Implicit Blending Research Articles

Sketch-based implicit blending

Implicit models can be combined by using composition operators; functions that determine the resulting shape. Recently, gradient-based composition operators have been used to express a variety of behaviours including smooth transitions, sharp edges, contact surfaces, bulging, or any combinations. The problem for designers is that building new operators is a complex task that requires specialized technical knowledge. In this work, we introduce an automatic method for deriving a gradient-based implicit operator from 2D drawings that prototype the intended visual behaviour. To solve this inverse problem, in which a shape defines a function, we introduce a general template for implicit operators. A user's sketch is interpreted as samples in the 3D operator's domain. We fit the template to the samples with a non-rigid registration approach. The process works at interactive rates and can accommodate successive refinements by the user. The final result can be applied to 3D surfaces as well as to 2D shapes. Our method is able to replicate the effect of any blending operator presented in the literature, as well as generating new ones such as non-commutative operators. We demonstrate the usability of our method with examples in font-design, collision-response modeling, implicit skinning, and complex shape design.

A symbolic-numerical method for computing approximate parameterizations of canal surfaces

A canal surface is the envelope of a one-parameter family of spheres centered at the spine curve m(t) and with the radii described by the function r(t). It was proved in Peternell and Pottmann (1997) [9] that any canal surface to a rational spine curve and a rational radius function possesses a rational parameterization. Then a symbolic method for generating rational parameterizations of canal surfaces was developed in Landsmann et al. (2001) [21]. Indeed, this method leads to the problem of decomposing a polynomial into a sum of two squares over reals, which is solved numerically in general. Hence, approximate techniques generating a parameterization within a certain region of interest are also worth studying. In this paper, we present a method for the computation of approximate rational parameterizations of canal surfaces. A main feature of our approach is a combination of symbolic and numerical techniques yielding approximate topology-based parameterizations of contour curves which are then applied to compute an approximate parameterization of the given canal surface. The algorithm is mainly suitable for implicit blend surfaces of the canal-surface-type.

A new implicit blending technique for volumetric modelling

Current implicit blending techniques are mostly designed for use in surface modelling, where only boundaries of the object defined by the implicit primitives are important. In contrast, in volumetric implicit modelling the interior of the object is also significant, which requires different and more suitable techniques for combining implicit primitives. In this paper, we first discuss irregularities that occur using the current techniques. Then, a new technique for blending implicit primitives, especially appropriate in volumetric modelling (e.g., cloud modelling), is introduced. It overcomes these abnormalities and gives us better results than current techniques.

Editorial for the special section on implicit surfaces

This special section is the first of two that will be devoted to a topic of growing importance, that is the field of implicit surfaces or implicit modelling as it is also known. The first of these papers, describes a fast adaptive algorithm for polygonizing an implicit surface. The central idea of the Shrinkwrap algorithm can be explained conveniently as an example of homotopy. Indeed, the purpose is to stabilize a numerical algorithm for finding points on an implicit surface, defined by a function f . Instead of doing this in one step, the shrinkwrap algorithm takes a series of functions fi that smoothly interpolate between a trivial first function f0 and the final function f . The functions fi constitute the homotopic relation between f0 and f . The demands on modelling systems increase as there is an expectation from users to have more expressive power to build more complex models. Therefore it is important that modeling techniques can cope with the complexity of both CSG and implicit blending. In this light, the second paper makes use of advantages offered by implicit surfaces of automatic blending and introduces a new way of combining blending and CSG operations. In the third paper, the authors put a new light on an old problem. Much research has gone into the problem of reconstructing a three-dimensional surface from data-sets. The method described, creates an implicit-surface from the input data.

Bulge Elimination in Convolution Surfaces

The relationship between surface bulge and several implicit blend techniques, particularly those based on convolution of a skeleton, is discussed. An examination of branching skeletons reveals that for two and three‐dimensional skeletons, the surface will be bulge‐free if skeletal elements are sufficiently large with respect to the convolution kernel.