The Dirty Little Secrets of NURBS

Stephen M. Hollister, New Wave Systems, Inc.

www.pilot3d.com

Introduction

There is nothing more flexible than a piece of paper, a pencil, and the human brain. Once you begin using a computer for shape design, you are forced into the limitations and idiosyncrasies of the program and its underlying mathematical geometry technique. In exchange, however, you get the following and more.

· Automatic matching of geometry in all views

· Automatic 2D view and section drawings

· Simple correction and modification of the design

· Accurate full-size templates/patterns

· Surface and volume calculations

· 3D Rendering

Many feel that because of these benefits and capabilities, there are few or no complications or difficulties in designing by computer. This could be blamed on something I call "the rendering effect." One look at a 3D rendered geometric model with all of the colors, shading, and all of the details defined in 3D, and it is hard to imagine anything a computer can't do. When I look at a picture like that, I think of the following:

· Is the geometry accurate enough for construction use?

· How long did it take to model the geometry? (Rendering is easy, modeling is difficult.)

· Is the geometry smooth and fair?

Renderings may look good in a brochure, but the 3D model used to generate the rendering may not be accurate enough for construction purposes. Most rendering programs hide unfairness in curved surfaces because they process the surfaces into triangles and the shading routines smooth the triangles to hide their edges.

This article discusses some of the flexibility you give up and some of the difficulties you encounter to take advantage of these benefits. These are the "dirty little secrets" that most designers learn about only after they buy and start to use a 3D curve and surface modeling program. This doesn’t mean that you should not use the computer for design. It means that designers should know more about the tools they use and understand what restrictions are placed on them by the program's user interface and what restrictions are placed on them by the underlying NURB geometry.

Surface Modeling

The field of geometric modeling is huge. Two-dimensional vs. three-dimensional geometry, wireframe vs. surface vs. solid modeling, relational vs. parametric vs. variational geometry, parametric vs. implicit equations, the list goes on. This article, however, will talk only about smooth curved surface design using NURB (Non-Uniform Rational B-spline) surfaces, since that is the dominant mathematical technique used by most curve, surface, and solid design programs, such as Autodesk's Mechanical Designer (all ACIS modelers), Intergraph's Solid Edge (all Parasolid modelers), and other independent programs, like New Wave System’s Pilot3D, McNeel’s Rhino3D, and Think3’s Solid Thinking. If you wish to transfer exact 3D curve and surface geometry to other CAD programs, you will have to use NURB surfaces (or a surface type which is mathematically compatible with NURBs). For example, there are numerous companies that will take your computer-generated 3D shape and numerically cut a full-size model out of various materials. These machines are controlled by one of several surface NC programs like SmartCAM and MasterCAM, using NURB surfaces. You will have to provide them with a file containing the NURB surface definition in a standard transfer file format, such as IGES (Initial Graphics Exchange Specification).

NURBs are the best techniques available for smooth curve and surface design, but you have to understand their capabilities to get the most out of your 3D modeling software.

What is a NURB?

NURBs are Non-Uniform, Rational, B-splines. They are equations used to define curves or surfaces that simulate the designer's batten in terms of stiffness and continuity. In the early 1970's, 'B' or Basis splines were created (There are an infinite number of splines.) as a convenient way to define smooth curves and surfaces interactively on the computer screen. Rather than have a curve pass through a set of points, like ducks on a batten, a B-spline creates a curve that is "influenced" by the positions of defining points called vertices (see Figure 1). The designer changes the shape of the curve by changing the positions of the defining vertex points. A curve influenced by B-spline vertex points is smoother than a curve passing through the defining vertex points, but the curve is by no means automatically fair. You can create some pretty ugly curves and surfaces with B-splines. In general, all flexible curve and surface modeling programs need some form of fairing tool.

Figure 1. NURB Curve Showing its Defining Vertex Points and Vertex Polyline

A NURB surface uniquely defines every square inch of a rectangular-like spline membrane. For editing purposes, it is usually represented on the screen by drawing a grid of lines on the surface called rows and columns. Like the NURB curve, the surface is defined by vertex points that do not lie on the surface, but are organized as a rectangular matrix or grid of vertex points.

Figure 2. NURB Surface Showing Defining Vertex Points and Vertex Grid or Mesh

To understand and use NURB surfaces effectively, you need to know what controls you have over their shape: vertices, degree, knots, and weights. Programs might hide this terminology from you, but you need to know how each of these variables affect the shape of a curve and surface.

For completeness, here is the parametric NURB surface equation:

P(i,j) Matrix of vertices: nrows=(k1+1) ncols=(k2+1)

W(i,j) Matrix of vertex weights: one per vertex point

b_i(s) Row-direction basis or blending polynomial functions of degree M1

b_j(t) Column-direction basis or blending polynomial functions of degree M2

s Parameter array of row-direction knots

t Parameter array of column-direction knots

Vertices - The basic shape of a NURB surface is defined by its rectangular grid of vertex or control points which do not necessarily lie on the surface. When you move one of the vertex points, the surface in that local area is pulled in the direction of the vertex movement as if it was attached to the point by a spring. (see Figure 3) The surface does not move as much as the vertex point and the change only affects a small area near the moved point. The range of influence depends on the degree of the surface: a smaller degree (see the next sub-section) means a smaller area of influence and a larger degree means a larger area of influence. The main drawbacks of these vertices are:

1. It is more difficult to get the surface to be positioned exactly where you want.

2. The vertex mesh hides the shape of the surface.

Figure 3. NURB Curve Example of Moving the Middle Vertex Point

Another approach is to use edit points that lie on the curve or surface. For every defining vertex point “floating in space” there is an associated point that lies on the curve or surface. Some programs allow you to control the shape of the geometry using these points instead of the vertex points. The two common ways to implement shape control using points that lie on the curve or surface are local control and global control. Local control using points that lie on the curve or surface means that when you move a point on the curve or surface, it is mathematically guaranteed that the change affects the shape only in a small area surrounding that point; the same as moving one vertex point. (see Figure 4) The problem is that when you move one of the points on the curve or surface using this technique, some of the surrounding defining points also move! This technique is the same as moving a vertex point, except that you are moving the associated point on the surface.

Figure 4. NURB Curve Example of Points on the Surface - LOCAL

Global curve or surface control means that when you move one of the defining points on the curve or surface, none of the other surrounding edit points move. This is called a full NURB curve or surface interpolation technique. (see Figure 5) As the selected defining point moves, the entire curve or surface shape is affected, although the shape change decreases dramatically as you move away from the changed point.

Figure 5. NURB Curve Exampe of Points on the Surface - GLOBAL

Although the curve or surface interpolation approach is global, it gives you the most accurate, direct control over the surface. This is because you control the shape directly and only the selected defining point moves. Some feel that this direct, interpolated control makes it difficult to smooth or fair the curve or surface. As you will see later, this is not a problem when you apply dynamic curvature feedback and fine tune shape control.

Vertices or surface points can be positioned anywhere in space and are the major shape control factor. For fairing and detailed shape control, you need to learn more about the following variables.

Degree - A NURB curve or surface is represented mathematically by piecewise polynomial blending functions of the form:

The highest exponent used in the polynomial (3 in this case) is the degree of the polynomial. (The degree is 3 and the order is 4.) NURB curves use one type of polynomial and NURB surfaces use two different polynomial types (one for each grid direction - row/column), although most programs use the same degree for both directions of the surface. The higher the degree of the polynomial, the more flexibility there is in the curve or surface, but the more uncontrollable wiggles it might have. Cubic, or degree 3 polynomials are considered to have the best trade-off in flexibility and stiffness. It is the degree used most often in CAD programs. Quadratic or degree 2 curves and surfaces will be stiffer and will be automatically more fair than cubic curves, but the range of shapes they can achieve will be much more limited. Many CAD programs fix the degree to always be 3.

Knots (Non-Uniform) - NURB curves and surfaces are polynomials pieced together using knots or a knot vector. Curves use one knot vector and surfaces use two, one for each grid direction. The knots define how the polynomial pieces are blended together with the proper smoothness. Most programs give users only indirect control over this shape factor. It is also the cause of many curve and surface shaping and fairing problems, especially if the surface shape is non-rectangular.

A uniform knot vector (with constant spacing between the knots) is the most common technique and works best for most situations. A program may automatically adjust the spacing of the knots in the knot vector (non-uniform spacing) to account for uneven spacing of the rows and columns of the surface, but this may not really help and may cause problems with adjoining surfaces. Some programs also use knots to define tension or knuckle control over the surface.

The problem with knots is that they have a semi-global effect on surface shape. If you adjust the knot spacing in the row direction, then the surface shape is changed for the full length of the surface, affecting several columns. In addition, if there is a surface attached to the end of the rows, then it must use the same knot spacing to maintain exactly the same edge shape. Otherwise, you have to make sure that the small differences in shape between the two surface edges are within building tolerances. This is a major problem with knots. Not only do they have a global affect on the surface, they can also force changes in adjoining surfaces! Many programs avoid these problems by fixing the knot spacing to be uniform for all surfaces. This is a reasonable solution, but it means that the user should create surfaces with evenly spaced rows and columns.

Figure 6. Difference Between Uniform (unfair) and Centripetal (smooth) Knot Spacing

The uniform knot spacing works best for evenly-spaced curve points or evenly-spaced surface rows and columns. If the spacing is not uniform, the alternate knot spacing, like chord-length or centripetal spacing works best. In figure 6, where the point spacing is un-even, the smoother curve is defined using centripetal knot spacing. However, non-uniform knot vectors for surfaces can cause bad fit problems between adjacent surfaces. This can be a real problem if you expect the surfaces to be transferred to solid modeling programs without gaps or “leaks”. That is why some programs stick with uniform knot spacing for NURB surfaces and recommend fairly even spacing for the rows and columns. Actually, nice spacing of the surface rows and columns should be the goal for all surfaces, no matter how the knots are spaced. This will make surface shaping and fairing easier. As long as you space the curve points and the surface rows and columns fairly evenly, then you shouldn’t have any problems.

Weights - For NURB curves and surfaces, each vertex point has an associated weight value that affects the shape locally near that point. This weight value is the result of the rational aspect of NURBs, which means that the equation is defined as a fraction or ratio of polynomials. The purposes of the weights are to allow for exact descriptions of conic shapes and to add more user control over the shape of the surface. Although each vertex point has its own weight and although each weight has a local effect on the surface, a change in a vertex weight (1.0 is the default weight) does not necessarily change the surface the way you want. In practice, changing a vertex weight value can cause more problems for curve or surface fairing. The best approach is to apply weight changes only near the end of the design process to achieve very specific local effects. Do not use weight control to create shapes that can be achieved by moving vertex points. Some CAD programs severely limit the user’s control over weight values because of these problems. Most curve and surface shapes can be defined and faired without changing the weight values.

Figure 7. Difference in Curve Shape Due to Weight Change

In figure 7, the weight factor of the middle point was changed from 1 to 10. No other changes were made to the vertex positions, the degree of the curve, or the knot vector definition. Note how the weight change flattens the curve on either side of the middle point and how it leaves a little curve “lump” right at the ends of the curve. Even though a weight change has a (somewhat) local effect, it tends to create unwanted flats and bumps.

Now that you have been introduced to the NURB shape controls (vertices, degree, knots, and weights), the next section will discuss many of the 3D surface design problems that you may encounter and some of their solutions.

Some of the more insightful readers might note that if a program uses uniform knot spacing and fixes the weights to 1.0, then this would be a simple uniform B-spline curve or surface. Then why does a 3D modeling program use NURBS? That is a good question and I can make a strong argument that NURBs are not necessary, even for conic shapes. I guess the reason is that NURBs form a superset of many different spline techniques and that if you wish to transfer surface geometry between programs, then it is a good solution.

While we’re at it, I might even say that NURB surfaces, because of their rectangular nature and the difficulty in smoothly connecting them together into complicated topologies, are seriously flawed. That is why there is so much interest lately in subdivision surfaces. These surfaces allow for very complicated topologies, like 3D polygon structures. Some subdivision techniques even incorporate C2 continuity or smoothness constraints. So far, however, subdivision techniques have only become popular for organic shapes that don’t have the same precision or smoothness needs as manufactured parts.

Dirty Little Secrets

These are some of problems you can expect to encounter when using one of the NURB surface design programs.

Rectangular Shape Spline Mesh

The NURB surface is a rectangular mesh of rows and columns with four hard corner points. The entire surface is continuous, except for these four points. You can stretch and distort the surface to any shape by repositioning the vertex points, but the more non-rectangular of a shape you make, the more difficult it will be to fair. It is the designer’s job to determine how to break down a full 3D model into a collection of rectangular-like pieces, how to deal with the four hard corner points, and how to maintain the proper continuity between surfaces. There is no one way of doing this.

For example, a full keel, wineglass shaped, sailboat hull is typical of one of the more difficult shapes to define. The hull is not rectangular and is difficult to model with one surface. Figure 8 shows how the J Boat Endeavor I was modeled by one NURB surface because the aft end of the keel made a smooth transition up to the transom. Even so, there are many subtle shape problems (bumps and wiggles) that have to be dealt with due to the extreme distortion of the one rectangular surface.

Figure 8. J Boat Endeavor I Modeled With One NURB Surface

Figure 9. Front and 3D View of One Non-Rectangular Surface

Figure 9 shows another example of one surface distorted into a very non-rectangular shape. It also shows the rows and columns of the surface and shows a number of planar contours of the surface. You can distort the NURB surface with good results, but I would look at the surface a little bit more where the contours change from a ‘V’ shape to a ‘U’ shape. This might indicate an area where the shape might not behave the way you want.

Be careful, however, since shallow planar cuts of surfaces always magnify surface unfairness. A change in the surface by less than the building tolerance might have a dramatic effect on the shape of these planar cuts. In the old days, when draftsmen had to draw the three different views of a surface by hand, they had to both fair the curves and to match up the dimensions in the three views. It is very difficult to do both at the same time. If they erred a little bit, it was always in the direction of making the curves look smooth and fair. If you measured and compared distances in the three views, you could always find small errors. (hopefully within building tolerances!) With modern NURB surface design programs, it is the other way around. The programs guarantee that all three views match up exactly and that all planar cuts are calculated or derived from the shape of the surface. This means that it will be difficult at times to make all of the planar surface cuts look perfectly fair, especially for very shallow surface cuts. Fair only to within building tolerances. It may be irritating to look at some of these curves, but you will drive yourself crazy trying to get them perfect.

Finally, if there are more than 4 distinct, hard points on a continuous, smooth surface, then you will probably have to break it into two or more connected surfaces and try to maintain curvature continuity.

Clutter of Vertex Mesh and Surface Shape

Since the defining vertex points do not lie on the NURB surface, most programs allow you to connect the vertices with straight lines to make them more visible. This grid of straight lines is often called a vertex mesh. If you do not connect the vertices with a mesh, then the vertices are floating in space and it becomes very difficult to know which part of the surface they influence, especially in certain views. If you display the vertex mesh, however, it clutters up the screen and hides the shape of the surface. For detailed surface shaping with many rows and columns, this is a real problem. (See Figure 10)

Figure 10. 3D View of Multiple Surfaces Showing Clutter of Vertex Lines

To avoid this problem, some programs use points that lie on the surface. This cleans up the clutter and makes the surfaces easier to see and manipulate. (See Figures 11 and 12)

Figure 11. Same 3D View Showing On-Surface Edit Points

Figure 12. End View Showing Edit Points on the Surfaces

Planar Surface Cuts Cannot be Controlled Directly

NURB surfaces are defined using a grid of row and column vertex points that can be moved anywhere. You cannot directly grab and alter the shape of any traditional planar line cut of the surface. These lines are considered to be derived lines or surface cuts which can be automatically calculated from the surface definition. Some programs, however allow you to display these lines while dragging a control point so that you can immediately see what happens when a vertex point is moved. To change the shape of a planar cut, for example, you have to move the vertex point that lies closest to the point on the cut you wish to change. This design by rows and columns is the biggest adjustment required by the experienced hand draftsmen.

Adding and Deleting Rows and Columns

One can start with a very simple NURB surface consisting of two rows and two columns, and then add in additional rows and columns to increase the flexibility and control of the surface. In doing so, the program must add the row or column in over the entire length or width of the surface. There is no allowance for partial rows or columns. This means that if you want to add rows and columns near one end of a surface to define some local detail, then these rows and columns must extend over the entire length and width of the surface, whether you want them to or not. Remember that the only way to add shape control is by adding in control vertices, and the only way to add in control vertices is by adding them in as a complete row or column.

Spacing of Rows and Columns

The smoothest surface shape depends on a relatively even spacing of the rows and columns. If there is more than a 2 to 1 ratio of consecutive row or column spacing, you will begin to see some very strange anomalies, such as hard bumps and loops. A program could adjust the knot vector to minimize the bumps and loops of uneven spacing, but this adjustment can cause other difficulties. For example, if two surfaces are to be joined exactly, edge-to-edge, then the two surfaces must have the exact same knot vector at that edge. If you have several surfaces attached together with common edges (a polysurface), then a change in the knot vector in one surface can affect the shape of all of the connected surfaces. One technique in dealing with attached surfaces is to use uniform knot spacing for all surfaces and always add in rows or columns by splitting the difference between two existing rows or columns. This will always maintain a reasonable 2 to 1 ratio in row and column spacing acceptable for uniform spacing.

Figure 13. Uneven Spacing of Columns

Figure 14. Added Column to Create Even Spacing

Piecing Together Surfaces

Rarely can you use one NURB surface to define a whole object. If two surfaces meet edge to edge, you need to understand the requirements of attaching two surfaces together. There are two types of attachment: exact and approximate. To form an exact attachment between two surfaces, the two surfaces must match up exactly along their entire length. The common surface edges must have the exact same edge vertex points, they must use the exact same polynomial degree at the edge, they must use the exact same knot vector at the edge, and they must use the exact same weight values at the edge. Otherwise, the mathematical definition and therefore, the shape of each surface at the edge will be different. If the two surfaces do not agree exactly at the common edge, then the attachment is considered to be approximate. It is then up to the user to manually make sure that the two approximate edges are within building tolerances of each other. If two surfaces match exactly and the user adds a column that runs into the attached surface, then the program must add a column to the attached surface or the two surfaces will not be joined exactly anymore.

Matching up Existing Designs

The original goal of B-splines was to create smooth curves and surfaces interactively on the computer screen. That is why they use vertices that do not lie on the curve or surface. But these floating vertices make it difficult to recreate shapes that currently exist. For example, if you have a set of measurement points (perhaps from a 3D digitizer) and wish to create a NURB surface representation of that shape, you will have difficulty trying to position the vertices to achieve that shape. Some programs allow you to digitize the sections of an existing object and to use that as an interactive guide for moving NURB surface vertex points. You move the vertex points until the surface section shape matches the digitized section shape. This is a very tedious process. A better approach is to automatically fit the NURB surfaces to the input data or the digitized sections so that the surface passes through the measurement points. You will still have to perform the final shaping and fairing, but the matching-up time will be greatly reduced.

For those interested in scanning in a 2D set of curves that define a 3D object, there is still quite a bit of work left to do. First, the scanner reads the whole drawing as a rectangular grid of dots (a bitmap). It doesn’t know anything about curves at this point – just dots. To convert the dots into curves, you can use one of two methods. The first is to use a raster to vector converter program to connect the dots to form polylines (not curves). Unfortunately, if the original curves overlapped each other or crossed each other, the raster to vector converter will not do a good job of creating one simple polyline entity per curve on the drawing. You will have to import the raw vector polylines into a CAD program and clean them up yourself. This could take some time. The other technique is to find a CAD program that can read the raw, scanned bitmap and display it on the screen. You would then define polylines or curves that overlay the bitmap curve images on the screen. People are better at raster to vector conversion than programs. Either way, however, you will have to spend some time getting the curves ready for skinning or lofting.

Rather than scanning a 2D drawing, you could use a digitizer tablet with a cross-hair puck or stylus. You would tape the drawing to the tablet and use a program to trace in each curve. This also takes time to set up the coordinate system and to trace in each curve. It is probably easier and faster to display the scanned bit map on the screen and to “digitize” the image on the screen. [Watch out for screen distortion of the bitmap.] A lot will depend on what devices and software you have. If you expect to do a lot of this “reverse engineering”, then it probably pays to research the different devices and software available to do the job.

In all cases, there is no free lunch. What you are trying to do is to go from less information (point or curve data) to a NURB surface definition, which is a complete, exact definition of the surface. There are an infinite number of solutions to this problem. It requires both software to do the surface fitting and a human to judge the results. The quality of the results depend on the amount of input shape data you have, the accuracy of the data, the organization of the data (curves, point cloud of data, etc.), and the ability of the software to organize and use this data. For more information about this subject, see the article on Reverse Engineering.

Detailed Surface Shaping

In the old, hand drafting days, you had many tools to create curved shapes: French curves, ship’s curves, templates, battens and ducks, and even freehand sketching. For designing one curve, you might use several, different curve templates. In addition, you might also have a number of battens of different lengths and flexibilities. Using a surface design program, you just have one tool: a NURB surface. A NURB surface is quite adaptable, but it really isn’t as flexible as the collection of original hand drafting tools. That is why it is very important for a surface design program to give users detailed shape control tools.

Some programs have difficulty with detailed surface shaping. They might have many ways to generate surfaces using lofting, extrusion, and sweeping commands, but few commands to control the resultant shape. There is no guarantee that if the shape and path curves of a sweep operation are perfectly fair, the resultant surface will be fair. Even if the surface is fair, you may still want to manipulate the resultant shape. This requires commands that allow you to move and fair the individual surface edit points. Surface evaluators, like Gaussian curvature are not the answer because the surface curvature cannot be displayed dynamically while you are moving the surface edit points. You also need some sort of fine tune shape control. Otherwise, you will have to zoom in very close to the edit point to get the resolution you need for detailed shaping. When you zoom in that close, however, you lose all sense of what you are shaping.

Some programs provide automatic fairing or smoothing tools, such as a command that equally respaces all of the points on a surface. This eliminates problems associated with the knot vectors and the uneven spacing of defining points. The resultant surface will be smoother, but it probably won’t be the shape you want. In old hand drafting days, designers were always told not to let the curves or batten define the shape of the object. Today, you must not let the 3D modeling “curves” or the program define the shape of the object. The joke nowadays is to ask who designed the shape of an object, the designer or the computer program. [i.e. Is that really the shape you wanted, or did you just give up?]

For complete control over your surfaces, you need detailed surface shaping tools. You need a way to eliminate the clutter of the vertex mesh, you need some form of dynamic curvature fairing feedback or display, and some sort of fine tune edit point move command.

Fairing and Smoothing Surfaces

In spite of what you may have heard, there is no mathematical definition of fairness. This does not mean that everyone would disagree on the fairness of a curve or that there are no mathematical tools to use to check for bumps and wiggles. It just means that there are no good ways to automatically fair a curve or surface by computer without some form of human input.

For example, one of the tricks in fairing a manual spline is to lift certain ducks for a moment to “relax” the batten. This smooths the curve, but it doesn’t necessarily maintain the shape that you want. With the computer, you have the same problem of getting the shape you want and making sure that the shape is fair. Automatic computer fairing techniques may smooth a surface, but they do not have the human judgment to make this shape vs. fairness trade-off.

Some programs say they guarantee fairness of a curve or surface, but this is only an opinion and is usually done by severely limiting the shape of the surface. For example, if someone gave you a batten and three ducks, it would be impossible to create a curve that was not considered to be fair. On the other hand, you would be severely limited in the range of shapes you could draw.

Most surface design and fairing programs provide ways to check the fairness of a surface by using some form of second derivative or curvature curve, or by using some form of surface curvature color mapping. Designers don’t need to know the mathematics behind these techniques. They just need to see how the technique magnifies all bumps, wiggles, and inflection points. Hopefully, the curvature information is displayed in real time while the user edits the shape of the surface. The designer interactively changes the shape of the surface to obtain a smooth derivative or curvature curve. As you can see, the designer is actively involved with the fairing process.

Figure 15. NURB Surface Curve With Unfair Overlaid Curvature Curve

Note that the points where the orange curvature curve crosses the surface curve are inflection points on the curve. The goal of fairing is to move the defining curve edit points to make the curvature curve smooth. If you make the curvature curve smooth, then the underlying curve is very smooth.

Figure 16. NURB Surface Curve With Faired Overlaid Curvature Curve

The top surface curve in figure 16 was faired to within .002 units over a length of 15.0 units. This means that the user can accurately see changes in the curvature curve when the surface curve is moved by 0.002 units. To do this, the program must provide some form of fine tune shape control without having to zoom in on a portion of the screen. For details of this specific fairing process, see the tutorial on fairing NURB curves and surfaces.

The problem with fairing NURBs is that the surfaces are so sensitive to the knot values. If a program uses uniform knot spacing, then the NURB surfaces need to have fairly evenly spaced rows and columns. When the knot spacing does not match the spacing of the rows and columns, you may have fairing problems. When a NURB surface is stretched into a very non-rectangular shape, even non-uniformly spaced knots do not help. In addition, non-uniform knots cause other problems, as mentioned before.

Another problem is that you cannot control the shape of the planar cut lines directly. You can only change the shape of the rows and columns of the surface. This means that it will be difficult in some cases to make shallow planar cuts of surfaces look just right. The slightest change in the shape of the surface will cause a noticeable change in the shape of the planar cut line. During the fairing process, you must fair only to within the building tolerance and not worry too much about how perfect the planar lines cuts look.

Vertex Weight Values

Although weights have a local influence on the shape of the surface, manipulating their values can be difficult. A common approach is to try to obtain the desired surface shape by only adding and manipulating rows and columns of vertices. After you have roughed out the surface shape, you can then change the weight values to achieve specific local effects. Most objects, however, can be defined without changing the vertex weight values. Only experienced users should try manipulating these values.

Relational vs. Trimmed Surfaces

There has been some discussion between two ways of attaching or connecting surfaces: trimmed surfaces vs. relational or attached surfaces. For trimmed surfaces, one surface is pushed through another and the program automatically trims off the overlap. An example of this is to push the root end of a wing through the side of an airplane fuselage. For relational or attached surfaces, the root edge of the wing surface is attached to the fuselage surface of the airplane. If the wing surface shape is changed, it will always stay attached to the fuselage surface, and if the fuselage surface changes shape, the end of the wing will stay attached.

The obvious advantage of attaching one surface to another is that you do not have to worry about whether the surfaces intersect if one or the other surface changes shape. The downside is that if the attached-to surface changes shape, then the attached surface changes shape. For the wing and fuselage surface attachment example, if the fuselage surface changes shape near the wing, then the attached wing surface changes shape! For airfoil-shaped wings, you do not want a change in fuselage shape to change the shape of the airfoil. For trimmed surfaces, you would just have to make sure that the wing root end penetrates the side of the fuselage surface by a little bit. The shape of the wing is independent of the shape of the fuselage. You can “pick up” the wing, shift its position, and re-penetrate the fuselage without changing its shape.

There are benefits and disadvantages to both approaches. Attachment or relational geometry works best when the two surfaces or geometry shapes relate and are dependent on one another, such as when you attach a structural rib to the side of a surface. Trimmed surfaces work best when the two surfaces must be joined, but their shapes are independent of each other. Trimmed NURB surfaces also work best if you need to convert the surface model to industry standard solid modelers. Relational attaching of one surface to another can leave very small gaps between the surfaces that are very difficult to deal with in solid modelers.

Calculations

Most of the specialized surface design programs use or allow open surface design geometry. This means that the programs do not requires a complete, closed solid geometry at all times. Solid modeling has some advantages when it comes to calculations, but it makes the process of modeling surface geometry much more difficult, especially in the initial concept design phase. This means that the surface design software has to do a lot of work to calculate the volumetric properties of an open surface model.

The first step is to cut all of the surfaces at a series of cross-section positions along the length of the object. The next step is to somehow combine these surface cuts into realistic, trimmed and closed cross-section shapes. This can be very difficult for intersecting and open surfaces, and for surfaces that do not quite match up exactly along common edges. Finally, some form of longitudinal integration can be performed to produce the final results. For all of these programs, the key to obtaining accurate results is to verify the shapes of the cross-section cuts before the calculations are done. If the cuts look good (and you have enough of them covering the length of the model), then the results should be accurate. Be aware that many problems can arise during the cross-section cutting and reconstruction phase.

To avoid this problem, the program could require that you always create a closed and trimmed surface model before the calculations are done. Then the program could triangularize the model and use solid modeling volume calculations techniques. The downside to this method is that it can be difficult to create the surfaces to close the object. This can be quite a distraction during the early stages of modeling an object.

Figure 17. Automatically Trimmed and Closed Cross-Sections Used For Volume Calculations

Figure 18. 3D View of Cross-Sections Used For Volume Calculations

Conclusion

NURBs are the dominant mathematical technique used for designing curves and surfaces of manufactured objects. In spite of this, there are many things you need to know to get the most out of your surface design or CAD program. This article discussed many of these hidden problems. Since all NURB-based programs vary greatly, the best advice is to get the demos of all of the surface design programs and spend some time with them trying to design various objects. The user interface of a program can have as much of an impact on your ability to get the job done as which mathematical technique is used to define the object. Don’t wait until after you buy the program to learn the truth.