Constraned Shape Modfcaton of B-Splne curves Mukul Tul, N. Venkata Reddy and Anupam Saxena Indan Insttute of Technology Kanpur, mukult@tk.ac.n, nvr@tk.ac.n, anupams@tk.ac.n ABSTRACT Ths paper proposes a methodology to change the shape of a B-Splne curve locally. Present algorthm modfes the exstng B-Splne curve and makes t pass through a gven pont P, constraned to le n regon R, wth lmted devaton from the orgnal curve. Proposed methodology splts the curve nto atmost three segments based on the curvature of the curve and then modfes only one segment usng knot nserton and control pont ncluson. Keywords: B-Splne curve; Knot Inserton; Curve splttng.. INTRODUCTION A B-Splne curve C(u) s defned by control ponts (also known as de-boor ponts) and the knot vector u, u,..., } as U { 2 u p + n n C( u) N, p( u) where p s the degree of the curve, n s the number of control ponts and N, p ( u) are the splne bass functons [4] defned over a knot vector U. Durng the past few decades, B-splne curves have ganed popularty n shape modelng and geometrc desgn []. Beyond creatng these types of curves, modfcaton of the exstng ones s also of great mportance. B-splne curves offer a unfed mathematcal formulaton for representng not only free-form curves, but also standard analytcal shapes. Through the manpulaton of control ponts, degree and knots, a user can desgn varety of shapes usng B-splne formulatons. In a system that uses B-Splne curves as shape desgn tool, one can mplement many algorthms such as poston and dervatve evaluatons, knot nserton [2], knot deleton and degree elevaton [4]. Based on these fundamental algorthms, one can perform geometrc modelng and processng. B-Splne follows a representaton based on control ponts and knots, and weghts n case of ratonal B-Splne curves. A desgner can modfy a local curve segment by teratvely movng a control pont, changng the knots or modfyng the weghts assocated wth the control ponts. To ths end, Pegl [3] presented a method for modfyng shape of a ratonal B-Splne curve by relocatng ponts on the curve. The modfcaton s realzed by repostonng one or more control ponts. Pegl and Tller [4] dscussed n detal curve wrappng, flattenng, bendng and twstng usng the same approach. In addton, Au and Yuen [5] nvestgated how the shape of a NURBS curve changes when the weghts and locaton of the control ponts are modfed smultaneously. A drawback that the above mentoned methods have s that they are not automatc. A desgner should predefne the control pont to be shfted or the correspondng weght to be changed to provde the desred shape modfcaton. Workng on the nteractve scheme for curve modfcaton, Fowler and Barter [6] had presented constraned based manpulaton of curve of arbtrary degree. A drect modfcaton of free form curve by dsplacement functons was proposed n [7], whch comprses of degree elevaton and control pont repostonng. Besdes the most basc tool for shape modfcaton, control pont or weght modfcaton, one can work on knot alteraton. Juhasz and Hoffmann [8] showed that modfcaton of a knot of B-Splne curve of degree p generates a one parameter famly of curves. The famly has an envelop whch s also a B-Splne curve wth the same control polygon and degree p-. The theoretcal results have been appled n [8] to modfy the shape of a cubc B-Splne curve. The approach used determnes three consecutve knots by fttng a curve of a parabolc famly based on the ()
desred constrants. The procedure enables a local shape modfcaton but s, however, restrcted to cubc B-Splne curves only. Some other aspects of knot modfcaton have also been studed by Lyche and Morken [9], where the effect of knot varaton s examned from numercal pont of vew. The choce of knot value n curve approxmaton and nterpolaton has also been nvestgated n several papers, the recently publshed [0] and references theren. There are not many attempts n the area of knot alteraton to get the desred shape change. The man reason for ths s the non ntutve nature of curve change by knot alteraton. The problem arses because wth each dfferent placement of knots one can derve a new set of control ponts, whch can exactly get the same curve []. Ths problem s further compounded by the fact that the estmaton of knot value s a non lnear process. Czuczor et al. [2] and Mamc et al. [3] used optmzaton procedures to overcome the problem faced n knot vector alteraton. The use of optmzaton s computatonally expensve. Present work s an attempt to fnd the B-Splne curve wth desred change wthout any teraton nvolved. The most effectve shape control tool of B-Splne curves s control pont repostonng. A remarkable advantage of knot modfcaton over control pont repostonng s that the modfed curve always remans n the regon determned by the convex hull of the orgnal curve. Thus, the knot based method s preferred f one wants the curve n that regon or when fne tunng n the curve s requred [8]. Present work nvolves modfyng the shape of an exstng B-Splne curve wth the shape preservng constrant and less devaton from the orgnal curve. The approach uses both knot vector alteraton by ntroducng new knots usng knot nserton [4] and control pont ncluson. Unlke the approach of fttng a B-Splne curve to the gven data [5], ths method works at makng the curve change locally. The B-Splne curve fttng problem s to produce a B-Splne curve to approxmate a target curve wthn a pre-specfed tolerance. Ths fttng technque uses an energy functon and then mnmzes t to fnd the new poston of the control ponts. On the contrary, the present work modfes the curve locally, by splttng t nto parts and then acheves the desred shape change wthout any optmzaton nvolved. 2. PROBLEM DESCRIPTION The man am of the present work s to make a B-Splne curve C(u) pass through a prescrbed pont P (Fg..) whch les n a constraned regon R (descrbed later n ths secton). The other objectve of ths work s to preserve the curve shape and to have lmted devaton from the orgnal curve. The shape of the curve s assumed to be preserved f the modfed curve has the same number of feature ponts as the orgnal curve. Fg.. Problem defnton, orgnal curve and the modfed curve Every curve has certan features that defne the shape of that curve, namely, maxma and mnma of curvature and sudden changes n the curvatures. These ponts are defned n the present work as feature ponts F. To fnd these feature ponts, dscrete curvatures [6] at regular nterval on the curve are evaluated. These feature ponts are dentfed usng fne and rough checkng technque proposed by Lu et al. [6]. Feature ponts are ntrnsc to the curve and are used n secton 3 to fnd the ends of the splt segment. To have lmted devaton from the
orgnal curve the shape modfcaton s performed locally, only n the curve segment between the splt ends (secton 3). The proposed algorthm for constraned shape modfcaton works when pont P s present wthn a constraned regon R as shown n Fg.2(b). To modfy the curve, the essental step s to fnd the value of knot to be nserted n the exstng knot vector usng the concept of knot nserton and mappng knots on the control polylne as dscussed by Sederberg [4]. Fg.2(a).Fndng the knot value to be nserted, 2(b). Hashed regon R 2. Admssble regon R for pont P Let P,P +, P + 2 be three consecutve control ponts of a degree p B-Splne curve (Eqn. ().). Consder knots u u, u u U. Defne, + + p, + p+ ( u+ u ) + + ( u+ p u+ ) u + p u and + ( u u + p+ + p) + ( + p + ) u + p+ to be the ponts on the polygonal leg L+{P+,P} and L+2{P+2,P+}, as shown n Fg.2. Here V and V+ correspond to the mapped postons of knot u+ on L+ and u+p on L+2 respectvely. The regon R has three boundares, two formed by the control polylne and one by the parabola wth startng and endng ponts as V and V+. 2.2 Algorthm for Constraned shape modfcaton The gven B-Splne curve s made to pass through a pont P R by nsertng a knot u n U, such that one of the newly formed control polylne {Qj-Qj} [4], passes through pont P (Fg.2(a).). Note that Qj- and Qj are the mapped postons of knot u on the polygonal legs L+ and L+2 respectvely. The steps nvolved n constraned shape modfcaton are: The value of knot u s found such that P α ( u) + ( α) ( u) where α [0,]. Usng knot nserton [4], new control ponts are found. Pont P s ncluded n the control pont set as a control pont of multplcty p-2. The use of knot nserton reduces the sze of the convex hull and brngs t nearer to the curve keepng the curve unchanged. The reducton n convex hull sze lmts the local devaton of the curve n a smaller regon (Fg..). A B-Splne curve s monotone f the control polygon s monotone and s convex f t s convex. The present algorthm fnds a control polygon by nsertng knot u, whch mantans the convexty. Next, ncluson of pont P as control pont does not change the convexty of newly formed control polygon, snce the polygon remans the same wth the reducton n convex hull sze. As there s no change n the convexty of the polygon, the shape of the modfed curve s preserved. Procedure of fndng new control ponts, a set of Qj such that P les on the polylne and ncludng P as a control pont of multplcty p-2 makes one the newly formed polylne tangent to the modfed curve at P. Ths produces a smooth curve. It can be seen n Fg.3(a) that polylne Q+Q+2 forms a tangent to the curve at P. There wll be a formaton of + u u + u + (2)
(a). Orgnal curve (), modfed curve usng knot nserton (2) (b) Formaton of Knk n curve (3) and wthout knot nserton (3). Fg. 3. Constraned shape modfcaton knk n the curve f the curve s modfed by drectly ncludng P as a control pont of multplcty p-2 wthout knot nserton, as shown for curve (3) n Fg.3.(b). 3. DETERMINATION OF SPLIT SEGMENT Modfcaton of the B-Splne curve affects the shape of the curve n p- convex hull [4]. To lmt the shape change to further smaller regon, the curve s splt nto atmost three segments dependng on the postons of pont P n space. The shape of the segment nearest to the pont P s changed. Feature ponts defned n secton 2 are used for the purpose of fndng the approprate postons of the splt ends. An ntrnsc local shape modfcaton method s developed usng these feature ponts. 3. Determnaton of Crtcal Feature ponts To determne the segment to be modfed, the shortest dstance of the pont P from the orgnal B-Splne curve s evaluated. Fg.4 shows a part of a B-Splne curve and pont P. The feature ponts F -3to F+2 are marked on the curve. Pont Q * denotes the pont on the curve nearest to pont P and let D be the dstance between them. Let d,+ denote the length of the curve between two consecutve feature ponts, F and F+. Ths length controls the placement of the feature ponts on the curve. It s clear that the curve wll have frequent changes n curvature f d,+ s small as compared to a large value of d,+. Consder that pont Q * les on the segment F F+, the length of segment F Q * and Q * F+ are denoted as l and l+ respectvely. Fg.4. Feature ponts on the curve
5 Let d m mn( l, l+ ) and l < l+ I m + otherwse (3) Thus denotes the feature pont nearest to pont P. In order to preserve the shape of the curve, t s mportant F to preserve the features n the neghborhood of the desred pont P. defned as crtcal features n ths work. Thus s one of the crtcal features. There could be a need to nclude more features as crtcal features to provde a F smooth transton of the curve at the ends of the splt segment. It can be observed n curve () of Fg.5 that ncluson of only one feature as crtcal feature may lead to an abrupt change n curve shape at splt ends. Curve (2) also shown n the same fgure, wth three crtcal feature ponts, shows a smooth transton of the modfed curve to the other segments at the splt ends. In order to determne the other crtcal features, feature ponts present n the neghborhood of P are dentfed usng a threshold dstance parameter ã() defned as 3 ( d ) ~ 2, + d, + a( ) (4) max( d,, d, + ) 2 + mn( d,, d, + ) Ths length parameter relates the relatve dstance between the neghborng features. To estmate the upper bound of ã(), consder d,+>d-,, 3 3 ( d ) ( ) ( ) ~ 2, + d, + d 2, + d, + d 2, + d, + ~ a( ) a( ) < ( d, +, ), greater than 3 greater than 2 d. d + + 2 + d, Based on the length parameter and the dstances d-, and d,+ there can be two cases: If d,+- d-, <ã() d,+ d-,, ths s defned as Case A Else f d,+- d-, ã() one of the two lengths, d,+ or d-, s small as compared to other, ths s defned as Case B For d,+d-, d,+- d-, 0 and ã() d-,, ths ndcates Case A. On the other hand f d,+>>d-, d,+- d, 3 d-, d,+. In Eqn.(4) usng lm 0, ã() s obtaned as d, and snce d,+>>d-,, the set of values of d, + 2 d-, and d,+ belong to Case B. Two examples consdered valdate the use of the above stated condtons. It can be observed from Fg.4 that lengths d-2,- and d-, belong to case A whle d-, and d,+ belong to case B. It can also be seen that d,+ and d+,+2 belong to case A (here the two dstances are almost equal n sze but large as compared to any of the dstances d-2,- or d-,). Thus case A can be further dvded nto two sub cases; case A wth almost equal but large lengths (d,+ and d+,+2) and case A2 wth almost equal but small lengths (d-2,- and d-,). In order to dentfy the neghborng crtcal feature ponts, three consecutve feature ponts and F FI m F + are consdered and the case to whch they belong s determned. The procedure carred out based on the case s explaned next. Case A: In ths case snce the lengths between the feature ponts s large, only F s consdered as the crtcal feature pont. Other feature pont may or may not get ncluded n the splt segment dependng on the mnmum dstance of pont P from the curve. If the dstance D s large there s a possblty that the algorthm mght nclude more feature ponts whle computng the splt length to avod sudden change n the curve. In fndng the splt ends, the splttng s performed at some dstance away from the extreme crtcal features. To make sure that the features at the ends are also preserved, t s mportant to fnd ths dstance. Ths dstance s dentfed by a length parameter defned as d d (5) I, + m, +
Fg.5. Need to fnd Crtcal Feature ponts Case B: In ths case one of the lengths and s small. Wthout loss of generalty t can be assumed that d I d, < m, + drecton of the smaller feature length, ths drecton ( F F d I m, I m d I m, I m +. Other crtcal feature ponts for ths case are dentfed by searchng for the features n the d I m, I m (Fg.6.). Through ths, all the features wth small dstance d -, n ) are ncluded as crtcal features. The ncluson termnates whenever a large dstance s encountered. The last feature ncluded s denoted as F end. Fg.6. Fndng Crtcal Feature Ponts n Case B Fg.7. Fndng Crtcal Feature Ponts n Case A2 Algorthm. Let J and Case(J)case for dj-,j and, + d I m I m 2. f {Case(J)B} FJ s ncluded as a crtcal feature pont JJ- Case(J)case for dj-,j and, + d I m I m 3. go to step 2 4. f the condton s no more satsfed IendJ, Fend s the last feature ncluded. 5. Termnate For ths case s defned as
7 mn(( d I, I + +... + d I, I ), d I, I + ) (6) end end m m m m Case A2: In ths case snce both the dstances are small, the search s extended n both the drectons untl a large length s encountered (Fg.7.). Algorthm (Frst the features before are dentfed) F. Let J- and Case(J)case for dj-,j and d I m, I m 2. f {Case(J)A} FJ s ncluded as a crtcal feature pont JJ- Case(J)case for dj-,j and d I m, I m 3. go to step 2 4. f the condton s no more satsfed, IendLJ, F L s the last feature encountered n the left drecton. 5. go to step 6 (Next the features after are dentfed) F 6. Let J+ and Case(J)case for dj-,j and d I m, m + 7. f {Case(J)A} FJ s ncluded as a crtcal feature pont JJ+ Case(J)case for dj,j+ and, + d I m I m 8. go to step 7 9. f the condton s no more satsfed, IendRJ, F R s the last feature encountered n the rght drecton. 0. Termnate For ths case s defned as d I, I + +... + d I, I + d I, I + +... + d endl endl m m m m IendR, I I + endr endl IendR The dentfcaton of the crtcal feature ponts gves a rough estmate of the splt length. The curve s splt a small dstance (determned by ) away from the extreme feature ponts ncluded as the crtcal feature. Defne ~, x where x (>2) s a factor that mnmzes the value of splt length. The optmal value of the splt length s evaluated such that both the objectves of lmted devaton and shape preserval are addressed. The shape of the curve s preserved by fndng the crtcal feature ponts that would be affected by the shape change. Lmted devaton s addressed by mnmzng the value of ~. If ~ >dm (Eqn.3.), the maxmum value of x s obtaned as max( x) dm where as f ~ max( d I, I, d ) m m, + d m t s obtaned as max(x) 2 +. mn( d, d, d ),, + m (7) Fg.8. Determnaton of Splt Segment
8 3.2 Determnaton of Splt Ends The splttng of the curve s carred out at some dstance from the extreme crtcal feature ponts, evaluated at dl and dr from the left and the rght extreme crtcal features (F L,F R ) respectvely (Fg. 8.). The procedure for obtanng the values of dl and dr s presented below. 2 Defne 2 D D +, DLlength(F L,Q * ) and DRlength(F R,Q * ). m d m Algorthm For left (L) or rght (R) f D< ~ then d( ~ +dm)/2 else d( ~ +Dm)/2 Determnaton of dr and dl completes the splt end dentfcaton. Fg.9. shows the splt ends for dfferent postons of P. Fg.9(a) to (c) ndcates dfferent cases based on the placement of P and the determned splt ends. In Fg.9(d) the curve s splt n only two segments. Ths s observed because one of the splt ends concdes wth the curve end. It s thus ndcatve that placement of pont P s an mportant parameter to decde the curve segment to be modfed. To splt the curve, the parametrc value of the pont at whch the curve has to be splt s evaluated as u *. The parameters correspondng to splt ends are ul * and ur * respectvely. Splt segment s obtaned when the above evaluated parameters (ul *,ur * ) are nserted as knots of multplcty p each n the knot vector U defned n Eqn.(). [4]. 4. RESULTS AND DISCUSSION To llustrate the curve modfcaton strategy nto practce, MATLAB TM s employed to develop the prototype system. The prototype system was developed usng personal computer wth 52 MB RAM. Fg.9. Splt Segment
The orgnal curve s splt at the splt ends, evaluated n secton 3.2. Algorthm for constraned shape modfcaton (secton 2.2) s appled to the segment between the splt ends. The splttng procedure explaned n secton 3, decdes the number of segments after splttng. In any gven curve and a gven pont P there could be atmost three segments. The segments n between the splt ends determned s modfed to acheve the desred objectve. Applcaton of constraned shape modfcaton algorthm makes the curve n the splt segment pass through the desred pont. After the shape of the curve segment s modfed, all the segments are combned to obtan the modfed curve. Ths combnng procedure nvolves removal of the knots nserted for splttng the curve (ul *,ur * ) wthout changng the shape of any of the segments [4]. The use of splttng methodology explaned restrcts the devaton of the curve n a smaller regon compared to the devaton whch occurs wthout splttng. Ths s demonstrated n Fg.0. The curve n Fg.0(a) s obtaned after splttng algorthm s appled wth constraned shape modfcaton where as n Fg.0(b) the curve s not splt. Fg.0(c) Fg.0. Intrnsc Local Shape modfcaton and Fg.0(d) show an enlarged vew of the curves. It can be clearly seen from Fg.0(b) and Fg.0(d) that there s an extra devaton n the curve whch s not observed when the curve s frst splt before shape modfcaton (Fg.0(a),(c)). Thus splttng of the curve ads n, shape preservaton and havng lmted devaton from the orgnal curve wthout applyng any optmzaton methods.
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