arXiv subjects
Publications and source records attributed to T. J. Peters.
Previously, numerical evidence was presented of a self-intersecting Bezier curve having the unknot for its control polygon. This numerical demonstration resolved open questions in scientific visualization, but did not provide a formal proof of self-intersection. An example with a formal existence proof is given, even while the exact self-intersection point remains undetermined.
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergence, with that discovery aided by computational experiments. Sufficient conditions to attain isotopic equivalence can be determined a priori. These sufficient conditions need not be tight bounds, providing opportunities for further optimizations. The results presented will facilitate further computational experiments on the theory of PL knots (also known as stick knots), where this theory is less mature than for smooth knots.
The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum of total curvature and derivative. High degree Bezier curves are often used as smooth representations, where computational efficiency is a practical concern. Subdivision can produce PL approximations for a given Bézier curve, fulfilling the above two conditions. The primary contributions are: (i) a priori bounds on the number of subdivision iterations sufficient to achieve a PL approximation that is ambient isotopic to the original Bezier curve, and (ii) improved iteration bounds over those previously established.
For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance. We prove that the exterior angles of control polygons under subdivision converge to 0 at the rate of $O(\sqrt{\frac{1}{2^i}})$, where $i$ is the number of subdivisions. This angular convergence is useful for determining topological features. We use it to show homeomorphism between a Bezier curve and its control polygon under subdivision. The constructive geometric proofs yield closed-form formulas to compute sufficient numbers of subdivision iterations to obtain small exterior angles and achieve homeomorphism.
Classical topological concepts are applied to understand high performance computing simulations of molecules writhing in three dimensional space. These simulations produce peta-bytes of floating point data, to describe 3 dimensional changes in molecular structure. A zero-th order analysis is achieved by viewing a computer animation synchronized with these changes. The performance demands for animation of this voluminous data can become problematic, but techniques from low-dimensional topology are helpful. The 3D molecule is reduced to a lower dimensional model of a 1-manifold, which undergoes a piecewise linear approximation for animation. An example is presented here to show how a 1-manifold and its PL approximation could come to have different embeddings as molecular writhing proceeds. This should serve as a cautionary warning to animators to respect established sufficient conditions for topological preservation so that the movies generated will faithfully reflect the topology of the underlying model. To obtain this result, techniques from low-dimensional topology were joined with experimental mathematics and numerical analyses.
When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We establish a criterion to preserve knot type under approximation that relies upon pointwise convergence and convergence in total curvature.
Properties of a parametric curve in R^3 are often determined by analysis of its piecewise linear (PL) approximation. For Bezier curves, there are standard algorithms, known as subdivision, that recursively create PL curves that converge to the curve in distance . The exterior angles of PL curves under subdivision are shown to converge to 0 at the rate of $O(\sqrt{\frac{1}{2^i}})$, where i is the number of subdivisions. This angular convergence is useful for determining self-intersections and knot type.
For applications in computing, Bezier curves are pervasive and are defined by a piecewise linear curve L which is embedded in R^3 and yields a smooth polynomial curve C embedded in R^3. It is of interest to understand when L and C have the same embeddings. One class of counterexamples is shown for L being unknotted, while C is knotted. Another class of counterexamples is created where L is equilateral and simple, while C is self-intersecting. These counterexamples were discovered using curve visualizing software and numerical algorithms that produce general procedures to create more examples.
The Boolean algebra of regular closed sets is prominent in topology, particularly as a dual for the Stone-Cech compactification. This algebra is also central for the theory of geometric computation, as a representation for combinatorial operations on geometric sets. However, the issue of computational approximation introduces unresolved subtleties that do not occur within "pure" topology. One major effort towards reconciling this mathematical theory with computational practice is our ongoing I-TANGO project. The acronym I-TANGO is an abbreviation for "Intersections - Topology, Accuracy and Numerics for Geometric Objects". The long-range goals and initial progress of the I-TANGO team in development of computational topology are presented.