B-spline Curve Generation and Modification based on Specified Radius of Curvature

Transcription

1 B-spline Curve Generation and Modification based on Specified Radius of Curvature TETSUZO KURAGANO AND KAZUHIRO KASONO Graduate School of Information Science Meisei Universit -- Hodokubo, Hino-cit, Toko, 9-86 JAPAN Abstract: - A method to generate a uintic which passes through given points is described In this case, there are four more euations than there are control point positions Two methods have been developed to compensate for the difference between the number of unknowns and that of the euations These are assuming that the s at both ends of the curve are zero, and assigning four gradients to the given points In addition to this method, another method to generate a uintic which passes close to given points, and which has the first derivative at these given points is described In this case, a linear sstem will be underdetermined, determined or overdetermined depending on the number of given points with gradients A method to modif a uintic shape according to the specified distribution to realize an aestheticall pleasing freeform curve is described The difference between the and the specified is minimized b introducing the least-suares method Eamples of curve generation are given Ke-Words: - generation, vector, curve shape modification, given points, given points with gradients, underdetermined sstem, overdetermined sstem Introduction A NURBS curve, which is commonl used in the field of CAD CAM and Computer Graphics, is used as an epression of a freeform curve In particular, cubic NURBS curves are widel used In the smoothing of curves, it ma be desirable to interpolate second derivative information at the knots This is not possible with cubic splines, and so splines of higher degree have to be used For smmetric boundar conditions, it is more convenient to work with uintic splines [] In this stud, ranging over multi segments of a NURBS curve is modified based on the specified In addition, the weights of the NURBS curve are set to one For these reasons, the curve used in this stud is a uintic The objective of this stud is to develop a uintic generation method using given points in seuence for practical use Positions and gradients are given to the B-spline curve euations and first derivative euations of the respectivel Then, a is generated Afterwards, if necessar, the shape of this is modified according to the target that has been smoothed Using the least-suares method, the shape of the designed curve is modified based on the target specified There are man related works for generation of a curve There are data fitting b interpolation [, ], and data fitting with B-spline [], and data fitting b principal component analsis following the statistical method [-] There are man related works for generation of a fair curve dealing with knots These are knot insertion [, ], knot removal algorithm for s [], knot removal algorithm for NURBS curves [], and fair curve generation b knot value and weight modification [] There are man related works for generation of fair distribution Fair distribution algorithms b modifing knot spacing [6, 7], and b removing and reinserting knots [8-] have been published Curve generation algorithms related to b modifing the control points have been published These use a clothoidal curve for specifing the [], and modif the shape of the curve ISSN: Issue, Volume, November 8

2 based on the target specified [-8] Curve generation algorithms related to b specifing distribution have also been published [9] Section of this paper describes a uintic B-spline curve, the derivatives of a uintic, vector,, and Section describes the generation of a uintic which passes through some given points in seuence and the generation of a uintic using the given points with gradients In section, shape modification based on the specified is described Section gives the engineering eamples of curve generation Freeform Curve Epression The objective of freeform curve design is to design the framework of surface patches Surface patches are defined as tensor products, which are bi-variate and normall defined b u and v In other words, one knot seuence in u direction, and another knot seuence in v direction are defined despite the compleit of the surface patches Therefore, knot spacing is fied in this stud Epression An n segments ( n 6) uintic is composed of n control points such as,,, n as n R() t = Ni,6 () t i, () i= where Ni,6 ( t) ( i =,,, n ) are B-spline basis functions These functions are defined b the de Boor-Co [] recursion formulas, and are recursivel defined b knot seuence t, t, t, t,, t n+ as ( ti t < t ) i+ Ni, () t = otherwise, () t ti ti+ M t NiM, () t = NiM, () t + Ni+, M () t ti+ M ti ti+ M t i+ where i =,,, n+ and M =,,,6, i corresponds to the knot number If the knot vector contains repeated knot values called multiple knots, then a division of the form NiM, ( t) /( ti+ M ti) = / (for some i ) ma be encountered during the eecution of the recursion Whenever this occurs, it is assumed that /= [] A one segment uintic with the knot vector {,,,,,,,,,,,6} is R { () t = ( t) (t t t t t 6) ( t t 6t 66) (t t t t t 6) ( t t t t t ) t } () The first derivative of a uintic shown in E() is epressed as dr( t) = { ( t) dt + (t 6t + t + 8t ) + t + t t ( ) + t t t + t + ( 6 8 ) + t + t + t + t+ ( 6 ) + t } () The second derivative of a uintic shown in E() is epressed as d R( t) = { ( t ) dt 6 + (t t + 6t + ) ( t 8t 6) + + (t t 6t ) + + ( t t t ) t } () The third derivative of a uintic shown in E() is epressed as d R( t) = { ( t) dt + (t 8t + ) ( t t) + + (t 8t ) + ( t t ) t } (6) The fourth derivative of a uintic shown in E() is epressed as d R( t) = ( t) dt + (t ) + ( t + 6) + (t ) + ( t + ) + t Curvature vector is epressed as (7) ISSN: Issue, Volume, November 8

3 ( R () t R () t ) R () t κ () t =, (8) ( R () t ) where R ( t) is the first derivative of a, and R ( t) is the second derivative of a Curvature is the magnitude of the vector, therefore, is epressed as κ ( t) = κ ( t) (9) B definition, the of a plane curve is nonnegative However, in man cases it is useful to ascribe a sign to the [] The choosing of the sign is commonl connected with the tangent rotation b moving along the curve in the direction of the increasing parameter The of the curve is positive when its tangent rotates counter-clockwise, the of the curve is negative when its tangent rotates clockwise The is the reciprocal number of, therefore, the is epressed as ρ () t = κ () () t B-spline Curve Displa The of a curve is the most significant descriptor of its shape [] To check the shape of a curve b displaing its or plots is widel known This is simpl the graph of κ () t or ρ ( t) Curvature information is plotted using straight lines drawn outward from and perpendicular to the curve, with the line length proportional to the amount of at that spot Radius of information is plotted using straight lines drawn inward from and perpendicular to the curve, with the line length proportional to the amount of at that spot It is hard to distinguish the shape of the two curves b just looking at their graphs If the plots are drawn for both, the difference between the two curves is recognized immediatel Curve shape is judged b looking at the lines coming out from the curve and seeing how their lengths change along the path, not along the parameter Therefore, or distribution must be drawn to the of the curve (a) with (b) and and plots distribution Point marks indicate knot position Fig, and distribution (a) modified with and (b) and distribution plots Point marks indicate knot position Fig Modified, and distribution A with and plots, and and distribution are shown in Fig(a), (b) respectivel A shape modified with and plots, and and radius of distribution are shown in Fig(a), (b) respectivel While both curves shown in Fig(a) and Fig(a) can hardl be distinguished b just looking at their curves, the curve with and plots tell the two curves apart immediatel The distribution of a uintic B-spline curve is not monotone as is shown in Fig(b) But and distribution of a shape modified is monotone as is shown in Fig(b) This displa techniue provides designers with the abilit to evaluate the ualit of a designed curve Seeing a with and radius of plots gives designers a deeper understanding of their design The relation of to is inverted Therefore, it can be seen in Fig(b) and Fig(b) that in a portion of a curve where the is small, at this portion the the will be large As shown in Fig(b), if the ISSN: Issue, Volume, November 8

4 to the is linear, the distribution will be parabolic On the contrar, if the to the is linear, distribution will be parabolic In case the curve shape is close to a straight line, the becomes infinit Therefore, a limit value should be assigned to the Both distribution and distribution displas are effective to eamine the shape of a curve Quintic B-spline Curve Generation In this section, methods to generate a uintic B-spline curve which passes through given points in seuence and to generate a uintic using given points with gradients in seuence are described One widel used form of data fitting is interpolation Sometimes the problem is to interpolate to positional data alone, and sometimes it is necessar to interpolate to derivatives as well as positions, at least at some of the points [] Generation of a Quintic B-spline Curve which Passes through Given Points in Seuence The parameter of E() is set to zero b defining the geometrical knot position corresponding to the knot of the knot vector Then, E() is epressed as R = ( ), () i i i+ i+ i+ i+ i =,,,,, m ( ) where i is the number of the given points in seuence, m is the total number of the given points The positional vectors of the given points in seuence are assigned to R i ( i =,,,,, m ) in E(), and i, i+, i+, i+, i+ ( i =,,,,, m ) are the control points of a uintic When the control points of a uintic are calculated using E(), the number of unknowns, which are the positions of the control points, are four more than the number of euations which are epressed b E() Assuming the s at both ends of the uintic are zero, the second derivatives at both ends of the uintic B-spline curve are set to zero, while setting the parameter of E() to be zero b defining the geometrical knot position corresponding to the knot of the knot vector Then, the following euations are obtained =, () n + n+ 6n+ + n+ + n+ = where n is the number of segments Accordingl, the fourth derivatives at both ends of the uintic are set to zero Then, in the same manner, the following euations are obtained =, () n n+ + 6n+ n+ + n+ = where n is the number of segments Using E() and E(), E() is obtained = = () n+ = n+ n+ n+ = n+ n In E(),,, n+, and n+ ( n is the number of segments) are epressed b the combination of known control points Thus,,, n+, and n+ are made known That is, the number of unknowns is decreased Therefore, the number of euations will be eual to the number of unknowns That is, this linear sstem is determined Using different boundar derivatives has a strong influence on the shape and approimation ualit of the curve [, 6] A uintic which passes through the given points in seuence, and,, n+, and n+ in E() are illustrated in Fig R R R n n+ R n n+ n+ n+ As another method, in addition to the given points in seuence, gradients are assigned to the given points to increase the number of euations For this, the location of the four gradients is assigned situationall E() is applied to the gradients b setting the parameter of E() to zero b defining the n uintic Fig Illustration for a uintic which passes through the given points in seuence R, R,, Rn, Rn are given points,,, n+, n+ are control points of a uintic ( n is the number of segments) ISSN: Issue, Volume, November 8

5 geometrical knot position corresponding to the knot of the knot vector dri = ( i i+ + i+ + i+ ), () dt ( i =,,,,, n ) where n is the total number of given gradients The i shown in E() corresponds to the i in E() and is assigned situationall The defined gradients are located at the beginning given point and its adjacent point, and at the end given point and its adjacent point in general As a magnitude of the first derivative, one third of the value of the distance between the two adjacent given points is assigned as a default value As for further adjustment, the magnitude of the first derivative is determined interactivel Using the given points in seuence and four location specified gradients, the linear sstem becomes determined That is, the number of unknowns is eual to the number of euations The concept of a uintic generation using the given points in seuence and four location specified gradients are illustrated in Fig The defined gradients are located at the beginning given point and it s adjacent point, and at the end given point and it s adjacent point That is, the i of E() are determined as,, n, n respectivel P, P, P, P, P,, Pm, Pm, Pm are the positional vectors of the given points in seuence, and d, d, d n, and d n are the four location specified gradients d P P d P P P d n- P m- d n- P m- P m- Fig Concept of a uintic generation using given points in seuence and four location specified gradients As eamples of a uintic which passes through the given points in seuence, three uintic s which simulate filleting curves are shown with their plots in Fig In Fig(a), and (b), the i of E() are determined as,,, and 6 In Fig(c), the i of E() are determined as,,, and In the case of a curve including a filleting segment, note that two gradients are placed at the start and end points of the filleting segment as well as the uintic beginning and end points Generation of a Quintic B-spline Curve using Given Points with Gradients In this sub-section, a uintic generation using given points with gradients in seuence is described The concept of generating a uintic using given points with gradients in seuence is illustrated in Fig6 d P filleting segment filleting segment filleting segment (a) (b) (c) Point marks indicate given points Arrow marks indicate given four gradients Fig Eamples of a uintic which passes through given points in seuence and four location specified gradients d P P d P d d n- P m- d d n- n- P m- P m- d n- Fig6 Concept of a uintic generation using given points with gradients in seuence P m- P, P, P, P, P,, Pm, Pm, Pm are the given points in seuence, while d, d, d, d, d,, dm, dm, dm are gradients assigned to the given points in seuence A uintic which passes close to the given points with gradients in seuence is generated A uintic is generated b solving E() and E() simultaneousl b making m in E() eual to n in E() In this case, the i in E() corresponds to the i in E() If the number of given points with gradients is, the number of euations (E()) is and the number of first derivative euations (E()) is As a linear sstem, the total number of euations is 8, whereas the total number of control points of a uintic B-spline curve is 8 Therefore, this linear sstem is determined That is, the rank of a coefficient matri of a linear sstem is eual to the number of unknowns The solution to this linear sstem is eact ISSN: Issue, Volume, November 8

6 However, in case the number of given points with gradients is, the number of euations (E()) which pass through the given points is, and the number of euations of the first derivative (E()) is In this case, as a linear sstem, the total number of euations is 6, whereas the number of control points of the uintic is 7 That is, the number of euations is less than the number of unknowns Therefore, this linear sstem is underdetermined [7] For an underdetermined sstem, while setting auiliar function, the linear sstem is solved under the constraint condition b selecting one solution from an infinite number of eact solutions using Lagrange's method of indeterminate multipliers In this case, this generated uintic passes through the given points with gradients in seuence In case the number of given points with gradients is, the number of euations (E(8)) is, and the number of euations of the first derivative (E(9)) is In this case, as a linear sstem, the total number of euations is, whereas the number of control points of the uintic is 9 That is, the number of euations eceeds the number of unknowns Therefore, this linear sstem is overdetermined [8] For an overdetermined sstem, the differences between the right and left sides of all the euations of the sstem are minimized The control points calculated are therefore an approimation Therefore, this generated uintic passes close to the given points with gradients in seuence For a sstem where the number of given points with gradients is or more, the linear sstem is overdetermined For these sstems, in accordance with the increments of the differences between the number of euations and the number of unknowns, the status of the approimation worsens The above mentioned are summarized in Table A determined linear sstem is shown b the cross hatching As an eample, in case the number of given points with gradients is, that is, m in E() and n in E() are, the uintic generated as an underdetermined sstem is shown in Fig7 with its first derivative vectors, which are drawn outward perpendicular to the curve b the straight lines The solution to this linear sstem is eact The length of Table Linear sstem condition (I) (II) (III) (IV) (V) 6 underdetermined eact thru 7 underdetermined eact thru 8 determined eact thru 9 overdetermined approimation close 6 overdetermined approimation close 7 overdetermined approimation close 8 overdetermined approimation close (I) number of given points with gradients (II) number of control points of a uintic (III) sstem condition (underdetermined, determined, overdetermined) (IV) solution status (V) pass through or close to given points with gradients the line is proportional to the first derivative at that spot This is an unusual wa of displaing the first derivative vectors Nevertheless, this helps visual recognition of the uintic shape and the magnitude variation of its first derivative Therefore, it is visuall recognized that in Fig7 the first derivative vectors shown are perpendicular to the assigned gradients d i at P i ( i =,, ) d P first derivative vector generated P d d given points assigned gradient Fig7 Quintic and its first derivative vectors, in case of underdetermined P Fig8 Quintic and its first derivative vectors, in case of overdetermined In case the number of given points with gradients is, that is, m in E() and n in E() is, the uintic is generated as a determined sstem The solution to this linear sstem is eact Furthermore, in case the number of given points with gradients is, that is, m in E() and n in E() are, the uintic is generated as an overdetermined sstem and is shown with its first derivative vectors in Fig8 The solution to this linear sstem is an approimation Therefore, in Fig8, it is visuall recognized that the first derivative vector shown is not perpendicular to the assigned gradient d at P In this manner, a uintic is generated based on the given points with gradients in seuence d P first derivative vector d P given points assigned gradient P generated d P d P d ISSN: Issue, Volume, November 8

7 Curve Shape Modification based on Specified Radius of Curvature Distribution In this section, a method to modif a uintic B-spline curve shape according to the specified distribution to realize an aestheticall pleasing freeform curve is described Radius of is suitable, because it conforms to our visual recognition of the shape of the curve In a case where curve shape is ver close to a straight line, the becomes infinit Also, at the point of infleion, value becomes zero Therefore, value becomes infinite For these reasons, value is converted to value for computation The concept of specification and uintic shape modification based on the specified distribution is shown in Fig9 A uintic and its plots are shown in Fig9(a) A method to modif the shape of the uintic shown in Fig9(a) to the curve shown in Fig9(b) is eamined Radius of plots shown in Fig9(a) are drawn inward from and perpendicular to the curve using straight lines The length of the line is proportional to the at that spot on current knot position (a) current and its plots O ˆi ρ δ i ρ i target specified distribution of current (c) difference between current and target specified shape modified knot position (b) shape modified and its plots O target specified distribution of shape modified (d) of shape modified and target specified (same as in (c)) Fig9 Concept of specification and uintic shape modification based on the target distribution specified the curve However, the straight lines are not parallel to each other and the beginning points of the individual straight lines are different Therefore, a curve with a displa is suitable to eamine the variation of as a whole But, it is not suitable to eamine the length of the straight lines and variation of in detail Therefore, considering the parameter of the uintic is different from the of the curve, the of a uintic as a straight line is set to the horizontal ais, and the radius of is set to the vertical ais as shown in Fig9(c) Then, the distribution to the is drawn After this, the target specified is superimposed on the current distribution Linear, uadratic, and cubic algebraic function is applied as the target radius of to the current distribution to modif the shape of the uintic B-spline curve In detail, the coefficients of the algebraic function are calculated b the least-suares method using the current distribution Then, the is specified b the determined algebraic functions To determine the target distribution b using algebraic functions has been published [6-8] In addition to this, the target is specified b smoothing the distribution using neighborhood values as in image processing Assuming f ( i ) is the value of position i of the current curve, the value g ( i ) of the smoothed corresponding to f ( i ) of the current curve is calculated b using one of the smoothing operators shown in E(6) g () i = ( ) ( ) ( ) f i + f i + f i+ g () i = ( ) ( ) ( ) ( ) ( ) f i + f i + f i + f i+ + f i+ g () i = ( ) ( ) ( ) ( ) ( ) 7 f i + f i + f i + f i + f i+ (6) + f ( i+ ) + f ( i+ ) g () i = f ( i ) f ( i ) f ( i ) f ( i ) f ( i) f ( i+ ) + f ( i+ ) + f ( i+ ) + f ( i+ ) The selection of an operator shown in E(6) and the iteration number of operations are determined ISSN: Issue, Volume, November 8

8 interactivel b looking at the profile of the distribution While increasing the number of iteration, the profile of the distribution becomes a straight line For an infinite number of iteration, it will be a straight line with no gradient This means that the shape of the curve will be an arc After the profile of the distribution is determined, the specified is superimposed on the current distribution An eample is shown in Fig9(c) The i th of radius of distribution of a perimetricall represented uintic is denoted as ρ i, the specified at the same spot is denoted as ˆi ρ, the difference δ i is shown b E(7) and is illustrated in Fig9(c) δ ( ˆ,,,,, i = ρi n n ) ρi (7) Where i =,,,, m, m is the number of specified, and n is the number of B-spline curve segments plus, which is the degree of the curve S (,, n,,, n ) which is the sum of the suared differences for all specified s in E(8) is minimized b introducing the least-suares method The epression is non-linear Therefore, b Talor's theorem, E(8) is linearized as in E(9) S (,, n,,, n ) m (8) = ρ ( ˆ i,, n,,, n ) ρ i i= + n + n + n + n m ρi ρi(,, n,,, n ) i= ρi ρi ρi ˆ n n ρi n n S (,,,,, ) = + + (9) E(9) is minimized b euating to zero all the partial derivatives of S ( +,,,,, n + n + n + with respect to r =,,, n as ) n and ( ) S = ( r =,,, n ) r S = ( r =,,, n ) r Using these simultaneous linear euations, r r () r and r ( r =,,, n ) are calculated Then, r and r are determined This kind of stud on the, or the to realize a fair curve is called a constrained non-linear minimization problem [9] For computation, ρ i and ˆi ρ are calculated based on the Then, the used is converted to the parameter to calculate the position of the control points of the uintic Thus, a uintic is generated The total length of the curve, which is the, is calculated and rescaled as Repeating these operations, positions of the control points of the uintic are determined while δ i ( i =,,, m ) are minimized for the entire Smoothed distribution, which is specified as target distribution is shown in Fig9(c) Using the above mentioned method, the shape of the curve is modified based on the distribution specified The dotted line shown in Fig9(d) is the smoothed distribution shown in Fig9(c) It is visuall recognized that the distribution of the shape modified curve shown in Fig9(d) matches to the specified Eamples of Curve Generation In this section, eamples of curve generation using the methods described in the previous section are given An eample of a uintic generation which passes through some given points in seuence, while setting the s at both ends to zero is shown in Fig(a) with its and plots Also, an eample of a uintic generation using some given points in seuence and four location specified gradients is shown in Fig(b) with its and plots (a) s at both ends are zero s are zero (b) given points and four gradients Fig Quintic s (a) s at both (b) given points and four (c) given points with ends are zero gradients gradients Fig Curvature and distribution (c) given points with gradients ISSN: Issue, Volume, November 8

9 In adding to these eamples, a uintic B-spline curve generation using some given points with gradients is shown in Fig(c) with its and plots Furthermore, to make the variation of and clear, distribution and distribution of these sample curves are shown corresponding to the generation methods, in Fig(a), (b), (c) These correspond to Fig(a), (b), (c) Notice that the at both ends of the curve are zero in Fig(a) The uintic s shown in Fig(a) and (b) are generated b solving the determined linear sstem Therefore, these two curves pass through their given points in seuence In Fig(c), the number of given points with gradients is si The number of control points of a uintic B-spline is ten In this case, the total number of euations is twelve, whereas the number of control points of the uintic is ten That is, the number of euations eceeds the number of unknowns Therefore, this linear sstem is overdetermined The solution, which is the position of the control points for this sstem, is an approimation The position of the given points is compared with the geometrical knot positions of the generated uintic The maimum positional difference is - 79 And, the mean positional difference is - 89 The maimum directional difference is - 87 degree And, the mean directional - difference is 98 degree The of the curve, which is the arc length, is 98 Eample of curve shape modification is shown The coefficients of the uadratic algebraic function are calculated b the least-suares method using the distribution shown in Fig(b) The coefficients calculated are applied to the distributions shown in Fig to modif the shapes of the curves Then, the shapes of these three curves shown in Fig are modified b using the shape modification algorithm described in the section four These three shape modified curves are shown in Fig with their and plots Fig(a) corresponds to Fig(a), Fig(b) corresponds to Fig(b), and Fig(c) corresponds to Fig(c) Curvature and distributions of these three shape modified curves are shown in Fig Radius of distributions of these three shape modified curves are shown with distributions of their original curves in Fig It is visuall clear that the shapes of the curves are modified significantl (a) (b) (c) Fig Shape modified curves corresponding to Fig (a) (b) Fig Curvature and distribution corresponding to Fig shape modified original shape modified original 6 Conclusion A method to generate a uintic which passes through the given points in seuence is described In this case, the number of unknowns which are the positions of the control points is more than the number of euations which epress uintic s In other words, there are four more euations than there are control point positions Two methods have been developed to compensate for the difference between the number of unknowns and that of euations These are assuming that the s at both ends of the curve are zero to decrease the number of unknowns, and assigning four gradients to the given points to increase the number of euations (a) (b) (c) Fig Radius of distributions of original curves and those of shape modified curves These correspond to Fig (c) shape modified original ISSN: Issue, Volume, November 8

10 In addition to these methods, another method to generate a uintic which passes through or close to the given points and which has the first derivatives at these given points has been developed In this case, a linear sstem will be underdetermined, determined or overdetermined depending on the number of given points with gradients For an underdetermined sstem, the linear sstem is solved under the constraint condition while setting auiliar function b selecting one solution from an infinite number of eact solutions using Lagrange s method of indeterminate multipliers For an overdetermined sstem, the differences between the right and left sides of all the euations of this linear sstem are minimized A method to modif a uintic shape according to the specified distribution to realize an aestheticall pleasing freeform curve is described The difference between the uintic and the specified is minimized b introducing the least-suares method A reverse computational techniue is applied to solve this problem This kind of stud on the, or the to realize a fair curve is called a constrained non-linear minimization problem Eamples of curve generation are given References: [] J Hoschek, D Lasser, and L L Schumaker, Fundamentals of Computer Aided Geometric Design, A K Peters, 99, pp9 [] H Akima, A New Method of Interpolation and Smooth Curve Fitting based on Local Procedures, Journal ACM, 7,, 97, pp89-6 [] J Hoschek, D Lasser, and L L Schumaker, Fundamentals of Computer Aided Geometric Desing, A K Peters, 99, pp67-7 [] E Cohen, R F Riesenfeld, and G Elber, Geometric Modeling with Splines: An Introduction, A K Peters,, pp9-9 [] T Kuragano, A Yamaguchi, and Y Arimitsu, A Fair Curve Generation Algorithm based on a Hand-drawn Sketch, WSEAS Transactions on Computers,,, 6 [6] A Yamaguchi, and T Kuragano, Fair NURBS Curve Generation using a Hand-drawn Sketch for Computer Aided Aesthetic Design, Proceedings of the 7th WSEAS Int Conf On Signal Processing, Computational Geometr & Artificial Vision, 7, pp9-7 [7] T Kuragano, and A Yamaguchi, A Method to Generate Freeform Curves from a Hand-drawn Sketch, Journal of Sstemics, Cbernetics and Informatics,,, 7, pp- [8] A Yamaguchi, and T Kuragano, Fair NURBS Curve Generation and Determination based on a Hand-drawn Sketch, NAUN International Journal of Computers,,, 7, pp8-88 [9] A Yamaguchi, and T Kuragano, Fair NURBS Curve Generation Using a Hand-drawn Sketch for Computer Aided Aesthetic Design, Proceeding of the International Conferences: The 7th WSEAS International Conference on Signal Processing, Computational Geometr and Artificial Vision (WSEAS ISCGAV 7), 7, pp9-7 [] T Kuragano, A Yamaguchi, and Y Arimitsu, A Fair Curve Generation Algorithm based on Hand-drawn Sketch, Proceedings of the WSEAS International Conference: The 6th WSEAS International Conference on Signal Processing, Computational Geometr and Artificial Vision (WSEAS ISCGAV 6), 6, pp-9 [] W Boehm, Inserting new knots into, Computer Aided Design, Vol, No, 98, pp99- [] W Boehm, and H Prautzsch, The insertion algorithm, Computer Aided Design, Vol7, No, 98, pp8-9 [] T Lche, and K Morken, Knot removal for parametric s and surfaces, Computer Aided Geometric Design,, 987, pp7- [] W Tiller, Knot-removal algorithms for NURBS curves and surfaces, Computer Aided Design, Vol, No8, 99, pp- [] M Hoffmann, and I Juhasz, Shape Control of Cubic B-spline and NURBS Curves b Knot Modifications, IEEE,, pp6-68 [6] J F Poliakoff, An improved algorithm for automatic fairing of non-uniform parametric cubic splines, Computer Aided Design, vol8, No, 996, pp9-66 [7] C Zhang, F Cheng, and K Miura, A method for determining knots in parametric curve interpolation, Computer Aided Geometric Design,, 998, pp99-6 [8] G Farin, and N Sapidis, Curvature and the Fairness of Curves and Surfaces, IEEE Computer Graphics and Applications, 989, pp-7 [9] N Sapidis, and G Farin, Automatic fairing algorithm for s, Computer Aided Design, Vol, No, 99, pp-9 ISSN: Issue, Volume, November 8

11 [] N Sapidis, and G Farin, Letter to the Editor: Shape Preservation and the Fairness of Curves, Computer-Aided Design, (6), 99, pp6 [] K Pigounakis, N Sapidis, and P Kaklis, Fairing Spatial B-spline Curves, Journal of Ship Research, (), 996, pp-67 [] N Sapidis, Toward Automatic Shape Improvement of Curves and Surfaces for Computer Graphics and CAD/CAM Applications, in : CL Sabharwal and GW Zobrist (eds), Progress in Computer Graphics, Able, 99, pp6- [] M Kuroda, M Higashi, T Saitoh, Y Watanabe, and T Kuragano, Interpolating curve with B-spline function, Mathematical Methods for Curves and Surfaces II, Vanderbilt Universit Press, 998, pp- [] T Kuragano, and A Yamaguchi, NURBS curve Shape Modification and Fairness Evaluation, WSEAS Transaction on Computers, Issue, Vol7, 8, pp7-8 [] T Kuragano, and A Yamaguchi, NURBS Curve Shape Modification and Fairness Evaluation, WSEAS Transactions on Computers, 7,, 8, pp7-8 [6] T Kuragano, and A Yamaguchi, Curve Shape Modification based on a Fairness Evaluation, Proceeding of the International Conferences: The 7th WSEAS International Conference on Signal Processing, Computational Geometr and Artificial Vision (WSEAS ISCGAV 7), 7, pp8- [7] T Kuragano, and A Yamaguchi, Curve shape Modification and Fairness Evaluation, Proceedings of the ASME 7 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, 7, CD-ROM [8] T Kuragano, and A Yamaguchi, Curve Shape Modification and Fairness Evaluation for Computer Aided Aesthetic Design, International Conference on Engineering Design, 7, CD-ROM [9] W Li, S Xu, J Zheng, and G Zhao, Target driven fairing algorithm for planar cubic s, Computer Aided Geometric Design,,, pp99- [] C de Boor, On calculating with B-spline, Jappro Theor, 6(), 97, pp-6 [] D Marsh, Applied Geometr for Computer Graphics and CAD, Springer-Verlag,, pp88 [] Eugene V Shikin, Hand Book and Atlas of CURVES, CRC Press, 99, pp9 [] G Farin, D Hansford, THE ESSENTIALS OF CAGD, A K Peters,, pp- [] E Cohen, R F Riesenfeld, and G Elber, Geometric Modeling with Splines: An Introduction, A K Peters,, pp9 [] J Hoschek, D Lasser, and L L Schumaker, Fundamentals of Computer Aided Geometric Design, A K peters, 99, pp96 [6] G H Behforooz, and N Papamichael, End Conditions for Interpolator Quintic Splines, IMA Journal of Numerical Analsis, 98, pp8-9 [7] W Boehm, and H Prautzsch, Geometric Concepts for Geometric Design, 99, pp6-7 [8] G Farin, and D Hansford, Practical Linear Algebra -A Geometr Toolbo-,, pp- [9] L Piegl, and W Tiller, The NURBS Book, Springer-Verlag, 997, pp ISSN: Issue, Volume, November 8