Hyeong In Choi

Clifford Algebra, Spin Representation, and Rational Parameterization of Curves and Surfaces

The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts gives rise to different combinations of polynomials that satisfy further complicated identities. We present a novel approach to the Pythagorean hodograph curves, based on Clifford algebra methods, that unifies all known incarnations of PH curves into a single coherent framework. Furthermore, we discuss certain differential or algebraic geometric perspectives that arise from this new approach.

New algorithm for medial axis transform of plane domain

Abstract In this paper, we present a new approximate algorithm for medial axis transform of a plane domain. The underlying philosophy of our approach is the localization idea based on the Domain Decomposition Lemma, which enables us to break up the complicated domain into smaller and simpler pieces. We then develop tree data structure and various operations on it to keep track of the information produced by the domain decomposition procedure. This strategy enables us to isolate various important points such as branch points and terminal points. Because our data structure guarantees the existence of such important points—in fact, our data structure is devised with this in mind—we can zoom in on those points. This makes our algorithm efficient. Our algorithm is a “from within” approach, whereas traditional methods use a “from-the-boundary” approach. This “from within” nature of our algorithm and the localization scheme help mitigate various instability phenomena, thereby making our algorithm reasonably robust.

Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves

Abstract We present a new approach to medial axis transform and offset curve computation. Our algorithm is based on the domain decomposition scheme which reduces a complicated domain into a union of simple subdomains each of which is very easy to handle. This domain decomposition approach gives rise to the decomposition of the corresponding medial axis transform which is regarded as a geometric graph in the three dimensional Minkowski space R 2,1 . Each simple piece of the domain, called the fundamental domain, corresponds to a space-like curve in R 2,1 . Then using the new spline, called the Minkowski Pythagorean hodograph curve which was recently introduced, we approximate within the desired degree of accuracy the curve part of the medial axis transform with a G 1 cubic spline of Minkowski Pythagorean hodograph. This curve has the property of enabling us to write all offset curves as rational curves. Further, this Minkowski Pythagorean hodograph curve representation together with the domain decomposition lemma makes the trimming process essentially trivial. We give a simple procedure to obtain the trimmed offset curves in terms of the radius function of the MPH curve representing the medial axis transform.

Euler-Rodrigues frames on spatial Pythagorean-hodograph curves ✩

Abstract We investigate the properties of a special kind of frame, which we call the Euler–Rodrigues frame (ERF), defined on the spatial Pythagorean-hodograph (PH) curves. It is a frame that can be naturally constructed from the PH condition. It turns out that this ERF enjoys some nice properties. In particular, a close examination of its angular velocity against a rotation-minimizing frame yields a characterization of PH curves whose ERF achieves rotation-minimizing property. This computation leads into a new fact that this ERF is equivalent to the Frenet frame on cubic PH curves. Furthermore, we prove that the minimum degree of non-planar PH curves whose ERF is an rotation-minimizing frame is seven, and provide a parameterization of the coefficients of those curves.

Clifford algebra, Lorentzian geometry, and rational parametrization of canal surfaces

We present a new approach toward the rational parametrization of canal surfaces. According to our previous work, every canal surface with rational (respectively polynomial) spine curve and rational (respectively polynomial) radius function is a rational (respectively polynomial) Pythagorean hodograph curve in R3,1. Drawing upon this formalism and utilizing the underlying Lorentzian geometry, the problem is reduced to simple algebraic manipulations.We also illustrate how our work related to the previous work of pottmann and Peternell. Finally, we give an outline of an approach toward the rotation minimizing parametrization of canal surfaces.

Utility-based load balancing in WLAN/UMTS internetworking systems

The complementary characteristics of 3 G cellular networks (such as UMTS) and IEEEs 802.11 wireless LANs (WLANs) have stimulated intensive research efforts to integrate UMTS and WLAN networks. In such integrated networks, a mobile station is able to switch between UMTS and WLAN networks based on various network parameters such as network throughput, packet delay and other network or user quality criteria. In this paper, we develop a utility-based access selection algorithm targeted to achieve load balancing between UMTS and WLAN networks. The performance of our algorithm is evaluated using OPNET with dual-mode mobile stations.

Scheduling Algorithms For Policy Driven QoS Support in HSDPA Networks

Increasing demand for high data-rate multimedia services has led to the emergence of high-speed data transfer features such as high-speed downlink packet access (HSDPA) for WCDMA networks. The role of the MAC-hs scheduler is vital in HSDPA in improving overall system performance. Using opportunistic scheduling to exploit multi-user diversity for efficient transmission of best effort services or considering user fairness constraints has been the main focus of most scheduling algorithms. However the need for strict QoS support for services such as streaming, gaming, and VoIP is growing. Service operators can enforce their own policies in meeting these QoS requirements. In this paper we introduce two QoS-aware policy driven scheduling algorithms. We developed an HSDPA system in OPNET, and implemented our scheduling algorithms along with other well-known algorithms. OPNET simulations show that unlike other schedulers, our strict and loose policy scheduling (SPS and LPS) algorithms comply with the policy constraints if allowed by radio conditions and cell capacity.

Digital watermarking of polygonal meshes with linear operators of scale functions

Digital watermarking is already used to establish the copyright of graphics, audio and text, and is now increasingly important for the protection of geometric data as well. Watermarking polygonal models in the spectral domain gives protection against similarity transformation, mesh smoothing, and additive random noise attacks. However, drawbacks exist in analyzing the eigenspace of Laplacian matrices. In this paper we generalize an existing spectral decomposition and propose a new spatial watermarking technique based on this generalization. While inserting the watermark, we avoid the cost of finding the eigenvalues and eigenvectors of a Laplacian matrix in spectral decomposition; instead we use linear operators derived from scaling functions that are generated from Chebyshev polynomials. Experimental results show how the cost of inserting and detecting watermarks can be traded off against robustness under attacks like additive random noise and affine transformation.

VALUATION AND HEDGING OF OPTIONS WITH GENERAL PAYOFF UNDER TRANSACTIONS COSTS

We present the pricing and hedging method for op- tions with general payofis in the presence of transaction costs. The convexity of the payofi function - gamma of the options - is an im- portant issue under transaction costs. When the payofi function is convex, Leland-style pricing and hedging method still works. How- ever, if the payofi function is of general form, additional assump- tions on the size of transaction costs or of the hedging interval are needed. We do not assume that the payofi is convex as in Le- land (11) and the value of the Leland number is less (bigger) than 1 as in Hoggard et al. (10), Avellaneda and Paras (1). We focus on generally recognized asymmetry between the option sellers and buyers. We decompose an option with general payofi into difierence of two options each of which has a convex payofi. This method is consistent with a scheme of separating out the sellers and buyers position of an option. In this paper, we flrst present a simple linear valuation method of general payofi options, and also propose in the last section more e-cient hedging scheme which costs less to hedge options.

Rational parametrization of canal surface by 4 dimensional Minkowski Pythagorean hodograph curves

Lorentzian geometry with a Minkowski Pythagorean hodograph (MPH) formalism in R/sup 3,1/ gives us a new and insightful method of rational parametrization of canal surfaces. Our previous works about MPH curves in R/sup 3,1/ shows that a curve /spl gamma/(t)=(x(t),y(t),z(t),r(t)) in R/sup 3,1/ can be represented by the PH representation map in Cl(3,1) which avoids the complex root finding algorithm. Our parametrization method gives us the flexibility to represent the canal surfaces within their fiber ambiguities. This paper constitutes the first step of our ongoing work which deals with the issues for canal surfaces in a truly new and intriguing manner such as finding rotation minimizing frames. We believe this is just the tip of the iceberg and the further work will yield many valuable applications in the area of canal surfaces.