This paper is concerned with algorithmic techniques for the incremental generation of continuous curves that can efficiently cover an abstract surface. We introduce the notion of low-discrepancy curves as an extension of the notion of low-discrepancy sequences—such that sufficiently smooth curves with low-discrepancy properties can be defined and generated. We then devise a procedure for liftin...

We introduce a space-filling curve for triangular and tetrahedral red-refinement that can be computed using bitwise interleaving operations similar to the well-known Z-order or Morton curve for cubical meshes. To store the information necessary for random access, we suggest 10 bytes per triangle and 14 bytes per tetrahedron. We present algorithms that compute the parent, children, and face-neig...

We introduce a table-based specification for spacefilling curves. This specification consists of a primitive curve representation and a movement specification table. We present an algorithm that uses these tables to enumerate points on a space-filling curve and discuss how to generate such tables automatically. Experimental results on three modern microprocessor based platforms show that our al...

We elucidate the relationship between space-llling curves and the Euclidean Traveling 2 Salesman Problem (TSP) by reference to a particular space-lling curve whose scaling behaviour is strongly related to the conjectured scaling behaviour of the optimal TSP tour. We suggest that space-lling curves can be used to generate testbed TSPs: sets of points which in the limit cover a planar surface and...

The Nearest Neighbor algorithm is one of the simplest and oldest classification techniques. A given collection of historic data (Training Data) of known classification is stored in memory. Then based on the stored knowledge the classification of an unknown data (Test Data) is predicted by finding the classification of the nearest neighbor. For example, if an instance from the test set is presen...

We present a novel approach to parameterised curve detection. The method is based on the generalised Radon transform, which is traditionally applied to a 2D edge/line map. The novelty of our method is the mapping of the original 2D image to a 3D orientation space, which then forms the input for the Radon transform. The orientation space representation can represent multiple intersecting structu...

We study the properties of points in [0, 1]d generated by applying Hilbert’s space-filling curve to uniformly distributed points in [0, 1]. For deterministic sampling we obtain a discrepancy of O(n−1/d) for d ≥ 2. For random stratified sampling, and scrambled van der Corput points, we get a mean squared error of O(n−1−2/d) for integration of Lipshitz continuous integrands, when d ≥ 3. These rat...

This paper presents a methodology to control motion planning of a hyper-redundant planar pace robot tracing a specified end-effector path. For this, a space robot of 6 DOF has been considered for study. A hyperredundancy enhances the capability to realize various kinds of task, i.e., obstacle avoidance, joint limits, singularity avoidance, etc. For full utilization of its potential capability, ...

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present algorithms for computing their defining polynomia...