International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:15 No:03 7 A matrix Method for Interal Hermite Cure Segmentation O. Ismail, Senior Member, IEEE Abstract Since the use of matrix forms (largely promoted in CAD/CAM) turns out to be both conenient practical in representing parametric cures surfaces. Furthermore, this implementation can be made extremely fast if appropriate matrix facilities are aailable in either hardware or software. We can break a cure down into smaller segments by truncating or subdiiding it. There are many reasons for doing this. For example, we may truncate to isolate extract that part of a cure suriing a model modification process, or subdiide it to compute points for displaying it. To truncate, subdiide, or change the direction of parameterization of a cure ordinarily requires a mathematical operation called reparamelerization. Ideally, this operation produces a change in the parametric interal so that neither the shape nor the position of the cure is changed. This effect is often referred to as shape inariance under parameterization reparamelerization. This concept has been applied on interal Hermite cures. An algorithmic method for interal Hermite cure segmentation in matrix form is presented in this paper. To split the interal Hermite cure defined oer the range at a point defined by means that the first the second interal segments are to be defined oer the ranges, respectiely. The four fixed Kharitono's polynomials (four fixed Hermite cures) associated with the original interal Hermite cure are obtained. The proposed algorithm is applied to the four fixed Kharitono's polynomials (four fixed Hermite cures) in order to obtain the fixed control points for the first second fixed segments, respectiely. Finally the interal control points for the first second interal segments of the gien interal Hermite cure are found from fixed control points for the four fixed Kharitono's polynomials (four fixed Hermite cures) of the first second fixed segments. A numerical example is included in order to demonstrate the effectieness of the proposed method. Index Term Matrix representation, segmentation, interal Hermite cure, CAGD. I. INTRODUCTION Cure design is important in computer graphics, animation, computer aided design. Unfortunately, cure design requires ery inoled mathematics een though many cure design concepts are intuitie. The cure is the most basic design element to determine shapes silhouettes of industrial products works for shape designers it is ineitable for them to make it aesthetic attractie to improe the total quality of the shape design. The Hermite cures surfaces hae been widely applied in many CAD/CAM systems. They hae become a stard tool in industry of geometric modeling because they proide a common mathematical form for analytical geometry free-form cures surfaces. The author is with Department of Computer Engineering, Faculty of Electrical Electronic Engineering, Uniersity of Aleppo, Aleppo, (e-mail:oismail@ieee.org). Geometric modeling computer graphics hae been interesting important subjects for many years from the point of iew of scientists engineers. One of the main useful applications of these concepts is the treatment of cures surfaces in terms of control points, a tool extensiely used in CAGD. There are seeral kinds of polynomial cures in CAGD, e.g., Bezier 1, 2, 3, 4, 5 Said-Ball 6, Wang- Ball 7, 8, 9, B-spline cures 10 DP cures 11, 12. These cures hae some common different properties. All of them are defined in terms of the sum of product of their blending functions control points. They are just different in their own basis polynomials. In order to compare these cures, we need to consider these properties. The common properties of these cures are control points, weights, their number of degrees. Control points are the points that affect to the shape of the cure. Weights can be treated as the indicators to control how much each control point influences to the cure. Number of degree determines the maximum degree of polynomials. In different cures, these properties are not computed by the same method. To compare different kinds of cures we need to conert these cures into an intermediate form. Parametric equations aoid many of the problems associated with nonparametric functions. They also best describe the way cures are drawn by a plotter or some computer graphics display screens. Parametric equations generate the sets of points defining most of the cures, surfaces. Parametric equations hae many adantages oer other forms of representation. Here are the most important ones: (1) They allow separation of ariables direct computation of point coordinates. (2) It is easy to express parametric equations as ectors. (3) Each ariable is treated alike. (4) There are more degrees of freedom to control cure shape. (5) Transformations may be performed directly on them. (6) They accommodate all slopes without computational breakdown. (7) Extension or contraction into higher or lower dimensions is direct easy without affecting the initial representation. (8) The cures they define are inherently bounded when the parameter is constrained to a specified finite interal. (9) The same cure often can be represented by many different parameterizations. Conersely, a parameterization scheme is sometimes chosen because of its effect on cure shape. Hermite cure 13, 14, 15, 16, 17, 18, 19, 20 consists of indiidual parametric cure segments joined to form a single composite cure whose continuity is controlled at the segment joints. Hermite cure is not restricted to points. In its stard form for cures it uses both points deriatie data. For example, a cubic
International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:15 No:03 8 Hermite cure is defined by its two end points the tangent ectors at those points. It interpolates all its control points is fairly easy to subdiide. Howeer, its lack of inariance under affine transformations can be troublesome if not accounted for. The cubic Hermite polynomial representation is the most interested representation, because this is the lowest degree capable of describing nonplanar cures. Howeer, the Hermite form offers a practical alternatie, allowing us to define a cure segment in terms of conditions at its endpoints, or boundaries. The Hermite basis functions first appear in the deriation of the geometric form from the algebraic form, each function is a cure oer the domain of the parameter in the interal. Three important characteristics of these basis functions are: (1) Uniersality-they hold for all cubic Hermite cures. (2) Dimensional independence-they are identical for each of the three model-space coordinates, because they are dependent only on the parameter. (3) Separation of boundary condition effects-they allow the constituent boundary condition coefficients to be decoupled from each other. This means that we can selectiely modify a single specific boundary condition to alter the shape of a cure without affecting the other boundary conditions. These functions blend the effects of the endpoints tangent ectors to produce the intermediate point coordinate alues oer the domain of. Matrix representation is ery useful in computer aided geometric design, since the matrix is an important basic tool in mathematics. Matrix formulas of Hermite cures surfaces hae the adantages of both simple computation of points on cures or surfaces their deriaties of easy analysis of the geometric properties of Hermite cures surfaces. Segmentation is a process of splitting a cure or a surface into a number of parts such that the composite cure or surface of all the segments is identical to the parent cure or surface. Segmentation is essentially a reparameterising transformation of a cure or a surface while keeping the degree of the cure or surface unchanged. This paper is organized as follows. Section contains the basic results, whereas section shows a numerical example the final section offers conclusions. II. THE BASIC RESULTS Segmentation or cure splitting is defined as replacing one existing cure by one or more cure segments of the same cure type such that the shape of the composite cure is identical to that of the original cure. Segmentation is a ery useful feature for CAD/CAM systems. Model cleanup for drafting purposes is an example where a cure might be diided into two at the line of sight of another. One of the resulting segments is then deleted. Another example is when a closed cure has to be split for modeling purposes. Mathematically, cure segmentation is a reparametrization or parameter transformation of the cure. Polynomial cures such as Hermite cures, Bezier cures, B-spline cures require a different parameter transformation. If the degree of the polynomial defining a cure is to be unchanged, which is the case in segmentation, the transformation must be linear. Let us assume that the interal Hermite cure is defined oer the range. To split the gien interal cure at a point defined by means that the first the second interal segments are to be defined oer the range, respectiely. A new parameter introduced for each interal segment such that its range is. The parameter transformation takes the form: In general, if the two interal endpoints the first pairs of deriaties at the extreme points are known (a total of items), i.e.,, this will gie, they can be used to calculate an interal interpolating polynomial of degree, if we assume, then the interal Hermite cure can be written in matrix form as follows: is Hermite basis matrix. The segmentation of an interal cure is needed when we want to utilize only a part of the interal cure discard the remaining. When this utilized part is to be merged into the total model of the part being built, reparametrization is required as shown in Fig. 1. The four fixed Kharitono's polynomials (four fixed Hermite cures) in matrix form associated with the original interal Hermite cure are:
International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:15 No:03 9 1 u u m 0 u u 0 0 u 1 u 1 Fig. 1: Segmentation of interal Hermite Cure. Now the segmentation of an interal Hermite cure can be conerted into the following problem. For a gien four fixed Kharitono's polynomials (four fixed Hermite cures) associated with the original interal Hermite cure as in equation, split the four fixed Kharitono's polynomials (four fixed Hermite cures) at a point defined by, find the fixed control points of the four fixed Kharitono's polynomials (four fixed Hermite cures) for first second fixed segments, finally the interal control points for the first second interal segments of the gien interal Hermite cure can be obtained from fixed control points for the four fixed Kharitono's polynomials (four fixed Hermite cures) of the first second fixed segments. Hence, for first fixed segment, ( ) For second fixed segment, Therefore, for first fixed segment, the four fixed Kharitono's polynomials (four fixed Hermite cures) in matrix form can be rewritten as: The modified geometry, or boundary conditions, ectors of the four fixed Kharitono's polynomials (four fixed Hermite cures) first segment, are gien by: The interal control points for the first interal segment of the gien interal Hermite cure can be obtained from fixed control points of the four fixed Kharitono's polynomials (four fixed Hermite cures) of the first fixed segments as: ( ) ( ) For second fixed segment, equation (8) can be written as follows: ( ) For first fixed segment, equation (7) can be written as follows:
International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:15 No:03 10 Therefore, for second fixed segment, the four fixed Kharitono's polynomials (four fixed Hermite cures) in matrix form can be rewritten as: ( ) ( ) for interal tangent ectors at the two interal points are: ( ) ( ) The modified geometry, or boundary conditions, ectors of the four fixed Kharitono's polynomials (four fixed Hermite cures) second segment, are gien by: The problem is to diide the gien interal Hermite cure into two interal segments at a point, find the interal endpoints interal slopes for each segment. As explained in section, the four fixed Kharitono's polynomials (four fixed Hermite cures) associated with the original interal Hermite are obtained, the matrices for first fixed segment are found as follows: The interal control points for the second interal segment of the gien interal Hermite cure can be obtained from the four fixed Kharitono's polynomials (four fixed Hermite cures) of the second fixed segments as: ( ) ( ) Interal Hermite cure segmentation can be done by the following steps: Algorithm for the interal Hermite cure segmentation 1. Obtain the four fixed Kharitono's polynomials (four fixed Hermite cures) associated with the original interal Hermite cure. 2. Split the four fixed Kharitono's polynomials (four fixed Hermite cures) at a point defined by means that the first the second fixed segments are to be defined oer the range, respectiely. 3. The fixed control points of the four fixed Kharitono's polynomials (four fixed Hermite cures) for first second fixed segments can be obtained using equations (12) (18), respectiely. 4. The interal control points for the first second interal segments of the gien interal Hermite cure can be obtained from fixed control points for the four fixed Kharitono's polynomials (four fixed Hermite cures) of the first second fixed segments using equations (14) (20), respectiely. III. NUMERICAL EXAMPLE Consider the interal Hermite cure interal endpoints:, defined with The interal control endpoints for the first interal segment are obtained as: with interal slopes at the interal endpoints, ( ) Similarly, the matrix found, for second fixed segment is The interal control endpoints for the second interal segment are obtained as: with interal slopes at the interal endpoints, IV. CONCLUSIONS Segmentation is a process of splitting a cure or a surface into a number of parts such that the composite cure or surface of all the segments is identical to the parent cure or surface. Segmentation is essentially a reparameterising transformation of a cure or a surface while keeping the degree of the cure or surface unchanged. Matrix representation is ery useful in computer aided geometric design, since the matrix is an important basic tool in
International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:15 No:03 11 mathematics. Matrix formulas of Hermite cures surfaces hae the adantages of both simple computation of points on cures or surfaces their deriaties of easy analysis of the geometric properties of Hermite cures surfaces. A simple matrix form for interal Hermite cure segmentation is presented in this paper. To split the interal Hermite cure defined oer the range at a point defined by means that the first the second interal segments are to be defined oer the ranges, respectiely. The four fixed Kharitono's polynomials (four fixed Hermite cures) associated with the original interal Hermite cure are obtained. The proposed algorithm is applied to the four fixed Kharitono's polynomials (four fixed Hermite cures) in order to obtain the fixed control points for the first second fixed segments, respectiely. Finally the interal control points for the first second interal segments of the gien interal Hermite cure are found from fixed control points for the four fixed Kharitono's polynomials (four fixed Hermite cures) of the first second fixed segments. Splitting an interal Hermite cure implies creating two new interal cures, one that represents the first interal half of the original interal Hermite cure, one that represents the second interal half. If the gien interal Hermite cure is to be diided into more than two interal segments the proposed algorithm can be applied successiely. REFERENCES 1 P. Bezier, "Definition Numerique Des Courbes et I", Automatisme, Vol. 11, pp. 625 632, 1966. 2 P. Bezier, "Definition Numerique Des Courbes et II", Automatisme, Vol. 12, pp. 17 21, 1967. 3 P. Bezier, Numerical control, Mathematics Applications, New York: Wiley, 1972. 4 G. Chang, Matrix foundation of Bezier technique, Comput. Aided Des. Vol. 14, No. 6, pp. 345-350, 1982. 5 P. Bezier, The Mathematical Basis of the UNISURF CAD System, Butterworth, London, 1986. 6 H. B. Said, "A generalized Ball cure its recursie algorithm", ACM. Transaction on Graphics, Vol. 8, No. 4, pp. 360 371, 1989. 7 A. A. Ball, CONSURF Part 1: Introduction to conic lofting tile, Computer Aided Design, Vol. 6, No. 4, pp. 243 249, 1974. 8 A. A. Ball, CONSURF Part 2: Description of the algorithms, Computer Aided Design, Vol. 7, No. 4, pp. 237 242, 1975. 9 A. A. Ball, CONSURF Part 3: How the program is used, Computer Aided Design, Vol. 9, No. 1, pp. 9 12, 1977. 10 G. J. Wang, "Ball cure of high degree its geometric properties", Applied. Mathematics: A Journal of Chinese Uniersities Vol. 2, pp. 126-140, 1987. 11 J. Delgado J. M. Pena, A linear complexity algorithm for the Bernstein basis, Proceedings of the 2003 International Conference on Geometric Modeling Graphics (GMAG 03), pp. 162 167, 2003. 12 J. Delgado J. M. Pena, "A shape presering representation with an ealuation algorithm of linear complexity, Computer Aided Geometric Design, pp. 1-10, 2003. 13 C. de Boor, K. Hollig, M. Sabin, High accuracy geometric Hermite interpolation, Computer Adied Geometric Design, Vol. 4, pp. 269-278, 1987. 14 K. Hollig J. Koch, Geometric Hermite interpolation, Computer Aided Geometric Design, Vol. 12, No. 6, pp. 567-580, 1995. 15 C. Manni, comonotone Hermite interpolation ia parametric cubics, J. Comput. Appl. Math., Vol. 69, pp. 143 157, 1996. 16 C. Conti R. Mori, Piecewise shape presering Hermite interpolation, Computing, Vol. 56, pp. 323 341, 1996. 17 C. Manni, Parametric shape presering Hermite interpolation by piecewise quadratics, in Adanced Topics in Multiariate Approximation, F. Fontanella, K. Jetter, P. J. Laurent, eds., World Scientific, Singapore, pp. 211 226, 1996. 18 U. Schwanecke, B. Juttler, A B-Spline approach to Hermite subdiision. In: A. Cohen, C. Rabut, L. L. Schumaker (Eds.), Cure Surface Fitting, Verbilt Uniersity Press, Nashille, pp 385 392, 2000. 19 S. Dubuc, Scalar Hermite subdiision schemes, Appl. Comp. Harmon. Anal., Vol. 21, pp. 376 394, 2006. 20 B. Y. Su, J. Q. Tan, Circular trigonometric Hermite interpolation polynomials applications, J. of Information Computational Science, Vol. 4, pp. 709-720, 2007. 21 V. L. Kharitono, "Asymptotic stability of an equilibrium position of a family of system of linear differential equations", Differential 'nye Urauneniya, ol. 14, pp. 2086-2088, 1978. O. Ismail (M 97 SM 04) receied the B. E. degree in electrical electronic engineering from the Uniersity of Aleppo, Syria in 1986. From 1987 to 1991, he was with the Faculty of Electrical Electronic Engineering of that uniersity. He has an M. Tech. (Master of Technology) a Ph.D. both in modeling simulation from the Indian Institute of Technology, Bombay, in 1993 1997, respectiely. Dr. Ismail is a Senior Member of IEEE. Life Time Membership of International Journals of Engineering & Sciences (IJENS) Researchers Promotion Group (RPG). His main fields of research include computer graphics, computer aided analysis design (CAAD), computer simulation modeling, digital image processing, pattern recognition, robust control, modeling identification of systems with structured unstructured uncertainties. He has published more than 69 refereed journals conferences papers on these subjects. In 1997 he joined the Department of Computer Engineering at the Faculty of Electrical Electronic Engineering in Uniersity of Aleppo, Syria. In 2004 he joined Department of Computer Science, Faculty of Computer Science Engineering, Taibah Uniersity, K.S.A. as an associate professor for six years. He has been chosen for inclusion in the special 25th Siler Anniersary Editions of Who s Who in the World. Published in 2007 2010. Presently, he is with Department of Computer Engineering at the Faculty of Electrical Electronic Engineering in Uniersity of Aleppo.