Jiří Kosinka

Computing exact rational offsets of quadratic triangular Bézier surface patches

The offset surfaces to non-developable quadratic triangular Bezier patches are rational surfaces. In this paper we give a direct proof of this result and formulate an algorithm for computing the parameterization of the offsets. Based on the observation that quadratic triangular patches are capable of producing C^1 smooth surfaces, we use this algorithm to generate rational approximations to offset surfaces of general free-form surfaces.

G 1 Hermite interpolation by Minkowski Pythagorean hodograph cubics

As observed by [Choi, H.I., Han, Ch.Y., Moon, H.P., Roh, K.H., Wee, N.S., 1999. Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves. Computer-Aided Design 31, 59-72], curves in Minkowski space R^2^,^1 are very well suited to describe the medial axis transform (MAT) of a planar domain, and Minkowski Pythagorean hodograph (MPH) curves correspond to domains, where both the boundaries and their offsets are rational curves [Moon, H.P., 1999. Minkowski Pythagorean hodographs. Computer Aided Geometric Design 16, 739-753]. Based on these earlier results, we give a thorough discussion of G^1 Hermite interpolation by MPH cubics, focusing on solvability and approximation order. Among other results, it is shown that any analytic space-like curve without isolated inflections can be approximately converted into a G^1 spline curve composed of MPH cubics with the approximation order being equal to four. The theoretical results are illustrated by several examples. In addition, we show how the curvature of a curve in Minkowski space is related to the boundaries of the associated planar domain.

On the injectivity of Wachspress and mean value mappings between convex polygons

Wachspress and mean value coordinates are two generalizations of triangular barycentric coordinates to convex polygons and have recently been used to construct mappings between polygons, with application to curve deformation and image warping. We show that Wachspress mappings between convex polygons are always injective but that mean value mappings can fail to be so in extreme cases.

On rational Minkowski Pythagorean hodograph curves

Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorean hodograph curve can be obtained in terms of its associated planar rational Pythagorean hodograph curve and an additional rational function. Moreover, both in the original polynomial and new rational case, we investigate the close relationship between these associated curves in Euclidean plane and Minkowski space.

MOS surfaces: medial surface transforms with rational domain boundaries

We consider rational surface patches s(u, v) in the four-dimensional Minkowski space IR3,1, which describe parts of the medial surface (or medial axis) transform of spatial domains. The corresponding segments of the domain boundary are then obtained as the envelopes of the associated two-parameter family of spheres. If the Plucker coordinates of the line at infinity of the (two-dimensional) tangent plane of s satisfy a sum-of-squares condition, then the two envelope surfaces are shown to be rational surfaces. We characterize these Plucker coordinates and analyze the case, where the medial surface transform is contained in a hyperplane of the four-dimensional Minkowski space.

C2 Hermite interpolation by Minkowski Pythagorean hodograph curves and medial axis transform approximation

We describe and fully analyze an algorithm for C^2 Hermite interpolation by Pythagorean hodograph curves of degree 9 in Minkowski space R^2^,^1. We show that for any data there exists a four-parameter system of interpolants and we identify the one which preserves symmetry and planarity of the input data and which has the optimal approximation degree. The new algorithm is applied to an efficient approximation of segments of the medial axis transform of a planar domain leading to rational parameterizations of the offsets of the domain boundaries with a high order of approximation.

Watertight conversion of trimmed CAD surfaces to Clough-Tocher splines

The boundary representations (B-reps) that are used to represent shape in Computer-Aided Design systems create unavoidable gaps at the face boundaries of a model. Although these inconsistencies can be kept below the scale that is important for visualisation and manufacture, they cause problems for many downstream tasks, making it difficult to use CAD models directly for simulation or advanced geometric analysis, for example. Motivated by this need for watertight models, we address the problem of converting B-rep models to a collection of cubic C 1 Clough-Tocher splines. These splines allow a watertight join between B-rep faces, provide a homogeneous representation of shape, and also support local adaptivity.We perform a comparative study of the most prominent Clough-Tocher constructions and include some novel variants. Our criteria include visual fairness, invariance to affine reparameterisations, polynomial precision and approximation error. The constructions are tested on both synthetic data and CAD models that have been triangulated. Our results show that no construction is optimal in every scenario, with surface quality depending heavily on the triangulation and parameterisation that are used. Watertight conversion of boundary representations to Clough-Tocher splines.Comparative study of the most prominent Clough-Tocher constructions and some novel variants.Comparison based on visual fairness, invariance to affine reparameterisations, polynomial precision and approximation error.

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G^1 Hermite data; however, one could also obtain higher order algorithms.

Barycentric interpolation and mappings on smooth convex domains

In a recent paper, Warren, Schaefer, Hirani, and Desbrun proposed a simple method of interpolating a function defined on the boundary of a smooth convex domain, using an integral kernel with properties similar to those of barycentric coordinates on simplexes. When applied to vector-valued data, the interpolation can map one convex region into another, with various potential applications in computer graphics, such as curve and image deformation. In this paper we establish some basic mathematical properties of barycentric kernels in general, including the interpolation property and a formula for the Jacobian of the mappings they generate. We then use this formula to prove the injectivity of the mapping of Warren et al.

On the linear independence of truncated hierarchical generating systems

Motivated by the necessity to perform adaptive refinement in geometric design and numerical simulation, the construction of hierarchical splines from generating systems spanning nested spaces has been recently studied in several publications. Linear independence can be guaranteed with the help of the local linear independence of the spline basis at each level. The present paper extends this framework in several ways. Firstly, we consider spline functions that are defined on domain manifolds, while the existing constructions are limited to domains that are open subsets of R d . Secondly, we generalize the approach to generating systems containing functions which are not necessarily non-negative. Thirdly, we present a more general approach to guarantee linear independence and present a refinement algorithm that maintains this property. The three extensions of the framework are then used in several relevant applications: doubly hierarchical B-splines, hierarchical Zwart-Powell elements, and three different types of hierarchical subdivision splines.