CC BY
25
Direct Manipulation of B-Spline
Youssef Ali Alhendawi
Abstract— This paper introduces a technique for the immediate controls of B-spline and non-uniform discerning B-splines (NURBS) bends utilizing geometric imperatives. A deformable model is produced to characterize the misshapening vitality utilitarian of B-spline and NURBS bends. The limited component strategy is utilized to limit the distortion vitality useful and explain for the disfigured state of bends subjected to imperatives. This approach brings about an arrangement of direct conditions for a B-spline bend and an arrangement of non-straight conditions for a NURBS bend. A point of view mapping is utilized to linearize the NURBS details. NURBS bends are first mapped from the 3D Cartesian arrange space to the 4D homogeneous organize space, and changed to 4D B-spline bends. After the control in the 4D homogeneous facilitate space, the changed NURBS bends are then mapped back to the 3D Cartesian arrange space. The approach is executed by a model program, which is composed in C, and keeps running under WINDOWS.
Keywords—B-spline surface, space, 4D B-spline, program, 4D homogeneous.
I. Introduction
Bend and surface outline have numerous applications in assembling enterprises, for example, vehicle body configuration, dispatch frame plan, and so on. In the course of recent years, diverse bends and surfaces portrayal shapes have been proposed. Right now, B-splines and NURBS (non-uniform judicious B-splines) are the most well known numerical structures. NURBS offers a brought together numerical frame not just for portrayal of freestyle bends and surfaces, yet in addition for the exact portrayal of close-shape shapes, for example, lines, conics, quadrics, and surfaces of upset. Further, the additional degrees of opportunity NURBS offers—the weights, enable an extensive assortment of shapes to be produced. Consequently, NURBS has turned out to be increasingly well known as of late. It has been an IGES standard since 1983, and numerous business CAD frameworks depend on the NURBS portrayal. Giving operations to altering the state of NURBS bends and surfaces is an essential issue in bend and surface outline. The coveted state of a bend or a surface is gotten through an intricate and intuitive process. It from time to time happens that the underlying plan meet the required details, and the procedure of alteration goes ahead until the point when the state of the bend and surface fulfills the necessities. As of recently, all intuitive outline plans proposed can be ordered into two classes. one is the control point based strategy, which controls bends and surfaces by controlling the DOF (degrees of freedom) characterizing them and the other is the supposed direct control technique.
In this technique chose focuses on a bend or a surface are moved to new client characterized areas. The primary technique [1- 3] is as yet the pervasive strategy for intelligent plan of bends and surfaces in business frameworks. For instance, B-splines are normally controlled by moving their control focuses, and for NURBS, the additional level of flexibility called weights can be utilized to control the state of bends or surfaces. One of the issues with this method is that the control focuses or weights are not straightforwardly identified with the changed state of bends or surfaces. Along these lines, intelligent plan utilizing this plan is frequently bulky. Now and then a substantial number of DOF must be controlled so as to alter even a little bit of a bend section. It is likewise uncertain which DOF ought to be controlled and how it ought to be controlled. The immediate control technique despite what might be expected gives the creator a larger amount interface and shape configuration is more instinctive. In this plan, clients are permitted to determine discretionary requirements on a bend or a surface. For instance, the creator can pick focuses on a bend and move them to wanted positions. The new DOF, which fulfill the predetermined requirements, are consequently processed. The immediate control strategy permits the creator a greatly improved feel between the info (indicated requirements) and the yield (altered bend or surface). Current direct control strategies can be additionally characterized into two classes: coordinate straight techniques and physically based techniques.
1.1 Direct Linear Method:
The objective of direct control techniques is to arrange the DOF of bends and surfaces with the goal that the twisted areas of the chose focuses coordinate the predetermined target areas. As there are numerous DOF arrangements that will yield a similar target areas, certain criteria must be built up so as to choose a sensible arrangement. In coordinate straight techniques, minimum squares are received as the criteria, and the last arrangement is the one that moves the control focuses, the slightest. For sey and Bartels built up a strategy for the immediate control of various leveled B-spline surfaces at B-spline joints. Bartels and Beatty stretched out this technique to permit B-spline bends to be controlled anytime on the bend. Fowler and Bartels stretched out this procedure further to permit first and second subordinates at any focuses to be controlled.
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Youssef Ali Alhendawi is with the Computer Department, Prince Sattam bin Abdulaziz University of Saudi Arabia (e-mail:
yosefyose@yahoo.com)
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Hsu et al. proposed a technique which enables various focuses to be moved to new areas all the while. As of late, Rappoport proposed a probabilistic point imperative where he utilized Kalman channels to comprehend the arrangement of probabilistic conditions. The delicate imperatives he utilized are less demanding to fulfill and therefore prompt smoother misshapenings. The portrayal frames utilized as a part of every one of these references are B-spline bends or surfaces. Coordinate straight techniques have not been produced for the control of NURBS bends and surfaces.
1.2 Physical Method
In a physically based technique, bends and surfaces are doled out some physical properties, for example, extending solidness, bowing firmness, and so on. They are permitted to extend and bowed as genuine items. By applying powers and imperatives, bends and surfaces are disfigured and controlled in an instinctive way. This strategy was first connected in the PC vision region. Terzopoulos et al. proposed flexibly deformable models, and utilized the versatility hypothesis to build differential conditions that model the conduct of non-unbending bends, and surfaces. Celniker and Gossard built up a model framework for intelligent freestyle plan by minimization of the vitality utilitarian utilizing limited component technique. Celniker and Welch utilized Bsplines as portrayal shapes and created deformable B-splines with straight requirements. Terzopoulos and Qin proposed dynamic NURBS, to show distortion of bends and surfaces. They created material science based models with properties, for example, mass dispersions, misshapening energies, and so on. The non-straight conditions of movement for a dynamic NURBS are inferred in view of Lagrangian flow and are coordinated through time to anticipate the state of a protest in light of outside powers. Faloutsos et al. created dynamic deformable models in light of standard free shape distortions.
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They utilized distinctive misshapening modes, for example, shear and bowing modes to move or quicken genuine articles.The target of this paper is to create, and linearize a physically based technique for coordinate control of NURBS bends and surfaces. NURBS bends and surfaces are first linearized by mapping them into 4D space before any condition settling happens. The subsequent conditions are then tackled for the new DOF in 4D space, lastly mapped back to 3D space. The target of this paper is to create, and linearize a physically based technique for coordinate control of NURBS bends and surfaces. NURBS bends and surfaces are first linearized by mapping them into 4D space before
any condition settling happens. The subsequent conditions are then tackled for the new DOF in 4D space, lastly mapped back to 3D space.
— : Original curve
: New curve aller I' is constrained to J'
Note: Ph is in hyperplune n,—w0.
II. User Interface
To exhibit and confirm the bend control conspire created in this paper, a program called DMC (Direct Manipulation of Curves) has been produced. The program was created utilizing Borland C 11 4.5, and keeps running under Windows condition. The program is sorted out in a layered structure. The center of this structure is the information structure of B-spline, and NURBS bends, separately. Upon every datum structure lie the "Record", "View", "Build" and "Control" modules. The outmost layer is the UI, which comprises of menus for info and yield of information, realistic show and different control operations. Fig. 10 delineates the program structure. The "Document" module gives four alternatives: "Open", "Spare", "Realistic record" and "Exit". By picking "Open" choice, the client opens an info record, and the information characterizing a bend or surface are perused into the memory. The bend to be adjusted is then shown on the screen, and the client can begin the plan procedure. By picking "Spare" choice, the information characterizing the altered bend will be composed to a yield record. In the event that "Realistic document" choice is picked, the picture of the bend will be composed to a DXF arrange information record, and this record can be foreign made into AutoCAD. Thusly the picture of a bend can be hard duplicated to a printer or a plotter. The "View" module gives clients adaptability in review a bend under plan. At the point when the client opens an information document, at first, the bend is constantly fitted to the screen. The client can zoom in, zoom out and change sees whenever by choosing the comparing choices. This will enable the client to settle on the correct choice in the outline to process. The "Develop" module is utilized to rotate or expel bends into surfaces. By utilizing this choices, surfaces of transformation and surfaces of expulsion can be helpfully made from a cross-segment bend, and surface outline is disentangled to bend plan. The operations for controlling bends are given in the "Control" module. By picking "Include requirement", the client applies a limitation on a bend under plan. In bend plan, the adjusted state of a bend is instantly gotten and the screen is revived to demonstrate the new bend as the client moves a point on a bend to another area. For NURBS bends, the "Weight" alternative is accessible to change their shapes. The client initially picks a requirement on a bend and afterward moves
the mouse in the upper right or lower left bearings. This outcomes in an expansion or decline for the heaviness of a chose point.An immediate control conspire for B-spline, and NURBS bends utilizing geometric limitations has been displayed. The twisting energies of these parametric bends are resolved and limited in light of the limited component strategy. Two deformable models are produced. One is a direct model for B-spline bends and the other is a non-straight model for NURBS bends. Point of view mapping is utilized to linearize the NURBS detailing. In view of a homogenous arrange portrayal, a 3D NURBS bend is changed into a 4D B-spline bend. The straight deformable model produced for a B-spline bend is then changed and utilized for a NURBS bend. By modifying the weights of the limitations, a huge assortment of bends can be created which fulfill the predetermined requirements. The deformable models utilized as a part of the present plan relates bends to genuine protests, and relegates them physical properties. A planner can without much of a stretch anticipate the state of the changed bend by envisioning what the state of a genuine protest will be in the event that it is subjected to similar limitations. In the immediate control plot, there is no compelling reason to control the individual degrees of opportunity of bends, for example, control focuses areas and weights and the outline procedure is more natural. B-splines and NURBS (i.e., Non-Uniform Rational B Splines) were once in a while specified in a run of the mill illustrations course 10 years prior. As of late, as the customer advertise was overwhelmed with superb illustrations frameworks that all help NURBS (e.g., 3D Studio Max, Light wave 3D, Maya and true Space) and even after many books on B-splines and NURBS have been distributed, designs reading material courses still don't cover these points well. Numerous course books pick a numerical approach that regularly obscures the starting point and natural significance of NURBS. Course readings utilizing the programming approach once in a while furnish understudies with adequate data for how to draw and utilize NURBS and don't supply a domain for understudies to envision essential properties and calculations and to hone bend and surface outline. To enable understudies to learn geometric preparing aptitudes that are crucial to illustrations, representation and geometric plan, we made a lesser level elective course Introduction to Computing with Geometry and built up an academic device Design Mentor Version 2 or DM2.
III. Geometric Study
We generally begin our talk with a test: requesting that understudies draw a circle utilizing a B-spline bend. This is unimaginable and fills in as a decent inspiration for ensuing dialogs. Figure 1 indicates four B-spline bends of degree 2, 3, 5 and 7 characterized by 8 control focuses. Indeed, even with degree 7, the B-spline bend still does not resemble a circle. Henceforth, we have to discover a technique that can make circles effectively, and this is the value of examining
(a)degree 2 (b)degree 3 (c)degree 5 (d)degree 7
Figure 1. B-splines Can Not Represent Circles
B-splines and NURBS (i.e., Non-Uniform Rational B Splines) were once in a while specified in a run of the mill designs course 10 years prior. As of late, as the shopper advertise was overflowed with fantastic illustrations frameworks that all help NURBS (e.g., 3D Studio Max, Light wave 3D, Maya and true Space) and even after many books on B-splines and NURBS have been distributed, designs reading material courses still don't cover these subjects well. Numerous course readings pick a numerical approach that frequently obscures the source and instinctive importance of NURBS. Reading material utilizing the programming approach once in a while give understudies adequate data for how to draw and utilize NURBS and don't supply a situation for understudies to envision imperative properties and calculations and to hone bend and surface plan. To enable understudies to learn geometric preparing abilities that are fundamental to designs, representation and Many understudies were shocked by the way that the effective B-splines can't be utilized to speak to circles. Without a doubt the unit circle can be spoken to in an alternate shape, x = 2t/(1 + t2) and y = (1 - t2)/(1 + t2), which is an outcome examined in math. Notwithstanding, this parametric shape is objective (i.e., the remainder of two polynomials) as opposed to polynomial. The resulting discourse is mostly to find a reasonable frame and examining its properties. geometric outline, we made a lesser level elective course Introduction to Computing with Geometry and built up an instructive device Design Mentor Version 2 or DM2. We generally begin our talk with a test: requesting that understudies draw a circle utilizing a B-spline bend. This is outlandish and fills in as a decent inspiration for resulting talks. Figure 1 indicates four B-spline bends of degree 2, 3, 5 and 7 characterized by 8 control focuses. Indeed, even with degree 7, the B-spline bend still does not resemble a circle. Subsequently, we have to discover a technique that can make circles effectively, and this is the value of talking about and utilizing NURBS. Visualization:
To enable understudies to comprehend and picture the "lifting" and "projection" ideas, a perception framework NURBSvis is incorporated into DM2 appropriation. NURBSvis is a remain solitary framework and can be utilized without DM2's help. Be that as it may, since it is hard to show 4D objects, NURBSvis lifts a set of 2D control focuses to 3D, develops a 3D B-spline bend, and tasks it back to a 2D NURBS bend. NURBSvis has two windows: the 2D NURBS Curve window and the 3D B-Spline Projection window. A client makes a NURBS bend in the 2D NURBS Curve window with right-snaps to include control focuses (Figure 3(a)). At first, each control point has weight 1 and the bend is a B-spline. A client chooses a control point with a left-snap and uses left-drag to change its position. The vertical slide is for bend following. The lower-right corner has two catches to zoom in and out the 3D B-Spline Projection window, and a catch to kill on and the show of the matrix in the 2D NURBS Curve window.
(a) (b)
Figure 3. Windows of NURBSvis
The 3D B-Spline Projection window demonstrates the connection between 3D B-spline bend and 2D NURBS bend (Figure 3(b)). Since at first the made bend is a B-spline, it is indistinguishable to the projection NURBS bend in the w = 1 plane. This window underpins trackball sort turn for a client to see the connection obviously and effectively. The heaviness of a chose control point can be changed utilizing the slide in the lower-left corner of the 2D NURBS Curve window, and the new weight is appeared over the slide. In the event that the heaviness of a control point isn't 1, the bend turns into a NURBS bend. As the weight changes, the 3D BSpline Projection window demonstrates the comparing point (xw,yw,w) moving into space. The space bend in red is a 3D B-spline, and its projection 2D NURBS bend in w = 1 is in blue. There are lines associating control focuses in w = 1 furthermore, their relating 3D focuses, and there is likewise a line between C(u) (i.e., the point on the NURBS bend) and Cw(u) (i.e., the point on the 3D B-spline bend). De Boor's calculation is a standout amongst the most vital calculations in B-splines think about [1]. It takes a u 6 [0,1] and figures the comparing point on a B-spline bend. Since C(u) is the projection of Cw(u), a use of de Boor's calculation to the 4D B-spline yields Cw(u), and the projection of all calculation ventures to w = 1 gives de Boor's calculation for the NURBS bend. Figure 4 demonstrates this calculation and the de Boor net. Thusly, a client will have the capacity to imagine the connection between the B-spline form and the NURBS variant of de Boor's calculation. We found that this "verification without-words" approach is very compelling in clarifying the de Boor's calculation for NURBS. Bunch addition and bend subdivision for NURBS can likewise be talked about a
similar
way.
neighborhood adjustment property. The solid raised frame property of a B-spline bend of degree p expresses that the bend fragment on [ui,ui+1), lies in the curved structure characterized by p + 1 control focuses Pi-p, ..., Pi. This property gives a productive method for finding a bend portion and ensures that a chose bend fragment or the entire B-spline bend lies in an anticipated area. Since the 4D "lifted" B-spline bend fulfills the solid raised structure property and in light of the fact that focal projections protect convexity, the 3D NURBS bend additionally fulfills this property. The nearby change property expresses that the B-spline premise work Ni,p(u) is non-zero on [ui,ui+p+1). Since Ni,p(u) is the coefficient of Pi, if Pi changes, Ni,p(u)Pi likewise changes. Since Ni,p(u) is non-zero on [ui,ui+p+1), the difference in Ni,p(u)Pi just influences the section on [ui,ui+p+1) and does not influence bend fragments somewhere else. With this property, we realize that changing the position of a control point just influences a part of a B-spline bend and the adjustment is neighborhood. In this way, adjusting control point Pi of a NURBS bend C(u) causes the "lifted" control guide wiPhi toward change, which, thusly, changes the state of Cw(u) on [ui,ui+p+1). Since this bend fragment activities to the NURBS bend section of C(u) on [ui,ui+p+1), the neighborhood adjustment property holds for NURBS bends. Figure 5 demonstrates a NURBS bend of degree 4 characterized by control focuses P0, ..., P15. On the off chance that P10 is moved from its best position to its new position close to the base, just the bend fragment on [u10,u15), appeared in light shading, is changed. Bend portions at the two finishes are not influenced.
(a) (b)
Figure 4. De Boor's Algorithm
Important Properties:
After understudies have gained foundation in projection, extra vital properties are talked about. Actually, as long as a B-spline property isn't metric related, it likewise holds for NURBS on the grounds that a focal projection, which is relative, changes metric measure yet saves the relative connection (e.g., requesting and cross-proportion). Two properties that are critical to both B-splines and NURBS are examined: the solid arched structure property and
Figure 5. Modifying a Control Point
Modifying Weights:
Notwithstanding control focuses, bunches and degree, a NURBS bend has weights, one for each control point, and giving one more level of flexibility for shape plan. Truth be told, this basic expansion makes NURBS bends more intense than B-splines. In this way, the effect of altering the heaviness of a chose control point is an absolute necessity know property. Assume weight wk of control point Pk is to be changed. On the off chance that wk = 0, the term wiPk vanishes from the condition of the bend, and control point Pk has no commitment to the state of the bend. Consider the possibility that wk increments from 0 to boundlessness. Isolating the bend condition by wk yields: Plainly, as wk approaches endlessness, wi/wk approaches zero and the condition has a point of confinement Pk. Henceforth, as wk approaches limitlessness, the bend is "pulled" toward control point Pk and in the long run goes through it. Then again, as wk lessens to zero, the commitment of Pk additionally decreases and the bend is "pushed" far from Pk. In the long run, when wk diminishes to zero, control point Pk has no commitment to the state of the bend. Be that as it
may, which bend section will be influenced by this "pulling" and "pushing"? It can without much of a stretch be dissected with the projection idea. From the neighborhood alteration property of B-splines, adjusting wk changes wkPhk, which, thusly, changes the bend section of the 4D B-spline bend on [uk,uk+p+1). Subsequently, just the part of the NURBS bend on [uk,uk+p+1) changes.
With DM2, a client may choose a control point and change its weight. As the weight changes, the influenced bend portion of the NURBS bend pushes toward or far from the chose control point. Figure 6 demonstrates a NURBS bend with control point P5 chose. The bend fragment inverse to P5 is level when w5 = 0 on the grounds that P5 has no commitment. As w5 expands, the level bit draws nearer to P5. Figure 6 demonstrates the bend fragments comparing to w5 being 0, 0.1, 0.5, 1, 2, 4 and 10. At the point when w5 = 10, the bend is near P5. Besides, DM2 enables a weight to be negative with the goal that a client can see the effect of a negative weight. By and large, when the negative weight is adequately little, the solid curved structure property relating control focuses.
Figure 6. Modifying Weights
Conic sections:
Point 1 Curve
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Y 70.24 |
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(a)
(b)
coveted hover with focus at the starting point is appeared on-the-fly as the range changes (Figure 10(b)).
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Figure 9. Complete Circles
Cross sectional:
A few roundabout bends can be hung together to frame a NURBS portrayal of a circle. Figure 9 (a) demonstrates the engraved hover of an equilateral triangle. It is characterized by 7 control focuses P0, ..., P6 = P0 (n = 6). With the exception of P1, P3 and P5 that have weight , all other control focuses have weights 1. This NURBS bend of degree 2 has ties 0, 0, 0, 1/3, 1/3, 2/3, 2/3, 1, 1, 1. A circle can likewise be recorded in a square. The circle has four round bends as appeared in Figure 9(b). This NURBS hover of degree 2 is characterized by 9 control focuses P0, ..., P8 = P0 (n = 8). The weights of P1, P3, P5 and P7 are 2 and the weights of the remaining are 1. This bend has hitches 0, 0, 0, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 1, 1, 1. At the point when DM2 is made a request to create a circle, a little window shows up (Figure 10(a)) for a client to pick the equilateral triangle form or square form and utilize the slide to set a span. The
Why are circles vital? There are two reasons: (1) a circle is the easiest bend and (2) circles are utilized as often as possible (e.g., in producing surfaces of insurgency). DM2 underpins a unique surface plan system for creating generally utilized surfaces, the cross-sectional outline. In cross-sectional outline, a client determines a profile bend and a direction bend with the goal that the previous will take after the last to clear out a surface. The outcome, as a rule, is a NURBS surface albeit the two bends are when all is said in done B-splines. Assume we wish to plan a vase shape. The initial step is to plan a B-spline profile bend as appeared in Figure 11(a). Since this is a surface of upheaval, the direction bend is a circle. DM2 creates this direction circle naturally. When this hover, spoke to as a NURBS bend, is close by, the rotating procedure includes the assurance of all control focuses in light of the circle portrayal. Figure 11(b) demonstrates the wireframe adaptation of created vase in which the circles and their control focuses are unmistakably show, and Figure 11(c) demonstrates the rendered result. Circles may likewise be utilized as profile bends. A client may choose various circles with different size (Figure 12(a)) for the cross-sectional framework module of DM2 to process a NURBS surface that contains every one of them (Figure 12(b)). A surface that "introduces" an arrangement of bends is alluded to as a cleaned surface. Points of interest are given in figures underneath.
1) Figure 11. A Surface of Revolution
(a) (b)
Figure 12. A Skinned Surface
We have displayed our approach of instructing the basics of NURBS to college understudies in an elective course Introduction to Computing Geometry. In this course, we burn through two weeks on B-splines took after by one week on NURBS. Understudy responses in the previous seven years have been extremely positive. Understudies particularly like Design Mentor in light of the fact that it causes them comprehend the ideas and envision the calculations. In an expansiveness initially course, one may overview critical ideas and utilize DM2 and NURBSvis to exhibit the internal working of essential calculations and to rehearse bend and surface plan aptitudes. Preparatory course assessment comes about utilizing pre-and post-tests and mentality overview were distributed in [1, 2]. B-spline basic function.
iaï
(b)
Point constraint
to describe the miss happening imperativeness utilitarian of B-spline and NURBS twists. The constrained segment technique is used to restrict the mutilation imperativeness valuable and clarify for the distorted condition of curves subjected to goals. This approach realizes a game plan of direct conditions for a B-spline twist and a game plan of non-straight conditions for a NURBS twist. A perspective mapping is used to linearize the NURBS subtle elements. NURBS twists are first mapped from the 3D Cartesian mastermind space to the 4D homogeneous sort out space, and changed to 4D B-spline twists. After the control in the 4D homogeneous encourage space, the changed NURBS twists are then mapped back to the 3D Cartesian mastermind space. The approach is executed by a model program, which is made in C, and continues running under WINDOWS.
References
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[3] Au CK, Yuen MMF. Brought together way to deal with NURBS bend shape alteration. PC Aided Design 1995;27(2):85- 93.
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[5] Bartels R Beatty J. A strategy for the immediate control of spline bends. Procedures of the Graphics Interface 89. Los Altos, CA: Morgan Kaufmann. 1989. p. 33- 9.
[6] Fowler B, Bartels R. Imperative based bend control. PC Graphics and Applications 1993:43- 9.
[7] Hsu WM, Hughes JF, Kaufmann H. Coordinate control of freestyle disfigurements. PC Graphics 1992;26(2):177- 84.
[8] Rappoport A, Hel-Or Y, Werman M. Intuitive plan of smooth items with probabilistic point imperatives. ACM Transactions on Graphics 1994;13(2):156- 76.
[9] Terzopoulos D, Platt J, Barr A, Fleischer K. Flexibly deformable models. PC Graphics 1987;21(4):205- 14.
[10] Terzopoulos D, Fleischer K. Deformable models. Visual Computer 1988;4(6):306- 31.
[11] Celniker G, Gossard D. Deformable bend surface limited components with the expectation of complimentary frame shape plan. PC Graphics 1991;25(4):257- 66.
[12] Celniker G, Welch W. Straight limitations for deformable B-spline surfaces. Procedures of the Symposium on Interactive 3D Graphics. New York: ACM, 1992. p. 165- 70.
[13] Terzopoulos D, Qin H. Dynamic NURBS with geometric requirements for intuitive chiseling. ACM Transactions on Graphics 1994;13(2):103- 36.
[14] Faloutsos P, Van de Panne M, Terzopoulos D. Dynamic freestyle distortions for activity union. IEEE Transactions on Visualization and Computer Graphics 1997;3(3):201- 12.
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F
IV. Conclusion
This part presents a procedure for the prompt controls of B-spline and non-uniform perceiving B-splines (NURBS) twists using geometric goals. A deformable model is created
