freeform: A Tool for Teaching the Mathematics of Curves and Surfaces

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1 257 freeform: A Tool for Teachig the Mathematics of Curves ad Surfaces James R. Miller Uiversity of Kasas, jrmiller@ku.edu ABSTRACT We describe the structure ad use of a iteractive modelig ad visualizatio tool used to help studets uderstad various importat cocepts i curve ad surface desig. This paper describes those aspects of the tool related to NURBS curves, but Bezier ad Hermite curves are also supported as are Bezier ad B-Splie surfaces. The focus of this paper is almost exclusively o those features of the tool used to visualize the mathematics of NURBS curves ad surfaces, especially focusig o a itegrated visualizatio of how weights ad kots affect the bledig fuctios ad, through them, the curve. It is curretly packaged as a stadaloe C++ program usig OpeGL, ruig o Macitosh ad liux operatig systems. A port to Widows is uderway, but as a part of a larger effort to redesig ad better package a Java-JOGL-Swig versio, access to which will be placed i the public domai usig Java Web Start. Keywords: CAD educatio, visualizig B-Splie fuctios, curve & surface desig. DOI: /cadaps INTRODUCTION Over the past several years, we have developed a iteractive tool called freeform that is used to demostrate the mathematics of curve ad surface desig primarily for studets i our graduate Geometric Modelig course. All of the studets i the course have programmig expertise as well as basic kowledge of the mathematics of 3D graphics (poits ad vectors i affie ad projective spaces, affie trasformatios, etc.). Most are also familiar with OpeGL, ad may, if ot most, have used oe or more ope source or commercial CAD desig tools supportig the creatio ad maipulatio of freeform curves ad surfaces. The itet of this course is to give them more i-depth uderstadig of the uderlyig mathematics ad algorithms that they have bee employig at a user level. They demostrate their kowledge with a combiatio of writte homework ad programmig projects. The course is taught as a graduate level computer sciece course, but it is also actively advertised to graduate studets i other departmets. It frequetly draws some from mechaical egieerig as well as occasioally from mathematics ad aeroautical egieerig. The egieerig studets appreciate the chace to uderstad the uderlyig mathematics ad algorithms of desig packages they typically use as it removes the veil of black magic ad helps them better uderstad what they ca ad caot do. At the same time, it helps them to become better users of CAD packages [8]. The computer sciece studets (ad, to a certai extet, the mathematics studets) are exposed to a realistic vehicle for explorig umerical programmig issues, mathematical fuctio spaces, ad other

2 258 such issues that they all too frequetly oly study i the abstract. May are simply motivated by a desire to better uderstad the techology. The class is usually sufficietly small so that all the studets get to kow each other, ad they are able to share their respective expertise. The freeform program supports parametric curves ad surfaces defied as bleded sums of cotrol poits: curves: C u F ( u) P (1) i 0 tesor product surfaces: S u w i i m, F ( u) F ( w) P (2) i j ij i 0 j 0 It also supports sketchig ordered collectios of poit-vector pairs to geerate piecewise Hermite curves. We try to emphasize the importace of uderstadig how bledig fuctios determie geeral properties of curves ad surfaces, while the cotrol poits determie the shape of a particular istace. This ituitio is sometimes easier to covey with some classes of curves ad surfaces (e.g., Bezier) tha it is with others (e.g., NURBS). For example, the fact that desig hadles such as kots ad weights affect the bledig fuctios themselves ca be a stumblig block. To help with the developmet of this uderstadig, freeform icludes a itegrated ad iteractive visualizatio of how adjustig kots ad weights affect the shape of the bledig fuctios as well as that of the curve. I side-by-side widows, users watch the shape of the fuctios ad the shape of the curve chage i respose to iteractive kot ad weight adjustmets. As will be illustrated below, other features of freeform preset visualizatios of the relatioship betwee spas ad cotrol poits as determied by the kot vector. As the scope ad features of this tool have evolved, we have bee usig it i a more fudametal way i the course. While doig so, we have observed much greater egagemet o the part of studets, better performace o homework ad exams, much stroger ability to hadle more complex ad detailed questios, ad, perhaps most iterestigly, much better questios raised i class. 2 OTHER VISUALIZATION-BASED SPLINE EDUCATIONAL TOOLS While we kow of o educatioal tool with the breadth of coverage of ours, others do exist which preset visualizatios of various sorts specific to B-Splies. Oes with which we are familiar are described below. Lerios describes a program called scurvy that was writte to illustrate the depedece of B-Splie curves o cotrol poit positios ad kot vector sequeces [5]. No attempt to visualize the basis fuctios is metioed, or does scurvy as described support ratioal B-Splies. There are also some visualizatio aids described that allow the user to see what portio of a B-Splie curve will be affected if a particular cotrol poit is moved. Chag developed a tool usig Mathematica that allows a sigle o-ratioal B-Splie curve to be maipulated [1]. A butto is used to toggle betwee showig the bledig fuctios ad showig the curve. Sliders ca be used to adjust the kot values. Demidov offers web pages with Java applets that demostrate qualitatively how kot placemet affects B-Splie basis fuctio shapes [2]. No quatitative kot values are show, ad it is difficult to experimet with kots of multiplicity greater tha 1. It is also difficult to relate bledig fuctios to cotrol poits or positios o curves to poits o the bledig fuctios or eve spas i parameter space. Ulike the previous two examples, Fisher describes a tool that focuses o cotrol poit weights i NURBS curves [4]. They preset weighted cotrol poits as poits i projective space ad illustrate how the curve is geerated by projectig the projective space curve poits back to the affie plae. There is little or o metio of kots, visualizatio of basis fuctios, or ability to alter kot spacig for their curves. By ad large, all these tools allow oly a sigle plaar curve ad permit visualizatios of limited aspects of the curves ad/or basis fuctios. Our goal was to develop a comprehesive set of

3 259 visualizatios that allowed the user to uderstad deeply the relatioship betwee kots, basis fuctios, cotrol poits, weights, ad curve shape. Moreover, we also wated to be able to illustrate commo algorithms i a cotext sufficietly rich so that ot oly their operatio, but also their applicatio could be appreciated. This requires the ability to store ad maipulate several 3D curve ad surface istaces at oce. 3 THE DESIGN OF FREEFORM The freeform program was desiged first ad foremost as a teachig tool, ot as a productio curve ad surface desig tool. Most otably, this meas certai features are very limited or missig altogether. For example, there are relatively few ab iitio curve ad surface creatio mechaisms. There is o support for matchig curvatures at specific positios or other similar precisio desig tools. There are oly limited sketchig iput facilities. Supported curve creatio operatios iclude: Iterpolatio: o Create a degree Bezier curve exactly iterpolatig a ordered collectio of +1 poits. (A few optios are provided for selectig how parameter values are associated with the poits.) o Create a piecewise cubic Hermite curve iterpolatig a ordered set of +1 poitvector pairs. Least squares approximatio: Create a degree m Bezier curve which is the least squares approximatio to a collectio of poits, where m<. (A few optios are provided for selectig how parameter values are associated with the poits.) Shape approximatio: Use a ordered set of +1 poits as the cotrol polygo for a degree Bezier or Ratioal Bezier curve; alteratively, use the ordered set as the cotrol poits for a degree p B-Splie, or ratioal B-Splie (NURBS) curve, p. The poits ad/or poit-vector pairs ca be digitized or read from a file. Supported surface creatio operatios iclude: Geerate a regular +1 x m+1 array of poits o a plae to be used as cotrol poits for a Bezier or NURBS surface. Create a NURBS surface as, for example, a ruled surface, sum surface, or revolutio surface. There is a simple text file format that ca be read by freeform to import geometry. Geometry ca be saved by freeform i this format for later re-loadig. Its format is also sufficietly simple so as to be readily geerated by other applicatios to allow geometry from those sources to be imported ito freeform. Several features are icluded i freeform that would likely ot be visible at the user iterface i productio desig systems, or at least ot as directly as they are here. Some of these features are listed below. We will discuss these i detail i the ext sectio. spier cotrols for settig the kot values of the curret B-Splie, icludig allowig kots to assume ay multiplicity; spier cotrols for adjustig weights of the curret B-Splie; separate side-by-side displays of ormalized ad uormalized B-Splie basis fuctios that, alog with the separate display of the correspodig curve, are dyamically updated as weights ad/or kots are adjusted; several passive highlightig techiques discussed below that are desiged to help the studet visualize several importat cocepts icludig the spas affected by a give cotrol poit as well as the set of cotrol poits affectig a give spa; The rest of the paper focuses almost exclusively o NURBS curves ad associated visualizatios from the perspective of a studet tryig to master the relatioships amog the bledig fuctios, cotrol poits, weights, ad kot vectors i order to better uderstad the uderlyig NURBS represetatios ad algorithms.

4 260 4 USING FREEFORM TO VISUALIZE NURBS CURVES Our geeral approach whe teachig this material is to cover the detailed mathematics ad proofs i parallel with the more ituitive ad iteractive visualizatios preseted i this sectio. The goal is to iterpret the visual properties we discover as geometric maifestatios of correspodig mathematical properties of the bledig fuctios ad curves that we derive. Evetually we try to build the ituitio that allows studets to look at a visualizatio of some aspect of a curve s geometry ad be able to describe it i terms of what the kots ad weights (ad hece bledig fuctios) must be; ad vice versa: be able to see some mathematical result or expressio ad be able to predict what the visual maifestatio of it will be. 4.1 Prelimiaries: Creatig a Curve; Basic Cotrol Poit Kot Bledig Fuctio Associatios A top-level dialog widow allowig the studet to create istaces of curves ad specify how certai maipulatios are to be iterpreted is show i Fig. 1. Fig. 1: Basic curve cotrol widow. Whe a studet clicks the Add disjoit NURBS butto, a NURBS curve with the idicated properties is created ad displayed. Usig the properties settigs as displayed i the Curve Cotrols widow of Fig. 1., a ope, periodic, cubic B-Splie with te cotrol poits will be created, all of whose weights are 1. As illustrated i Fig. 2(a)-(c)., side-by-side widows are the preseted so the studet ca see the curve, the kot vector ad weights, ad the basis fuctios. Fig. 2: After creatio of a cubic NURBS curve, side-by-side widows preset (a) the curve, (b) the weight ad kot cotrols, ad (c) the Basis Fuctios. We shall use the followig otatio ad covetios whe discussig NURBS curves ad the various visualizatios supported by freeform: the curve is of degree p, order k=p+1; it employs +1 cotrol poits ad utilizes a kot vector, u U,, u 0. k I the basis fuctio widow, a solid white rectagle outlies the parametric domai. The alteratig light ad dark gray vertical bads emphasize the parametric spas. I Fig. 2(c) this is u p u 3 3 u u 1 u This parametric domai geerally makes sese to the studet at this poit because we have already derived the proof that the bledig fuctios N j, k, i k 1 j i are

5 261 the oly o-zero order k bledig fuctios defied over the parametric iterval u i u u i 1, ad that these k fuctios sum to 1. Betwee the proof ad this visualizatio, studets come to uderstad the idea that every spa of a order k B-Splie curve has exactly k o-zero bledig fuctios active, ad hece the shape of each spa is determied solely by the correspodig set of k cotrol poits. Moreover, it is the clear to the studet that the parametric domai must start at kot u p because that is whe we first have k active bledig fuctios. It eds at kot u 1 because we do ot create a (+1)-th bledig fuctio (there is o (+1)-th cotrol poit for it to bled), hece we o loger have k o-zero bledig fuctios for u u 1. Fig. 3: Some passive highlightig i the side-by-side widows: (a) the curve with its break poits, (b) a segmet highlighted i brow with cotrol poits colored to match their bledig fuctios, ad (c) the active bledig fuctios colored to match the cotrol poit colors. To make this coectio more cocrete as well as to better illustrate exactly what cotrol poits affect a give spa, we employ a passive highlightig scheme operatig i uiso across the curve geometry ad bledig fuctio widows. As a prelude, we request that spa break poits be displayed. (See the check box i Fig. 2(b)) The studet the sees the image of Fig. 3(a). As the studet passively moves the cursor over a spa, that spa is highlighted i brow, ad the portio of the cotrol polygo defied by the k cotrol poits correspodig to the k o-zero bledig fuctios is highlighted usig a color scheme tied to that of the bledig fuctios (Fig. 3(b)). We ca also see i Fig. 3(b) that all other cotrol polygo legs i the curve geometry widow have bee grayed out; similarly, all portios of all bledig fuctios i the bledig fuctio widow outside of the correspodig spa have bee grayed (Fig. 3(c)) all to emphasize the coectio betwee the parameter ad coordiate spaces. Oe use of the display i the curve geometry widow is to make clear to the studet that ay chage to the highlighted cotrol poits will chage the shape of the highlighted spa; coversely, i order to adjust the shape of that highlighted sectio, it is ecessary to adjust the positio of oe or more of the highlighted cotrol poits. The closely related cocept is that all cotrol poits (other tha the first p ad the last p) have ifluece over k cosecutive spas. Additioal passive highlightig schemes are used to covey this idea as well as to visualize how much of the curve is affected by the first p ad last p cotrol poits. I Fig. 4(a)., the studet has positioed the cursor over cotrol poit P 5. All elemets of the display have bee grayed out except the idicated cotrol poit ad the k spas it affects. Meawhile, the bledig fuctios i the adjacet bledig fuctio widow have all bee grayed out except for the correspodig bledig fuctio. The fuctio ad the poit have bee assiged the same color to make the coectio obvious. Fig. 4(c)-(d) illustrates the same idea for a iitial cotrol poit whose ifluece exteds over oly 2 spas sice its bledig fuctio starts two spas before the start of the active parametric domai.

6 262 Fig. 4: Passive highlightig i the side-by-side widows to illustrate the coectio betwee cotrol poits ad affected spas: (a) a iterior cotrol poit affectig all k spas, (b) its bledig fuctio whose domai is etirely withi the curve parametric domai, (c) oe of the first p cotrol poits with the two spas it iflueces highlighted, (d) its bledig fuctio whose domai icludes two spas outside the parametric domai of the curve. 4.2 Closed Curves To this poit, studets have see oly uiform kot spacig ad the periodic bledig fuctios they geerate. It geerally oly takes a little explaatio ad a simple demostratio of movig the positios of the fial p cotrol poits o top of the positios of the first p to show them how a closed C p 1 curve ca be created. Selectig the closed radio butto show i Fig. 1 ad creatig a ew curve the illustrates how this ca be doe automatically. Fig. 5: Illustratig the effect of kot adjustmet: (a) kot u 8 is decreased i magitude, icreasig the maximum value of N 5 while decreasig the maximum value of earby bledig fuctios, (b) the curve draws closer to cotrol poit P 5 as a result, (c) usig the bledig fuctio probe to examie the ew maximum value of bledig fuctio N Nouiform Kot Spacig ad Clamped Curves As a first step towards explorig ouiform spacig, the slider of a iterior kot, u 8, is adjusted. The studets watch as various curve spas grow ad shrik i arc legth; simultaeously they ca see

7 263 how the spas i the bledig fuctio widow grow ad shrik (Fig. 5(a)-(b).). More sigificatly, the relative ifluece of various cotrol poits i the adjacet spas of the curve rise ad fall at the same time. Fig. 5(c)., for example, shows how cotrol poit P 5 has more ifluece i those spas after the kot adjustmet sice its bledig fuctio ( N 5, k ) grows while others (e.g., N 4, k ) dimiish. Studets observe the curve drawig closer to P 5 as a result. Fig. 6: Illustratig the effect of kot multiplicity: (a) u 7 u 8, (b) u 6 u 7 u 8, (c) cotrol poit P 5 is iterpolated as a result. We the drop u 8 eve more util it equals the value of u 7 (Fig. 6(a).). A zero-legth spa has bee itroduced, ad eve though the passive cotrol poit spa highlightig (e.g., Fig. 4(a).) appears to idicate that cotrol poit P 4 ow oly iflueces three spas, it is obvious to the studets that it still affects four because they have watched oe of the spas gradually shrik util its legth is zero. We also poit out to the studets that this kot vector cofiguratio has caused the curve to become taget to the P 5 P 6 cotrol polygo leg this result beig the ecessary geometric cosequece of the fact that oly N 5 ad N 6 are o-zero at the duplicated kot. We the demostrate icreasig the multiplicity to p k 1 3, this time iteractively icreasig u 6 util u 6 u 7 u 8. The studets watch as bledig fuctio N 5 spikes to a value of 1 at the repeated kot while the curve pulls ito ad iterpolates P 5. (See Fig. 6(b)-(c).) It is atural to ask ad studets typically do at this poit what happes if the kot is repeated agai (ad agai ad agai ). After tryig to exted their uderstadig to guess the aswer, we demostrate the effect by settig u 5 equal to the kot whose multiplicity was previously three to yield the displays of Fig. 7. Studets see bledig fuctio N 4 spike to 1 at u u 5, the immediately drop to 0; meawhile, N 5 starts at 1 at u u 5 ad the gradually tapers dow to 0. The resultig broke curve is the ituitive. Fig. 7: Creatig a iterior kot with multiplicity k, breakig the curve: (a) two bledig fuctios spike to 1 at the repeated kot, (b) the curve is broke ito two pieces.

8 264 Describig how to geerate B-Splies that iterpolate their iitial ad fial cotrol poits is fairly straightforward at this poit. Combiig a uderstadig of how duplicated kots allow iterpolatio of arbitrary cotrol poits with the fact that the parameter rage of the curve starts at the k-th kot ad eds at the (+1)-th kot, it is ot surprisig that k iitial ad k fial duplicate kots lead to edpoit iterpolatio, ad we itroduce the commo term clamped B-Splie to describe B- Splies with this property. We geerally start with a uclamped B-Splie ad iteractively adjust the iitial ad fial kots to have multiplicity k so that studets ca watch the curve gradually draw ear the first ad last cotrol poits, ultimately iterpolatig them. Oe slightly cofusig issue that arises at this poit relates to the specific cotrol poit that gets iterpolated whe kots are repeated the appropriate umber of times. Whe first itroducig the idea i the cotext of repeated iterior kots, we discover the rule that says: if u i is the first of p k 1 repeated kots, the cotrol poit P i 1 is iterpolated at parameter value u u i. But thigs are a little differet at the start sice we have u L u, but cotrol poit P is iterpolated. There 0 k 1 0 are various ways to explai this, but we have foud it coveiet to explai it i the cotext of aother mystery: exactly how may kots are really eeded to describe a order k B-Splie with +1 cotrol poits? Studets are frequetly assiged readigs from differet sources, ad the how may kots are required? questio ofte arises i that cotext. For example, a studet will read i Fari s book [3] that a order k B-Splie with +1 cotrol poits requires +k 1 kots. They the read i Piegl & Tiller [6] that there are +k+1 kots! Is there a typographical error? Is oe of them wrog? We demostrate that either is really wrog by showig that the actual value of the first ad last kot are irrelevat to the shape of the curve. We demostrate this fact i freeform by observig that, as we drag the first ad last kot aroud, o part of the bledig fuctios iside the parametric domai chage; similarly, the curve shape is completely uaffected as we modify those two kot values. The geometry of Fig. 8. illustrates a clamped order 5 B-Splie alog with its bledig fuctios. Fig. 8(a). shows the curve, ad Fig. 8(b). shows the correspodig bledig fuctios as determied by the kot vector whose first ad last kots have multiplicity k=5. As we iteractively decrease the first kot ad/or icrease the last kot, studets see that we modify the iitial (fial) spa of the correspodig bledig fuctios, but we do ot modify ay bledig fuctio i ay way withi the defied parametric domai of the curve. It is the o surprise whe they also observe that the shape of the curve remais ualtered as those two kots are modified. Fig. 8: The actual value of the first ad last of the +k+1 kots of a B-Splie do ot affect the shape of the B-Splie curve: (a) bledig fuctios for a order 5 clamped B-Splie, (b) the correspodig curve, (c) the bledig fuctios after decreasig the first ad icreasig the last kot. I summary, strict iterpretatio of the recursive defiitio of the basis fuctios most aturally leads to the use of +k+1 kots. O the other had, typical implemetatios of evaluatio algorithms that start by locatig the oly o-zero order 1 basis fuctio at a give parameter value ad the work up to the appropriate order k fuctio will ever actually use kot 0 or kot +k. This observatio, coupled with the fact that the curve shape is uaffected by the first ad last kot provide the justificatio for sayig oly +k 1 kots are required. Of course care must be take whe iterfacig with programmig APIs like OpeGL or geometric data base represetatios (IGES, STEP; commercial oes like ACIS; etc.) to make sure that relevat covetios are observed. Some of these expect oly +k 1 kots; others require +k+1.

9 Weights ad Ratioal B-Splies The B-Splie cotrols show i Fig. 2(b). show spiers allowig weights of the cotrol poits to be modified. This ituitio is easy to covey: icreasig weights draws the curve projectively towards the give cotrol poit. We itroduce the stadard curve represetatio: C u N u w P i, k i 0 i i (3) N u w i,k i 0 i Particular values of the weights i cojuctio with specific kot vector ad cotrol poit placemet allow the coic curves to be exactly represeted. Fig. 9 illustrates matchig iety degree coic sectio arcs with the idicated weight assigmets. (These are actually ratioal Bezier curves, but studets uderstad by this time how to geerate a ratioal B-Splie represetatio of a ratioal Bezier curve.) Fig. 9: Matchig a hyperbola, parabola, circle, ad ellipse with ratioal curves. The mai additioal ituitio at this poit is to uderstad the basis fuctios. We itroduce two stadard equivalet mathematical represetatios for the NURBS curve. The first employs what we call uormalized bledig fuctios (the covetioal bledig fuctios multiplied by the weights): N iu, k u wi N i, k u (4) The curve represetatio remais defied i affie space ad becomes: C u N u P i 0 U i,k i (5) N u i 0 U i,k The secod represetatio employs what we call ormalized bledig fuctios ad are oly applied to cotrol poits embedded i projective space usig their weights: Pi P wi xi, wi yi, wi zi, wi N in, k u (6) wi N i, k u (7) w j N j,k u j 0 Usig Eq. (6) ad Eq. (7), the NURBS curve is defied i projective space as: C P u N in, k u Pi P (8) i 0 I freeform, we allocate oe display widow for the collectio of N N i,k U N i,k fuctios, ad oe for the set of. These widows, displayed i Fig. 10, are titled Uormalized B-Splie Basis Fuctios ad Normalized B-Splie Basis Fuctios, respectively.

10 266 Fig. 10: The geometry of Fig. 8 after icreasig the weight of P3 ad decreasig the weight of P7. From Eq. (4) it is clear that the U N i,k fuctios will ot i geeral sum to 1. I fact, idividual fuctios may exceed 1 as ca be see i Fig. 10. Modifyig weight wi affects oly fuctios remai uchaged. By cotrast, the ormalized N N i,k U N i,k ; all other fuctios allow studets to see exactly how the ifluece of other poits decrease (icrease) as wi icreases (decreases). I the example of Fig. 10., studets ca observe how cotiuous icreases to w3 cause correspodig cotiuous decreases i all bledig fuctios that are o-zero over a parametric iterval which overlaps that of N N 3,k. Therefore, the ifluece of the correspodig cotrol poits o the curve decreases, somethig that is readily observed by watchig the curve as the weight w3 icreases. 4.5 Other Features Several other features of freeform permit studets to see the operatio of importat algorithms like kot isertio, coversio of a B-Splie to a equivalet piecewise Bezier, ad others. There are also various iteractive tools for restrictig mouse-based draggig of poits to give axes, coordiate plaes, or cotrol polygo legs. A marker ca be moved through the Basis fuctio widow while the correspodig B-Splie poit is traced alog the curve. While doig so, successive spas alog with the cotrol poits determiig their shape are highlighted as i earlier examples. 5 CURRENT STATUS AND PLANS The freeform program is used to a limited extet late i our udergraduate seior-level graphics class i which the focus is less o the mathematics ad more istead o ituitio. The goal is to teach the basics of curve ad surface modelig at a user level ad possibly iterest some studets i takig the graduate Geometric Modelig class i which the program is used extesively. I that course, the goal is to uderstad the mathematics ad algorithms much more deeply. The curret implemetatio has evolved over several years ad is writte i C++ usig OpeGL ad the GLUT for basic display, ad the glui [7] for most user iterface widgets. A effort is curretly uderway to clea up the user iterface, make some thigs more cosistet ad move the higher level code to Java, Java OpeGL (JOGL), ad Swig. The goal is to the deliver the applicatio over the iteret via Java Web Start.

11 267 6 SUMMARY We have described the use of a portio of freeform, a curve ad surface modelig tool developed first ad foremost as a educatioal tool to help studets uderstad the mathematics of curves ad surfaces, especially B-Splies ad ratioal B-Splies. Reactio from studets i various classes has bee positive, ad performace o homeworks, projects, ad exams has exhibited sigificat improvemet as the tool has evolved ad bee used more extesively. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] Chag, Y.-S.: B-Splie Curve With Kots, BSplieCurve WithKots/. Demidov, E.: A Iteractive Itroductio to Splies, Fari, G.: Curves ad Surfaces for CAGD: A Practical Guide, fifth editio, Morga Kaufma Publishers, Sa Diego, CA, Fisher, J.; Lowther, J.; Shee, C.-K.: If You Kow B-Splies Well, You Also Kow NURBS!, Proceedigs SIGCSE 2004, March 3-7, 2004, Norfolk, VA, pp Lerios, A.: B-Splie Curve Visualizatio, Piegl, L.; Tiller, W.: The NURBS Book, secod editio, Spriger, New York, Rademacher, P., GLUI User Iterface Library, Saakes, D.: Hit ad Reder: Teachig CAD Visualizatio to Product Desigers, Computer-Aided Desig & Applicatios, Vol. 3, Nos. 1-4, 2006, pp