Jamaludin Md Ali

Monotonicity-preserving C2 rational cubic spline for monotone data

Abstract Designers in industries need to generate splines which can interpolate the data points in such a way that they preserve the inherited shape characteristics (positivity, monotonicity, convexity) of data. Among the properties that the spline for curves and surfaces need to satisfy, smoothness and shape preservation of given data are mostly needed by all the designers. In this paper, a rational cubic function with three shape parameters has been developed. Data dependent sufficient constraints are derived for one of these shape parameters to preserve the inherited shape feature like monotonicity of data. Remaining two shape parameters are left free for designer to refine the shape of the monotone curve as desired. Numerical examples and interpolation error analysis show that the interpolant is not only C 2 , local, computationally economical and visually pleasant but also smooth. The error of rational cubic function is also calculated when the arbitrary function being interpolated is C 3 in an interpolating interval. The order of approximation of interpolant is O ( h i 3 ) .

An Automatic Generation of G^1 Curve Fitting of Arabic Characters

An Arabic font is difficult to fit as it is cursive in character, having varying curves and cusps. Here, the Arabic character is represented as an outline font fitted with G1 rational Bezier cubic curves. As a method in reverse engineering, the Arabic character is created by way of digitizing an image that already exists and then fitting G1 curves automatically to the outline of the digitized image. The outline font representation is done in several phases - contour extraction of font image, corner points detection and lastly contour segment fitting. Image is considered as binary and boundary is obtained accordingly. Eigenvalues of covariance matrix and the concept of region of support are employed to search for the corners of the Arabic characters which are of varying degrees of smoothness. G 1 rational Bezier cubics, iteratively determined, are used in the last step. The weights are adjusted automatically to get curves that are as close as need be to the digitized data points. This technique can be extended to visualizing outlines of other contour-based images automatically

Fuzzy Geometric Modeling

Fuzzy geometric modeling provides a useful tool to introduce uncertainty into mathematical spline model. In this paper a new concept of a geometric modeling is presented, based on the theory of fuzzy numbers. By the notion of fuzzy number we introduce fuzzy control point for fuzzy curve and fuzzy surface model for CAGD. We study the properties concerning approximation of fuzzy control points by means of fuzzy Bezier, fuzzy B-spline and fuzzy NURBS.

Fuzzy Interpolation Rational Bezier Curve

Recently, interpolation methods especially in curves design are widely used for modeling data points. In this paper a new interpolation method for modeling fuzzy data using fuzzy interpolation rational Bezier curves is introduced based on fuzzy set theory. This method is able to model uncertainty data with definition of fuzzy data via fuzzy number concept. The fuzzy interpolation method is modeled using fuzzy interpolation rational cubic Bezier curve (in further will be referred as FIRCBC). For fuzzy n-data cases, segments curve constructed in order to interpolate fuzzy data piece wisely. For the illustration as hypothetical example, FIRCBC has been applied in verification of offline handwriting signature where the signatures become fuzzy cases.

Positivity-preserving rational bi-cubic spline interpolation for 3D positive data

This paper deals with the shape preserving interpolation problem for visualization of 3D positive data. A required display of 3D data looks smooth and pleasant. A rational bi-cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic/quadratic involving three shape parameters. Simple data dependent constraints for shape parameters are derived to conserve the inherited shape feature (positivity) of 3D data. Remaining shape parameters are left free for designer to modify the shape of positive surface as per industrial needs. The interpolant is not only local, C 1 but also it is a computationally economical in comparison with existing schemes. Several numerical examples are supplied to support the worth of proposed interpolant.

Data Visualization Using Rational Trigonometric Spline

This paper describes the use of trigonometric spline to visualize the given planar data. The goal of this work is to determine the smoothest possible curve that passes through its data points while simultaneously satisfying the shape preserving features of the data. Positive, monotone, and constrained curve interpolating schemes, by using a piecewise rational cubic trigonometric spline with four shape parameters, are developed. Two of these shape parameters are constrained and the other two are set free to preserve the inherited shape features of the data as well as to control the shape of the curve. Numerical examples are given to illustrate the worth of the work.

The G2 and C2 rational quadratic trigonometric Bézier curve with two shape parameters with applications

The rational quadratic trigonometric Bezier curve with two shape parameters is presented in this paper, which is new in literature. The purposed curve inherits all the geometric properties of the traditional rational quadratic Bezier curve. The presence of shape parameters provides a control on the shape of the curve more than that of traditional Bezier curve. Moreover the weight offers an additional control on the curve. Simple constraints for shape parameters are derived using the end points curvature so that their values always fall within the defined range. The composition of two segments of curve using G^2 and C^2 continuity is given. The new curves can accurately represent some conics and best approximates the traditional rational quadratic Bezier curve.

Towards G/sup 2/ curve design with Timmer parametric cubic

There has been vast research in the area of computer aided geometric design in order to identify algorithms to design objects. Timmer parametric cubic or Timmer PC was introduced by Harry Timmer as a follow up to the legendary Bezier curves. However it was then not utilized as it does not satisfy convex hull property, CHP. Since then, there lack studies to explore the possible usages and discover the uniqueness of Timmer PC. This paper discusses comprehensively the steps involved to generate a piecewise Timmer curve with G/sup 1/ and G/sup 2/ continuity. In the last section, a wine glass is rendered via surface of revolution as an example of implication to designing environment using the Timmer curve.

Local convexity-preserving C2 rational cubic spline for convex data.

We present the smooth and visually pleasant display of 2D data when it is convex, which is contribution towards the improvements over existing methods. This improvement can be used to get the more accurate results. An attempt has been made in order to develop the local convexity-preserving interpolant for convex data using C 2 rational cubic spline. It involves three families of shape parameters in its representation. Data dependent sufficient constraints are imposed on single shape parameter to conserve the inherited shape feature of data. Remaining two of these shape parameters are used for the modification of convex curve to get a visually pleasing curve according to industrial demand. The scheme is tested through several numerical examples, showing that the scheme is local, computationally economical, and visually pleasing.

Spur Gear Design with an S-shaped Transition Curve Application Using Mathematica and CAD Tools

The aim of this paper is to introduce the application of an S-shaped transition curve in designing and generating a spur gear. In engineering field, the spur gear usually designed by using an involute curve where this curve needs to use a tracing point method to obtain a curve while an S-shaped curve can obtain a curve directly. The comparative study between an S-shaped method and direct method in Computer Aided Design (CAD) tool also covered.