Leo Dorst

Length estimators for digitized contours

The estimation of the length of a continuous curve from discrete data is considered. For ideal straight chaincode strings, optimal estimators are given. Comparison are performed with known methods and recommendations given. A sampling density vs. accuracy trade-off theorem is presented. The applicability to nonstraight strings is discussed. For curves that may be bonsidered to be composed of circular arcs good length estimators are found.

Geometric Algebra: A computational framework for geometrical applications Part 1

Every vector space with an inner product has a geometric algebra, whether or not you choose to use it. This article shows how to call on this structure to define common geometrical constructs, ensuring a consistent computational framework. The goal is to show you that this can be done and that it is compact, directly computational, and transcends the dimensionality of subspaces. We do not use geometric algebra to develop new algorithms for graphics, but hope to demonstrate that one can automatically take care of some of the lower level algorithmic aspects, without tricks, exceptions, or hidden degenerate cases by using geometric algebra as a language.

First order error propagation of the Procrustes method for 3D attitude estimation

The well-known Procrustes method determines the optimal rigid body motion that registers two point clouds by minimizing the square distances of the residuals. In this paper, we perform the first order error analysis of this method for the 3D case, fully specifying how directional noise in the point clouds affects the estimated parameters of the rigid body motion. These results are much more specific than the error bounds which have been established in numerical analysis. We provide an intuitive understanding of the outcome to facilitate direct use in applications.

Modeling 3D Euclidean geometry

This article compares five models of 3D Euclidean geometry-not theoretically, but by demonstrating how to implement a simple recursive ray tracer in each of them. Its meant as a tangible case study of the profitability of choosing an appropriate model, discussing the trade-offs between elegance and performance for this particular application. The models we compare are 3D linear algebra, 3D geometric algebra, 4D linear algebra, 4D geometric algebra, and 5D geometric algebra.

The Inner Products of Geometric Algebra

Making derived products out of the geometric product requires care in consistency. We show how a split based on outer product and scalar product necessitates a slightly different inner product than usual. We demonstrate the use and geometric significance of this contraction, and show how it simplifies treatment of meet and join. We also derive the sufficient condition for covariance of expressions involving outer and inner products.

Honing geometric algebra for its use in the computer sciences

A computer scientist first pointed to geometric algebra as a promising way to ‘do geometry’ is likely to find a rather confusing collection of material, of which very little is experienced as immediately relevant to the kind of geometrical problems occurring in practice. Literature ranges from highly theoretical mathematics to highly theoretical physics, with relatively little in between apart from some papers on the projective geometry of vision [143]. After perusing some of these, the computer scientist may well wonder what all the fuss is about, and decide to stick with the old way of doing things, i.e. in every application a bit of linear algebra, a bit of differential geometry, a bit of vector calculus, each sensibly used, but ad hoc in their connections. That this approach tends to split up actual issues in the application into modules that match this traditional way of doing geometry (rather than into natural divisions matching the nature of the problem) is seen as ‘the way things are’.

Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra

Conformal geometric algebra is a powerful tool to find geometrically intuitive solutions. We present an approach for the combination of compact and elegant algorithms with the generation of very efficient code based on two different optimization approaches with different advantages, one is based on Maple, the other one is based on the code generator Gaigen 2. With these results, we are convinced that conformal geometric algebra will be able to become fruitful in a great variety of applications in Computer Graphics.

Guide to Geometric Algebra in Practice

This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.

The morphological equivalent of gaussian scale-space

An image is the result of a physical measurement, e.g. the luminance on the retina or the distance from the observer to the “depicted” objects. All physical measurements are the result of the interaction of a measurement probe of finite spatial and temporal size with the physical world. The size of the measurement probe determines at what scale the world is observed. This observation scale is often called the inner scale as it is proportional to the size of the smallest details that can be meaningful distinguished in the image.

The geometrical representation of path planning problems

Abstract The path planning problem for arbitrary devices is first and foremost a geometrical problem. For the field of control theory, advanced mathematical techniques have been developed to describe and use geometrical structure. In this paper we use the notions of the flow of vector fields and geodesics in metric spaces to formalize and unify path planning problems. A path planning algorithm based on flow propagation is briefly discussed. Applications of the theory to motion planning for a robot arm, a maneuvering car, and Rubiks Cube are given. These very different problems (holomic, non-holomic and discrete, respectively) are all solved by the same unified procedure.