DEPARTMENT OF TECHNOLOGY

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1 STUDIES FROM THE DEPARTMENT OF TECHNOLOGY AT ÖREBRO UNIVERSITY ISSN Implementation of NURBS curves into the VARKON CAD system Sören Larsson, Johan Kjellander No:8 July 14, 2003 Örebro universitet Institutionen för teknik Örebro Örebro University Department of technology SE Örebro, Sweden

2 Abstract This paper describes the implementation of NURBS-curves into the VARKON CAD-system. NURBS is an acronym for Non Uniform Rational B-Spline and is widely used in Computer Aided Design (CAD) to represent both curves and surfaces. The software sources of the VARKON CAD system is released under GNU/GPL licence, and is now maintained by the CAD-research group at Örebro University. The document gives a background to the area of curves in Computer Aided Design, different curve types, their main properties etc. This introduction is followed by a detailed explanation of the theory behind NURBS and the implementation in VARKON. During the implementation an effort has been made to benefit from the structure of previously implemented curve types. This made it possible to reduce the need of new code mainly to the evaluation routines. The possibility to make geometrical analyzes on a curve is an important part of a CAD-system. The theories and implementations of derivatives, normals and curvature are explained here as well as the calculation of an analytical offset curve. The document can serve as an introduction to curves in Computer Aided Design as well as an example of what kind of problem one has to deal with when implementing geometric entities in a CAD-system.

3 CONTENTS 1 Contents 1 Introduction 3 2 The use of curves Early curve modeling techniques Curves in Computer Aided Design Polynomial curve Offset curve Intersect curve Silhouette curve Geodesic curve Polynomial curve by approximation Planar curves Non planar curves Parametric curve representation 8 4 Polynomial curves Conic sections Cubic polynomial segments Higher degree polynomial curves Rational polynomials Splines Bezier curves B-splines and NURBS Curve continuity The VARKON CAD system Background The use of VARKON Curves in VARKON Data structure Creation and evaluation Introduction to NURBS Degree and order The knot vector Weights Summary Implementation in VARKON Data structure Evaluation of positions Definition of B-spline The evaluation steps Rational curves Implementation Calculation of basis values Derivatives

4 CONTENTS Theory Implementation of the first derivative Implementation of the second derivative Curve Geometry Curvature Tangents and normals Offset curve Position in offset First derivative in offset Second derivative in offset List of figures 36 References 37 Appendix A: GEevalnc.c 38 Appendix B: Manual page 49

5 1 INTRODUCTION 3 1 Introduction In mechanical engineering we design and manufacture real world objects. To do that we need to model their geometry so that we can simulate their behavior and control the machines needed to manufacture them. Geometric modeling has always been of great importance in mechanical engineering, even in the old times when everything was done manually. With the introduction of computers geometric modeling became easier and more accurate. In recent years Computer Aided Design has radically increased the use of geometric models. We are now approaching a point when it will be very difficult to do any design work at all without a CAD-system. It should be remembered though that most of the mathematics on which todays tools are based was developed a long time ago and in use by engineers long before computers existed. The line and the circle are probably the most common shapes in geometric modeling. Try to imagine though a world built with straight lines and circles only as the basic shape elements. It is rather obvious that we need more complex shapes to describe the world we live in. Think of the spiral curve that forms the shell of a snail or the helix curve that is used by the DNA molecule. Think of the shape of a car or a ship or an aeroplane, or think of the shape of a nice looking art object. In fact, the straight line and the circular arc are relatively uncommon shapes in real life. This paper is concerned with the modeling of curves that are not straight lines or circles. There are many such curves with specific names like the spiral or the helix but in modern geometric modeling we also use curves like conics and splines and offsets and many others. From a users point of view these curves have different geometric properties and require different kind of input and understanding. From the systems implementers point of view they usually have different mathematical representation and may need different data structures and algorithms to be implemented. Since we are primarily interested with the problem of implementing curve modeling functions in a CAD system we would of course like to find one single mathematical representation general enough to cover all curve types, ie. a mathematical model that would be capable to exactly represent a helix or a spiral or an ellipse or an offset or an intersect or a spline etc. Unfortunately this is not possible but a curve representation which covers many of the basic curve types and is commonly in use is the NURBS curve. When the VARKON system was originally developed in NURBS curves were not commonly used in commercial software and the NURBS representation was therefore not implemented but as NURBS have become more and more popular we see a need for NURBS also in VARKON. Örebro university is currently the maintainer of the GNU/GPL sources of the VARKON system and it was therefore decided to implement support for NURBS curves in VARKON, mainly as an exercise. This paper describes the background and the actual implementation done by Phd student Sören Larsson with help from his supervisor Dr. JohanKjellander during 2002.

6 2 THE USE OF CURVES 4 2 The use of curves 2.1 Early curve modeling techniques Ship design is an early application of mechanical engineering, at least some thousand years old. In those days drawings were not common, most of the modeling was done in one mans head and transferred to the next generation by the spoken word only. Still, many ships were built and all of them used the curve as its main shape element. It was only a few hundred years ago that engineers started to use paper drawings and mathematical methods to describe the geometry of their products. A ships hull was often described by a longitudinal curve representing the keel or backbone (spine) of the hull along with a set of curves in the transverse direction representing the frames or sections (ribs) of the hull. See figure 1. Figure 1: Keel curve with sections. The actual construction of a hull was quite similar. First you build the backbone of the hull and then you set up the frames like ribs on the backbone. Finally you add planking outside the frames to form the actual 3-dimensional surface of the hull. The backbone and frames are planar curves. See figure 2. With the introduction of paper drawings the shape of a hull could be described by simply drawing these curves on paper, usually not in full scale of course. To build the hull from the drawings require that you somehow transfer the shape from the paper to the construction material (usually wood). One way to do this is to measure the coordinates of points on the drawing, scale them to full size and plot them on the construction material. If you transfer a large number of points for each curve you can then connect them into a full scale curve with reasonable accuracy, at least after some training. It was now that tools like the duck and the spline was invented. This process required a flat surface large enough to fit the actual size of the part to shape. The loft floor of the building was an ideal place and so the process was named lofting. The technique developed for ships was easy to adopt to new products as they were invented. First automobiles and then aeroplanes were modeled and built using the same lofting technique. With mass production and the introduction of new manufacturing processes however the low accuracy of manual lofting slowly became a problem. In the middle of the 1900:s the first electronically controlled plotting machines came to use. These could draw curves in full scale with very high accuracy if fed with accurate data. So, the paper drawing was replaced by the mathematical model. Instead of drawing a section of the hull on paper a loftsman would now define the section using the exact mathematical representation of the curve to be used. He would

7 2 THE USE OF CURVES 5 Figure 2: Backbone with frames (upside down). then calculate points on the curve using a mechanical calculating machine and send these points to the electronic drawing machine for full size plotting. This was a big step forward as far as accuracy is concerned. On the other hand it required loftsmen with good knowledge in the mathematics of geometry. The invention of the computer made life much easier for loftsmen. Very soon they started to develop software for mathematical modeling of curves and surfaces. At the same time the first electronically controlled manufacturing machines came to use and the time was now ready to let the lofting programs control the manufacturing processes directly. The aeroplane and automobile industry were pioneers in this development. Lofting programs developed by specialists for specialists slowly became easier and easier to use. With the introduction of interactive computer graphics some time around 1970 it became possible to develop software that could be used even by non specialists, at least for limited applications. This development is still going on and today we see large international companies compete in the development of the most powerful and easy to use software for Computer Aided Design. Geometric modeling has become a multi billion dollar industry! 2.2 Curves in Computer Aided Design Modern CAD-systems usually include a number of different ways to create and manipulate curves of several different types. Curves are also very often created indirectly by the system as the result of more complex modeling operations Polynomial curve This is the obvious way to create a curve. Depending on what kind of polynomial curve you want to create you will have to define a number of data that controls the shape of the curve.

8 2 THE USE OF CURVES 6 With an ellipse for example, this could be the coordinates of the center of the ellipse and it s minor and major radius. With a spline it could be a number of points, optionally with tangents. Splines, Bezier curves and NURBS are all polynomial curves. Even conics, circles or circular arcs can be represented by polynomials if these are permitted to become rational, ie. one polynomial divided by another polynomial. Chapter 4 describes polynomial curves more in detail Offset curve An offset curve is a curve that is defined by the size of its offset distance to another curve. Offset curves are very important when modeling objects of constant thickness and to generate tool paths for numerically controlled machines. Offset curves are best represented by their offset distance and a reference to the original curve. Chapter 7.5 describes the implementation of offset curves in VARKON Intersect curve Intersect curves are curves defined by the intersection of two surfaces. Intersect curves can be defined directly by intersecting one surface with another or indirectly as the result of an operation between two solid bodies or a solid and a surface. Intersect curves can be used to define some interesting part of an object but are also often used to define a part of a surface to remove by trimming. Intersect curves are usually computed as a set of points along the curve. For practical purposes these points are then often interpolated (splined) to a continuous curve and stored in the database as such Silhouette curve This curve is defined by the visible edge of a surface as viewed from a specific direction. Computation and storage is similar to the intersect curve Geodesic curve This curve represents the shortest possible path between two points on a surface. Computation and storage is similar to the intersect curve Polynomial curve by approximation Due to the different mathematical representation of the curves described above they are not always compatible with each other or with surface geometry. This may result in situations where the system will not let you use a curve to define a new curve or surface the way you might have expected. In one system for example it may not be possible to create an offset curve if the original curve is not a polynomial curve. In other systems it may not be possible to define a surface if the limiting curves are not polynomial. To overcome this you can usually define a polynomial curve by approximating a curve which is not polynomial and use the approximate curve where the non polynomial curve did not work. Some systems do this automatically. It should be noted though that uncontrolled use of this functionality may impose problems with accuracy in later stages of the design work.

9 2 THE USE OF CURVES Planar curves Most curves can be planar. This means that they lie entirely in one plane and can be described in terms of 2-dimensional coordinates X and Y only. Some curves, like conics are planar by definition but most curves need not be planar in all cases Non planar curves Many curves need not be planar. The spline for example may very well twist out of the plane and become truly 3-dimensional. The general conic defined by a third degree rational polynomial may also be non planar.

10 3 PARAMETRIC CURVE REPRESENTATION 8 3 Parametric curve representation The first decision to make when implementing a new curve type into a CAD system is its mathematical representation. This will influence the data structures needed to store the curve in the CAD database and it will also influence the way to implement all the routines needed to use and to manipulate the curve. The most important routine is the one used to evaluate coordinates and derivatives of different points on the curve. This routine is used by all higher level routines needed for visualization as well as large numbers of geometric operations. It is of great advantage from the implementors viewpoint if all curves in the CAD system have a similar representation. This makes it possible to treat all curves in a similar manner, to a large extent using the same code. To achieve this a two layer implementation is needed where the low layer is curve dependant and the higher level is the same for all curves. A mathematical representation which has showed to be very useful in this context is the parametric representation. A parametric curve is defined by a parametric equation P (u) where P (u) = (x(u), y(u), z(u)) (3.1) and the parameter u is defined in some interval, 0 to 1 for example. Polynomial curves are easy to represent parameterically. A cubic polynomial curve for example has the following parametric representation x(u) = k 0x + k 1x u + k 2x u 2 + k 3x u 3 (3.2) y(u) = k 0y + k 1y u + k 2y u 2 + k 3y u 3 (3.3) z(u) = k 0z + k 1z u + k 2z u 2 + k 3z u 3 (3.4) or with vector notation P (u) = k 0 + k 1 u + k 2 u 2 + k 3 u 3 (3.5) where k i are 12 scalar constants that define the shape of the curve. Evaluating the coordinates for a point on the curve is straight forward and computationally very fast. It is also easy to evaluate the curve tangent P and higher order derivatives used by higher level routines. Offset curves, intersect curves and other curves are also possible to represent parameterically. With the parametric representation as the common factor the high level routines may not need to know the exact equations or algorithms that implement each type of curve. This is taken care of by low level routines. The important thing here is that the low level routines are relatively few. The high level routines are numerous and often very complex. When VARKON was developed it was decided to use this technique, mainly because it would make it possible to add new curve types at a later stage with a minimum of programming. First

11 3 PARAMETRIC CURVE REPRESENTATION 9 version of VARKON included support for cubic polynomial and offset curves but not NURBS curves. Some years later the intersect and silhouette curve was added. Since NURBS curves are relatively easy to represent parameterically it was then decided, more than 15 years after the original design of the system, to add NURBS curves to VARKON. We then expected only to have to implement routines on the lower level and reuse all code on the higher level. This turned out to be almost true, but to get everything working we had to modify one or two routines on the higher level as well. Still, this is a good result that shows that the initial design of the system is well justified.

12 4 POLYNOMIAL CURVES 10 4 Polynomial curves Different polynomial curve types have different mathematical representation as well as different geometrical properties. This chapter briefly describes the most common polynomial curve types used in geometric modeling. 4.1 Conic sections A conic section is a curve defined by the intersect between a plane and a cone. Depending on the orientation of the plane the section will become a circle, an ellipse, a parabola or a hyperbola. See figure 3. pictures/conics Figure 3: Different conic sections, from left to right: circle, ellipse, hyperbola, parabola Conic sections are planar curves without inflexion and thus well suited as section curves in lofting. Many ships and aeroplanes have been designed with conic sections defining their hulls. A common parametric representation for conic sections is the rational quadratic polynomial, see below. 4.2 Cubic polynomial segments The cubic parametric polynomial is defined by 12 coefficients, see equations (3.1) - (3.5). If we prescribe the x, y and z coordinates for the two points P0 and P1 where the curve starts and ends we still have 6 degrees of freedom left to control the shape of the curve. One way to do this is by supplying a start tangent T0 and end tangent T1 each with x, y and z coordinates, see figure 4. If we use these 4 boundary conditions to solve equation (3.5) for the constants k i we get the following form of the cubic polynomial P (u) = P 0(1 3u 2 + 2u 3 ) + P 1(3u 2 2u 3 ) + T 0(u 2u 2 u 3 ) + T 1(u 3 u 2 ) (4.1) which is sometimes referred to as the geometrical form of equation (3.5). It can be shown [1] that the cubic polynomial curve as defined above is a good approximation of an elastic beam

13 4 POLYNOMIAL CURVES 11 Figure 4: Cubic parametric curve segment under load. For an engineer this is an important property of the cubic polynomial. It is easier to work with a curve that behaves similar to some well known phenomena of the real world than a curve which is a pure mathematical construction. A cubic polynomial as defined by equation (3.5) or (4.1) is often referred to as a cubic segment. The reason for this is that several cubic polynomials (segments) are usually needed in order to define a complex shape. The spline for example is a composite curve defined by two or more cubic segments. Cubic segments differ from conic sections in that they may have at most one inflexion point. This means that locally curvature may be zero. See figure 5. Figure 5: Cubic parametric curve segment with inflexion 4.3 Higher degree polynomial curves An alternative to the composite curve defined by two or more cubic segments is to use only one segment but raise the degree of the polynomial to a higher value than 3. The CAD system CATIA for example uses this technique and allows for polynomials all the way up to degree 21. It is commonly not recommended though to use polynomials of high degree due to the risk for numerical instability in calculations. 4.4 Rational polynomials A rational polynomial is defined by one polynomial divided by another polynomial. Rational polynomials of second degree (quadratics) can be used to represent conic sections. A common

14 4 POLYNOMIAL CURVES 12 way to simplify the expression of rational segments is in terms of 4-dimensional homogenous coordinates, ie. P = (xw, yw, zw, w) where w is the 4:th coordinate. With this notation the rational segment becomes similar to the non rational segment with the difference that all operations include 4 coordinates instead of three. A 3-dimensional real world coordinate is derived from the 4-dimensional homogenous coordinates by dividing xw, yw, and zw respectively with w. x = xw w (4.2) y = yw w (4.3) z = zw w (4.4) For details on rational quadratics, rational cubics and homogenous coordinates see Faux and Pratt [2] chapters 3 and Splines A spline curve is a curve composed of two or more segments where each segment is defined by its own polynomial. To create a spline the user specifies three or more points for the spline to interpolate. The CAD system then automatically sets up the right equations to calculate the tangents at all points and then stores the spline in the database as a number of consecutive polynomial segments. The exact way to calculate tangents can differ between systems. VARKON includes three algorithms, Ferguson, Chordlength and Stiffness that give slightly different results. It is also sometimes possible for the user to control the shape of a spline by supplying a geometrical constraint at each end of the spline. Splines are easy to create when you have a number of points you want to connect with a smooth curve. If cubic segments are used a spline is a good approximation of the old wooden spline used for lofting. See Kjellander [1] for more details on cubic splines. 4.6 Bezier curves The Bezier curve is a curve that is defined by a number of points but it only interpolates the first and last point. The rest of the points are control points that influence the shape of the curve but the curve does not pass through them as the spline does. The Bezier curve is an approximating curve with a lot of smoothness and well suited for sketching. A Bezier curve is always described by one single polynomial even if it is defined by a large number of points. The polynomial is produced by a sum of terms. See Kjellander [3] chapter 4 or Faux and Pratt [2] chapter 5. For each new control point a new term with higher degree is added. A special situation arises if two control points have the same coordinates. Such a point is said to have multiplicity and it has a stronger effect on the curve than normal points, ie. the curve passes closer to this point than other points.

15 4 POLYNOMIAL CURVES B-splines and NURBS The B-spline reminds of the Bezier curve. It is an approximating curve and it is created and used in a similar way as the Bezier but it is not based on a single polynomial of high degree. The B-spline is more like the ordinary spline in this sense, ie. composed of separate pieces each described by its own polynomial. An effect of this is that if you move a B-spline control point only the portion of the curve close to the control point will alter its shape. This is often referred to as the local control property of the B-spline. If you move a Bezier control point the entire Bezier curve will change. As stated above, the control points of a B-spline are normally not passed by the curve. It is possible though, to calculate node points on the curve and it is also possible from node points on the curve to calculate the corresponding control points. This makes it possible to use the B-spline either as an interpolating curve or as an approximating curve. The user can specify points to interpolate and the system can calculate and store the corresponding control points instead. If the user needs sketching capabilities, he can use the control points directly instead. This makes the B-spline formulation for curves very flexible. B-splines can be rational just like other polynomials and can then be used to represent also conic sections. The capability of the B-spline to be used as an approximating or interpolating curve and to represent conic sections as well as cubic splines or polynomials of higher degree has made it popular from the implementors point of view. Many CAD systems have the rational B-splines implemented and they are then usually called NURBS. Chapter 6 and 7 include more details about B-splines and NURBS. 4.8 Curve continuity Depending on the type of curve used and how it it created it can have a higher or lower degree of continuity, ie. C0, C1, C2 etc. C0 continuity means that the curve is continuous with no gaps but not necessarily continuous with respect to its first derivative (tangent). A C0 curve may thus have sharp kinks. A C1 curve has continuous first derivative and a C2 curve is also continuous in its second derivative. C2 also implies continuous variation of curvature. For higher degree polynomials C3 and more may also apply. Single polynomial segments are usually C2 continuous or higher but a curve defined by two or more segments like the spline (and even the B-spline) is not necessarily C2 or even C1 continuous. This depends on the algorithm used to create the spline. The Ferguson spline in VARKON is C2 continuous but the Chordlength and Stiffness splines are only C1 continuous. NURBS may or may not be C2 or more but are usually at least C1 although even C0 is possible by the use of multiplicity. In some systems composite curves can be created by assembling segments of different types. A curve could then start out as a B-spline then turn into a high degree polynomial and finally end as a Bezier or even an offset or intersect. The total degree of continuity for such a curve depends on the lowest degree of continuity of the individual pieces and how they are connected.

16 5 THE VARKON CAD SYSTEM 14 5 The VARKON CAD system 5.1 Background VARKON was originally developed by a group at the University of Linkoping in Sweden during under the leadership of Dr. Johan Kjellander who was then the president of Microform AB. From 1986 the system was owned, marketed and further developed by Microform AB. The source code of VARKON is now maintained by the CAD research group at the Department of Technology at Örebro University in Sweden. VARKON is written in ANSI portable C and has been compiled and successfully executed on many different platforms. The UNIX version has a user interface based on X-Windows and the PC version uses Microsoft WIN The use of VARKON VARKON is a powerful geometric modeller. Basic 3D entities are points, lines, arcs, curves, surfaces, coordinate systems and transformations. Several representations of parametric curves are implemented including rational polynomial, analytical offset and curves on surfaces so called UV-curves. Surface representations include rational polynomial, lofted procedural, analytical offset and a faceted surface for approximations. Operations include intersects, closest point, silhouette, curvatures, transformation, trimming, export, import and approximation. Though VARKON can be used as a traditional CAD-system with drafting, modelling and visualization, the real power of the system is in parametric modelling and CAD applications development. VARKON includes interactive parametric modelling in 2D or 3D but also the MBS programming language integrated in the graphical environment. More information about VARKON and MBS can be found at VARKON s home page: Curves in VARKON VARKON uses two different representation schemes for curves. The first is the rational cubic parametric multisegment. This representation is used for splines and conics and 3D circular arcs. A spline is created using positions and optional tangents in the following way... cur_spline(id,type,pos_1,tan_1,pos_2,tan_2... pos_n, tan_n); There is no upper limit for the number of points but a minimum of two is always needed. Where no tangents are given VARKON calculates (splines) tangents automatically using the algorithm specified by the type parameter. You can choose between Ferguson, Chordlength and Variable Stiffness. End tangents if not specified are calculated using the free end constraint (curvature = 0). A multi segment conic is created using positions, tangents and intermediate points or P-values (I/P)... cur_conic(id, "FREE", pos_1, tan_1, I/P_1, pos_2, tan_2, I/P_2... pos_n, tan_n);

17 5 THE VARKON CAD SYSTEM 15 A cur conic becomes a true planar quadric if possible otherwise a general twisted cubic rational. For those who wish to define cubic rationals using other methods or to import them from other systems there is also a low level routine cur usrdef where you set all coefficients yourself. For curves on surfaces VARKON uses a representation based on multi segmented cubic rationals in the UV-plane of the surface. An intersect between two surfaces for example... cur_int(id,surface_id_1,surface_id_2); creates a UV-curve. Other methods are cur iso for curves of constant U or V and cur sil for silhouette curves. The UV-representation is computationally much heavier than the cubic rational but necessary to achieve accuracy enough for complex modelling. The final representation is the analytical offset which is actually an extension of the two schemes described above. Any plane curve, cubic rational, UV-curve or offset-curve can be used to create a new curve on a constant offset using... cur\_offs(id,other\_curve,offset); VARKON treats offset curves as true analytical offsets to the original curves by first evaluating the original curve and then adding the desired offset. If the original curve is a UV-curve this is a very heavy computation but the only way to act if you want highest possible accuracy. If you want to change an offset curve or a UV-curve into an ordinary cubic you can do this using cur approx. You will then have to supply a maximum value for the error of approximation that you can tolerate. 5.4 Data structure All curves in VARKON is built from segments of one of the types: 1. Rational cubic segment. Also the uv-curves is of this type (they are 2-dimensional rational cubic segments on the parameter plane). 2. Offset segment. In VARKON the offset curve is evaluated by first evaluating the desired point on the original curve and thereafter adding the offset. I.e. no approximating curve is generated to represent the offset curve. Therefore the original curve + the offset value is enough to define the offset curve. What is mentioned above implies that all curve segments could have a common data structure. We need basically only elements for the coefficients (4 x 4), an optional offset value, and a pointer to the next segment. The implementation of the data structure (before adding the NURBS curves) is listed below. There is some user defined data-types, but the names of them and and the comments should explain them sufficiently. The different segments in a curve is connected via a linked list by means of the pointer in nxt_seg. For each curve there is also a header with some common information about the whole curve, and a pointer to the first segment in the list. The structure is visualized in figure 6 and the type definition in C-code is found here.

18 5 THE VARKON CAD SYSTEM 16 Curve segment: typedef struct DBfloat c0x,c0y,c0z,c0; Segment-coefficients. DBfloat c1x,c1y,c1z,c1; UV_SEG only use 12. DBfloat c2x,c2y,c2z,c2; DBfloat c3x,c3y,c3z,c3; DBptr nxt_seg; GM-Pointer to next segment DBfloat ofs; Optional offset DBfloat sl; Arclength in R3 DBshort typ; Segment type CUB_SEGU or UV_SEG DBshort subtyp; Sub type DBptr spek_gm; Optional GM-pointer to surface DBptr spek2_gm; Optional GM-pointer to surface GMSEG; Curve header: typedef struct GMRECH hed_cu; Header DBshort fnt_cu; Font DBfloat lgt_cu; Dash length DBfloat al_cu; Arclength in R3 DBshort ns_cu; Number of geometric segments DBTmat csy_cu; Optional curve plane tbool plank_cu; TRUE if curve is planar DBptr pcsy_cu; GM-pointer to coordinate system DBshort nsgr_cu; Number of graphical segments DBfloat wdt_cu; Linewidth GMCUR; \bigskip pictures/databas Curve header Segment data Segment data Curve #1 No off segments=2 Segment #1 Coefficients (16 pce) Segment #2 Coefficients (16 pce) First segment Next segment NULL Figure 6: The database structure for curves in VARKON. 5.5 Creation and evaluation What is important to understand is the difference between the creation of an entity, and the evaluation of it. When an entity (e.g. a curve) is created the system checks the input from the user, perform suitable calculations to get the segment coefficients and stores the entity in the data base. If the input is wrong, the system returns suitable error message. So far no calculation of positions etc. is actually needed. The system do not know the positions in 3D space on the curve unless they are evaluated.

19 5 THE VARKON CAD SYSTEM 17 When the curve is evaluated the system retrieves the segment data from the database and calls suitable functions to evaluate position, tangent, normal or whatever is needed. In this stage the system do not know how the curve segment was created (Ferguson spline, conic section etc.). An evaluation of a position on the curve could be initiated by the user with a MBS-statement such as on(#10,0.3) or tang(#10,0.3). It could also be initiated by the system when for example painting the curve on the screen.

20 6 INTRODUCTION TO NURBS 18 6 Introduction to NURBS NURBS is an acronym for Non Uniform Rational B-Spline. Those words shall be deeper explained further ahead in this text. For the time being we can state that the NURBS curve is a curve, widely used in Computer Aided Design (CAD). It has the ability to exact represent most of the common more specialized curve types e.g. B-spline and bezier-curves, circular arcs, conics and straight lines. The NURBS curve can be used in arbitrary dimension, the definition remains the same. Sketches and initial explanations in this text is for the sake of simplicity made in two dimensions. As we shall discover in... an extension to three dimensions is straight forward. In this section NURBS-curves are presented from a users point of view. I.e., we are discussing how the different parameters that defines the curve will affect the shape of the curve. The exact procedure for the evaluation of he curve, and its mathematical definition will be further explained in section 7 Se also the NURBS book [4], in which detailed descriptions of various implementation tasks is explained. pictures/nurbs_1 Figure 7: A NURBS curve and its corresponding control polygon. A NURBS curve is briefly defined by the user by means of so called control points. These points can be regarded as vertices in a polygon, the control polygon. The curve does not necessarily interpolate the points (which is the case for an ordinary spline-curve). Normally the NURBS-curve is given such parameters that actually makes it interpolate the first an last point. The intermediate points operate as magnets on the curve, see figure 7. The control points will from here be designated P i, the number of control points will be designated n. Other parameters that control the shape of the curve is: the degree oft the curve, the weights of the control points and the so called knot vector. Those designations shall be deeper explained below. 6.1 Degree and order Although a NURBS curve is not evaluated in the same way as a polynomial curve, we can still talk about the the degree of a NURBS curve. The higher degree the smother curve we get. The lowest degree, 1, gives straight lines between the control points. Degree 2 gives an approximating curve that will touch and share tangent with the control polygon between the

21 6 INTRODUCTION TO NURBS 19 control points. Higher degrees gives more smooth curves. As an alternative, we can talk about the order of the curve. The order is always the degree+1. We will from here be designating the degree of the curve: p (power). pictures/nurbs_deg p=1 p=2 p=5 p=3 From the polygon it- Figure 8: NURBS curves with same control polygon, but different degree. self;degree=1, to the smoothest curve, degree=5. If a control point is moved, it will not affect the whole curve, but a limited region around that control point. How much of the curve that will bee affected depends on the degree. A lower degree give more local control (only part near the point is moved). This property of B-spline curves is called the local control property. Maximal allowed degree for a NURBS curve is p = number of control points +1. Such a curve is equivalent to a Bezier curve with the same degree and same control points. Moving a control point in a Bezier curve will affect the whole curve, i.e. the local control property is lost when the NURBS degree is increased to its maximal value. pictures/loc_prop a b Figure 9: The local control property of NURBS curves. The figure shows an order 4 NURBS curve with 12 control points. Moving one control point only affects the curve between a and b. 6.2 The knot vector The knot vector is a part of the definition for a NURBS-curve. To fully understand the meaning of the knot vector, it may be easiest to look at the implementation of the evaluation routines in section 7.2. The knotvector affects the parametrization of the curve, and also the shape of it. The knotvector is a one-dimensional array of float values.

22 6 INTRODUCTION TO NURBS 20 The knot vector is built up from n+p+1 values. A normal knotvector for a NURBS curve of degree 3, with 6 control points could bee: 0, 0, 0, 0, 1, 2, 3, 3, 3, 3. The first and the last knot value is repeated p+1 times. The steps between the other knot values are uniform. A B-spline with these properties is said to be uniform. The repeated values at both ends make the curve interpolate the start and end control points. It is actually the size-relations between the intervals in the knot vector that affects the shape of the curve. The knotvector 0, 0, 0, 0, 1 3, 2 3, 1, 1, 1, 1 and the one given above gives identical curves. In some CAD systems the knotvector is always generated in this way, since then the whole curve will be parameterized from 0.0 to 1.0. The knotvector could also have irregular intervals, such as 0, 0, 0, 0, 1, 2.5, 3, 3, 3, 3, we then have a non uniform knot vector and hence a Non Uniform B-spline. The part of a NURBS curve that is defined by means of a parameter interval between two knot values i.e. 1.0 < u < 2.0 is called a span. Inside a span the set of control points that affect the curve is intact. When the parameter enter next span, the curve lose contact with one control point and begin to get influenced by a new. 6.3 Weights Each control point could, beside its cartesian coordinates, also have a weight. The weight is normally 1.0, but a higher weight for any point will move the curve closer to that point. A curve with one or more weights unequal is rational. The weight is evaluated by means of so called homogenous coordinates. This is not unique for the NURBS curve, the method is used for other rational curves as well. What happens is that the 3 dimensions is mapped to a 4-dimensional space, the evaluation is carried out as for a non-rational curve and thereafter the result is mapped back to 3D. Rational segments is needed to be able to represent circular arcs exactly. e:sln urbs_artikelpictures urbs_wei w=5.0 w=1.0 w=1.0 w=1.0 w=1.0 w=1.0 Figure 10: The effect of weights 6.4 Summary We have introduced the properties of the NURBS-curve from a users point of view. The curve is defined by following parameters: A set of n control points,p i in 2D or 3D, forming the control polygon of the curve.

23 6 INTRODUCTION TO NURBS 21 A weight, w bound to each control point. If all weights are equal, the curve is nonrational, else it is rational. The degree,p of the curve. As an alternative we use the order of the curve. Order = p + 1. A knotvector, an array of n + p + 1 float values. A uniform knotvector has equal distance between knot values.

24 7 IMPLEMENTATION IN VARKON 22 7 Implementation in VARKON 7.1 Data structure As described in section 5.4, all curve segments in VARKON share the same data structure. It was desirable to keep it that way during the implementation of NURBS curves. If this could be achieved a lot of functions would not need to be changed. The reason for this is that these functions do not know what type of segments they are operating on. The key is to handle each NURBS span as a curve segment, and to let the parametrization point out what segment (span) we are working on. This is the same as for ordinary multisegment polynomial curves in VARKON, where parameter values between 0 and 1 implies first segment, parameter values between 1 and 2 implies second segment etc. The implementation of NURBS in VARKON follows this rule of parametrization regardless of the values in the knotvector, but the shape of the curve is still maintained the correct for a NURBS curve. pictures/databas1 Curve header Curve #1 No off segments=2 Segment data Segment #1 Coefficients First segment Control points (4D) Control points p1 p2 p3 p4 p5 knots k1=0.0 k2=0.0 k3=0.0 k4=0.0 k5=1.0 k6=2.0 k7=2.0 k8=2.0 k9=2.0 No off control pts=5 Offset in c-pts=1 Knotvector No off knots=9 Offset in knotvector=5 NURBS order=4 Next segment Segment data Segment #2 Coefficients Control points (4D) No off control pts=5 Offset in c-pts=0 Knotvector No off knots=9 Offset in knotvector=4 NURBS order=4 NULL Figure 11: The database structure for NURBS segments, note that the space for coefficients (tot 16 pce) are not used for NURBS curves. All segments (spans) in the NURBS curve is given access to the knotvector and the control points, but they only use some limited part of them. Therefore the knotvector and the control points is stored only once for each curve. Each curve segment points to these arrays, but do also store offset values for their own position in the arrays. To handle this some data elements has been added to the data structure of a curve segment, see code below (compare to section 5.4).

25 7 IMPLEMENTATION IN VARKON 23 typedef struct DBfloat c0x,c0y,c0z,c0; Segment-coefficients. DBfloat c1x,c1y,c1z,c1; UV_SEG only use 12. DBfloat c2x,c2y,c2z,c2; NURB_SEG use none DBfloat c3x,c3y,c3z,c3; DBptr nxt_seg; GM-Pointer to next segment DBfloat ofs; Optional offset DBfloat sl; Arclength in R3 DBshort typ; Segment type CUB_SEGU or UV_SEG DBshort subtyp; Sub type DBptr spek_gm; Optional GM-pointer to surface DBptr spek2_gm; Optional GM-pointer to surface DBptr cpts_db; DB-pointer to 4D controlpoints DBHvector *cpts_c; C-pointer to 4D controlpoints DBint ncpts; Number of cpts DBint offset_cpts; Offset for this span DBptr knots_db; DB-pointer to knotvector DBfloat *knots_c; C-pointer to knotvector DBint nknots; Number of knots DBint offset_knots; Offset for this span DBshort nurbs_degree; Degree (order-1) for NURB span GMSEG; This change of the data structure for curve segments made it necessary to adjust the database functions for storing and reading them. These changes where made, but is no further described here since our main concern in this paper is the implementation of the evaluation routines for NURBS segments which will be described in forthcoming sections. 7.2 Evaluation of positions The theory of NURBS will be presented here and in following sections together with corresponding algorithms and our implementation. The code-snips is cut-outs from the file geevalnc.c. A printout of the whole file is found in appendix A. The evaluation of a point could for example be initiated by the user via an on() -statement in MBS, or it could be one of the sample points needed to draw the curve on the screen. The calling function demands what to be evaluated e.g: position, 1:st derivative, 2:nd derivative, curvature etc. This is done by means of the global variable (structure) p_evalc ->evltyp Definition of B-spline In this section the definition and theory of NURBS will be presented. Following designations will be used: p P i w i n u i N i,p u - the degree of the curve - the control points - the weights - number of control points - the knot vector - base value number i for degree p - the parameter value to be evaluated We shall initially cover non rational B-splines. When we are familiar with its definitions and algorithms, it will be straight forward to let our theories also comprehend the rational curves. More reading about B-splines can be found in several books, see for example references [3],[4], [5] and [6].

26 7 IMPLEMENTATION IN VARKON 24 The important thing is to understand the fundamentals of B-splines, the step from there to the more general NURBS-curves is not far. A B-spline curve of degree p is defined as: C(u) = n N i,p (u)p i a u b (7.1) i=0 where P i is the control points, n is the number of control points and N i,p is the basis values for degree p. There exist one basis value for each control point (index i). The basis functions sometimes is denominated blending functions. The basis values is derived from the knot vector by means of the basis functions. Basis functions for degree 0 is defined as: N i,0 (u) = 1 if ui u < u i+1 0 in other cases (7.2) Basis function for higher degrees is defined recursively as: N i,p (u) = u u i N i,p 1 (u) + u i+p+1 u N i+1,p 1 (u) (7.3) u i+p u i u i+p+1 u i+1 where u is actual parameter on the curve, u i is the knot value with index i. Hence we must always calculate the basis values of all degrees up to and including the degree of the curve. In the equation (7.1) above, it is however only the basis values of degree p that is used. The other basis values are needed indirectly by the recursively definition. To calculate the basis values you need the degree of the curve, actual parameter u, and the knot vector. The control points does not affect the basis values. The sum of all basis values for a certain parameter, u is always 1. Some basis values will, for a certain parameter value, be zero. This means that corresponding control point shall not influence the curve. According to the definition above it then arises a division 0/0. The result of this division is defined to be zero. In practice it is possible to know in forehand where this will happen. Hence this division need never to be done The evaluation steps As input to the evaluation you need to perform following steps: 1. Determine in what span of the curve you are working. 2. Calculate the basis values for that span. (the other basis values are zero and need not to be calculated). 3. Loop through the control points and sum up the influence of each point on the curve (only points in actual span need to be summarized).

27 7 IMPLEMENTATION IN VARKON 25 I the NURBS book[4] these steps corresponds to the algorithms A2.1, A2.3 and A3.1 respectively. As we shall see in section 7.2.5, our implementation will be slightly different since step 1 is already performed by the system when the evaluation reaches our function. Furthermore the basis values for position and if needed, for the derivatives, is calculated at the same time, in the same algorithm, therefore the exact algorithm will be presented in section 7.3 together with the derivatives Rational curves Evaluation of a point is done in the same way for each of the components x,y and z respectively. The same basis values i used, the only difference is which component x, y or z that is summarized. Rational curves is evaluated by means of Homogeneous coordinates. This means that each control point in 3D is transformed to a 4-dimensional space by means of the weight, w. The 4 components x,y,z and w is then each of them evaluated in exactly the same way as if we had an non rational 2D or 3D curve as described before. When the evaluation is made the position on the curve is transformed back from homogeneous coordinates back to 3D space. This is done by dividing x,y and z with the calculated w-coordinate. This technique is used for other curve types as well, it is not unique for NURBS, see section Implementation Following code is a part of the function GEevalnc() in the file geevalnc.c. A printout of the whole file can be found in appendix A. The evaluation of a position on the curve can be divided into following steps. When the evaluation process reaches this function it has formerly pointed out the correct segment (span) of the NURBS curve. 1. Calculate basis values. Note that the calling function has already determined in what segment of the curve the actual point is situated. This is done earlier in the evaluation process in the same way regardless of the curve type. As the segments here corresponds to the NURBS spans (see section.??), and the correct offset in the knotvector was stored together with this actual span, we already know the correct offset in the knot vector offs_u. Compare with the NURBS book [4], where corresponding task is performed by algorithm A2.3 The function DersBasisFuns is explained in section *** Calculate basis values status=dersbasisfuns (offs_u,u,degree,no_der,p_seg->knots_c,ders); 2. Determine if the position is to be calculated. It is possible that the calling function has only ordered calculation of, let say the 1:st derivative. Then there is no need to calculate the position. The desired evaluation (position, derivatives etc.) is situated in the global variable p_evalc-> evltyp. *** Calculate position *** (always needed for a rational curve) if ((p_evalc ->evltyp & EVC_R) (ratseg == TRUE))