Callum Galbraith

Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms

Implicit objects have gained increasing importance in geometric modeling, visualisation, animation, and computer graphics, because their geometric properties provide a good alternative to traditional parametric objects. This book presents the mathematics, computational methods and data structures, as well as the algorithms needed to render implicit curves and surfaces, and shows how implicit objects can easily describe smooth, intricate, and articulatable shapes, and hence why they are being increasingly used in graphical applications. Divided into two parts, the first introduces the mathematics of implicit curves and surfaces, as well as the data structures suited to store their sampled or discrete approximations, and the second deals with different computational methods for sampling implicit curves and surfaces, with particular reference to how these are applied to functions in 2D and 3D spaces.

Pen-and-Ink for BlobTree Implicit Models

We present new techniques for rendering hierarchical, skeletal implicit models in several pen-and-ink styles. Our approach extracts silhouette strokes and lines following local shape features, such as those caused by CSG junctions. To this end, it uses particle systems to find interesting areas on the surface and perform stroke stylization guided by local shape features. Also, it removes hidden lines either by applying a surfel technique for rapid prototyping, or more accurately by using ray tracing. Examples from simple to complex models illustrate the capabilities of our system.

Implicit Visualization and Inverse Modeling of Growing Trees

A method is proposed for photo‐realistic modeling and visualization of a growing tree. Recent visualization methods have focused on producing smoothly blending branching structures, however, these methods fail to account for the inclusion of non‐smooth features such as branch bark ridges and bud scale scars. These features constitute an important visual aspect of a living tree, and are also observed to vary over time. The proposed method incorporates these features by using an hierarchical implicit modeling system, which provides a variety of tools for combining surface components in both smooth and non smooth configurations. A procedural interface to this system supports the use of inverse modeling, which is a global‐to‐local methodology, where the local properties of plant organs are described in terms of their global position within the tree architecture. Inverse modeling is used to define both the tree structure at any time, and a continuous developmental sequence for the tree from a seedling. These techniques provide an intuitive paradigm for the definition of complex tree growth sequences and their subsequent visualization, based solely on observed phenomena. Thus, a key advantage is that they do not require any knowledge of, or simulation of, the underlying biological processes.

BlobTree trees

In recent years several methods for modeling botanical trees have been proposed. The geometry and topology of tree skeletons can be well described by L-systems; however, there are several approaches to modeling smooth surfaces to represent branches, and not all of the observed phenomena can be represented by current methods. Many tree types exhibit nonsmooth features such as branch bark ridges and collars. In this research a hierarchical implicit modeling system is used to produce models of branching structures that capture smooth branching, branch collars and branch bark ridges. The BlobTree provides several techniques to control the combination of primitives, allowing both smooth and nonsmooth effects to be intuitively combined in a single blend volume. Irregular effects are implemented using precise contact modeling, constructive solid geometry and space warping. We show that smooth blends can be obtained, without noticeable bulging, using summation of distance based implicit surfaces. L-systems are used to create the branching structure allowing botanically based simulations to be used as input

Efficient use of the BlobTree for rendering purposes

One of the major applications of implicit surface modeling systems has been the generation of cartoon-like characters. Recently, additional modeling methods have been combined with implicit surfaces to create much more complex models. These methods include constructive solid geometry (CSG), warping, and two-dimensional texture mapping (among others). The BlobTree has been introduced to organize all of these elements into a single structure which allows both local and global applications of each of these techniques in a general and intuitive fashion. The BlobTree lends itself well to rapid and direct specification of complex models, however current implementations of the BlobTree have not been engineered for efficiency, and perform poorly when attempting to render large models. In this work we apply established techniques, such as spatial subdivision and tree optimization, to the BlobTree. The objective is to increase efficiency during rendering without restricting the functionality of the BlobTree as a modeling tool.

Modeling Murex cabritii sea shell with a structured implicit surface modeler

Implicit surface modeling systems have been used since the mid-1980s for the generation of cartoon like characters. Recently implicit models combined with constructive solid geometry (CSG) have been used to build engineering models with automatic blending. This work is built on a structured implicit modeling system which includes CSG, warping, 2D texture mapping and operations based on the BlobTree, and its application to the generation of a complex and visually accurate biological model of the sea shell Murex cabritii. Since the model is purely procedurally defined and does not rely on polygon mesh operations, it is resolution independent and can be rendered directly using ray tracing. An interface has been built to the BlobTree using an interpreted programming language (Python). The language interface readily allows a user to procedurally describe the shell based on numeric data taken from the actual object.