Tae-wan Kim

Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

Log-aesthetic curves (LACs) have recently been developed to meet the requirements of industrial design for visually pleasing shapes. LACs are defined in terms of definite integrals, and adaptive Gaussian quadrature can be used to obtain curve segments. To date, these integrals have only been evaluated analytically for restricted values (0,1,2) of the shape parameter @a. We present parametric equations expressed in terms of incomplete gamma functions, which allow us to find an exact analytic representation of a curve segment for any real value of @a. The computation time for generating an LAC segment using the incomplete gamma functions is up to 13 times faster than using direct numerical integration. Our equations are generalizations of the well-known Cornu, Nielsen, and logarithmic spirals, and involutes of a circle.

Challenges in computer applications for ship and floating structure design and analysis

This paper presents a review on the key research areas in the design and analysis of ships and floating structures. The major areas of computer application are identified in several stages of ship/floating structure design and analysis with the principal emphasis on the methodologies, the modeling, and the integration of the design and analysis process. The discussion addresses some of the key challenges in computer applications for ship and floating structure design and analysis, and reports on the emerging trends in the research, design and industrial application.

Adaptive triangular-mesh reconstruction by mean-curvature-based refinement from point clouds using a moving parabolic approximation

We use a moving parabolic approximation (MPA) to reconstruct a triangular mesh that approximates the underlying surface of a point cloud from closed objects. First, an efficient strategy is presented for constructing a hierarchical grid with adaptive resolution and generating an initial mesh from point clouds. By implementing the MPA algorithm, we can estimate the differential quantities of the underlying surface, and subsequently, we can obtain the local quadratic approximants of the squared distance function for any point in the vicinity of the target shape. Thus, second, we adapt the mesh to the target shape by an optimization procedure that minimizes a quadratic function at each step. With the objective of determining the geometrical features of the target surface, we refine the approximating mesh selectively for the non-flat regions by comparing the estimated curvature from the point clouds and the estimated curvatures computed from the current mesh. Finally, we present various examples that demonstrate the robustness of our method and show that the resulting reconstructions preserve geometric details.

Fitting G2 multispiral transition curve joining two straight lines

In this paper, we describe an algorithm for generating a C-shaped G^2 multispiral transition curve between two non-parallel straight lines. The G^2 multispiral is a curve that consists of two or more log-aesthetic curve segments connected with curvature continuity, and it has inflection endpoints. Compound-rhythm log-aesthetic curves are not directly applicable to the generation of transition curves between two straight lines, which is important in highway and railroad track design, because both endpoints are required to be inflection points. Thus, a new approach for generating transition curves is necessary. The two log-aesthetic curve segments with shape parameter @a<0 are connected at the origin, and they form a multispiral. The problem is to find a similar triangle, as in the given data. Depending on the parameter @a, the G^2 multispiral transition curve may have different shapes; moreover, the shape of the curve approximates a circular arc as @a decreases. The obtained curves also find applications in gear design and fillet modeling.

Dimensions and bases of hierarchical tensor-product splines

We prove that the dimension of trivariate tensor-product spline space of tri-degree (m,m,m) with maximal order of smoothness over a three-dimensional domain coincides with the number of tensor-product B-spline basis functions acting effectively on the domain considered. A domain is required to belong to a certain class. This enables us to show that, for a certain assumption about the configuration of a hierarchical mesh, hierarchical B-splines span the spline space. This paper presents an extension to three-dimensional hierarchical meshes of results proposed recently by Giannelli and Juttler for two-dimensional hierarchical meshes.

Moving parabolic approximation of point clouds

Using moving parabolic approximations (MPA), we reconstruct an improved point-based model of curve or surface represented as an unorganized point cloud, while also estimating the differential properties of the underlying smooth manifold. We present optimization algorithms to solve these MPA models, and examples which show that our reconstructions of the curve or surface, and estimates of the normals and curvature information, are accurate for precise point clouds and robust in the presence of noise.

Maneuvering control simulation of underwater vehicle based on combined discrete-event and discrete-time modeling

When designing or acquiring underwater vehicles such as submarines and torpedoes, it is necessary to predict their performance precisely and perform tests repeatedly using modeling and simulation at both the engineering level and the tactical engagement level. For simulation performed for analysis purposes at the engineering level, which requires a considerable amount of computation power, a discrete-time system simulation that computes significant values at every single unit time using the established mathematical model or engineering model is mainly employed. To simulate a complex or complicated task such as a traffic analysis or tactical measure of effectiveness (MOE) analysis at the engagement level, it is appropriate to use a discrete-event system simulation that causes transition between model states through the triggering of events on the basis of the passing of messages between simplified mathematical models coupled in various ways. In this paper, we studied a maneuvering control of underwater vehicle from the perspective of a combined discrete-event and discrete-time system simulation; the simulation model is established on the basis of discrete-event system specification (DEVS) formalism, which is a representative modeling formalism of a discrete-event system simulation. In detail, the simulation includes DEVS modeling implementations of simulation execution time control and discrete-time step size control in real time at the time of performing a discrete-time system simulation for the purpose of three-dimensional visualization or carrying out a performance analysis using the DEVS model. This hybrid approach makes possible to build a simulation-based expert system which supports the decision making for the acquisition of an underwater vehicle.

High-order approximation of implicit surfaces by G1 triangular spline surfaces

In this paper, we present a method for the approximation of implicit surface by G^1 triangular spline surface. Compared with the polygonization technique, the presented method employs piecewise polynomials of high degree, achieves G^1 continuity and is capable of interpolating positions, normals, and normal curvatures at vertices of an underlying base mesh. To satisfy vertex enclosure constraints, we develop a scheme to construct a C^2 consistent boundary curves network which is based on the geometric Hermite interpolation of normal curvatures. By carefully choosing the degrees of scalar weight functions, boundary Bezier curves and triangular Bezier patches, we propose a local and singularity free algorithm for constructing a G^1 triangular spline surface of arbitrary topology. Our method achieves high precision at low computational cost, and only involves local and linear solvers which leads to a straightforward implementation. Analyses of freedom and solvability are provided, and numerical experiments demonstrate the high performance of algorithms and the visual quality of results.

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.

Finding the best conic approximation to the convolution curve of two compatible conics based on Hausdorff distance

We consider the convolution of two compatible conic segments. First, we find an exact parametric expression for the convolution curve, which is not rational in general, and then we find the conic approximation to the convolution curve with the minimum error. The error is expressed as a Hausdorff distance which measures the square of the maximal collinear normal distance between the approximation and the exact convolution curve. For this purpose, we identify the necessary and sufficient conditions for the conic approximation to have the minimum Haudorff distance from the convolution curve. Then we use an iterative process to generate a sequence of weights for the rational quadratic Bezier curves which we use to represent conic approximations. This sequence converges to the weight of the rational quadratic Bezier curve with the minimum Hausdorff distance, within a given tolerance. We verify our method with several examples.