n(Gn), and for objects of arbitrary parametric dimension p. Two equivalent characterizations of geometric continuity are developed: one based on the notion of reparametrization, and one based on the theory of differentiable manifolds. From the basic definitions, a set of necessary and sufficient constraint equations is developed. The constraints (known as the Beta constraints) result from a direct application of the univariate chain rule for curves and the bivariate chain rule for surfaces. In the spline construction process the Beta constraints provide for the introduction of freely selectable quantities known as shape parameters. For polynomial splines, the use of the Beta constraints allows greater design flexibility through the shape parameters without raising the polynomial degree. The approach taken is important for several reasons. First, it generalizes geometric continuity to arbitrary order for both curves and surfaces. Second, it shows the fundamental connection between geometric continuity of curves and that of surfaces.

Numerical estimation of the curvature of surfaces

Abstract: Meusnier's and Euler's theorems relating to surface curvature are given as is a brief discussion on the Dupin indicatrix. The connection between Gaussian curvature and growth ratio is outlined. A method for approximating the Dupin indicatrix from surface-point data is given and, by way of comparison, a method of obtaining surface curvature information from a local quadric interpolant. Numerical examples are given and discussed

A Note on Planar Hexagonal Meshes

Abstract: We study the geometry and computation of free-form hexagonal meshes with planar faces (to be called P-Hex meshes). Several existing methods are reviewed and a new method is proposed for computing P-Hex meshes to approximate a given surface. The outstanding issues with these methods and further research directions are discussed.

A discrete curvature-based deformable surface model with application to segmentation of volumetric images

Abstract: In this paper, we present a new curvature-based three-dimensional (3-D) deformable surface model. The model deforms under defined force terms. Internal forces are calculated from local model curvature, using a robust method by a least-squares error (LSE) approximation to the Dupin indicatrix. External forces are calculated by applying a step expansion and restoration filter (SEF) to the image data. A solution for one of the most common problems associated with deformable models, self-cutting, has been proposed in this work. We use a principal axis analysis and reslicing of the deformable model, followed by triangulation of the slices, to remedy self-cutting. We use vertex resampling, multiresolution deformation, and refinement of the mesh grid to improve the quality of the model deformation, which leads to better results. Examples of the model application to different cases (simulation, magnetic resonance imaging (MRI), computerized tomography (CT), and ultrasound images) are presented, showing diversity and flexibility of the model.

Adaptive unstructured grid for three-dimensional interface representation

Abstract: Moving-boundary problems arise in numerous important physical phenomena, and often form complex shapes during their evolution. The ability to track the interface in such cases in two dimensions is well established. However, modifying the grid representing the interface as it evolves in three-dimensional space introduces additional issues. In the current work, three-dimensional interfaces are represented by adaptive unstructured grids. The grids are restructured and refined based on the shape and size of the triangular elements in the grid that forms the interfaces. As the interface deforms, points are automatically added to ensure that the accuracy of interface representation remains consistent. Results are presented to show how complex interface features, including surface curvatures and normals, can be captured by modifying an existing method that uses an approximation to the Dupin indicatrix.