Trimmed Spline Surfaces with
Accurate Boundary Control

Trimmed NURBS surfaces are the standard for industrial surface modeling, today. This class of surfaces combines computational efficiency, geometric flexibility, and ease of use for the designer. All current CAD/CAM systems and various file exchange formats support this–and typically only this–method. While the advantages of trimmed NURBS are undisputed, there is one severe and persistent shortcoming that has not found a commonly accepted solution in decades: multi-patch arrangements of trimmed NURBS surfaces are typically discontinuous along common boundary curves of neighboring pieces. The problem is the following: Given a tensor product spline surface $\mathbf{s}:\mathbb{R}^{2}\to\mathbb{R}^{3}$, current CAD systems provide two ways of representing the boundaries of a trimmed NURBS surface. Either one can use a planar spline curve $\gamma:\mathbb{R}\to\mathbb{R}^{2}$ to bound the domain or one can use a spatial spline curve $\mathbf{c}:\mathbb{R}\to\mathbb{R}^{3}$ to bound the trace. The first variant leads to the boundary curve $\mathbf{s}\circ\gamma$ in three-space. This is a spline curve, again, but typically, it has high degree and many knots, stemming not only from the breaks of $\gamma$ but also from all points where the trace of $\gamma$ intersects a knot line of $\mathbf{s}$. The resulting knot structure is uneven and hardly predictable¹¹1 Basically, the only case which does not lead to exploding complexity is the special situation where $\gamma$ parametrizes a straight line parallel to one of the coordinate axes.. This fact makes it virtually impossible to match $\mathbf{s}\circ\gamma$ with any other surface boundary at reasonable expense. The second variant is equally discouraging. It seems to be impossible to find a surface $\mathbf{s}$ and a curve $\mathbf{c}$ such that the trace of $\mathbf{s}$ contains the trace of $\mathbf{c}$ exactly. The makeshift for this problem is to accept disjoint traces, and to project $\mathbf{c}$ onto $\mathbf{s}$ in order to determine the actual trimming curve. In general, however, this curve is no longer a spline and again not suitable for a watertight match with a neighboring patch.

To avoid the described problem, one can try to find arrangements of patches for the desired shape which do without trimming. This approach leads to methods like geometric continuity [Pet02], singular parametrization [Rei97], or subdivision [PR08], and also to a variety of constructions for $n$-sided patches [PGVACN06, VSK16]. As a matter of fact, none of these concepts has found widespread acceptance in industrial CAD/CAM applications. It may be speculated that a certain perseverance of designers and executives is responsible for that situation. However, there is also a more solid explanation. The flow of parametric lines is most important for the design process²²2This observation is a blatant contradiction to the concepts of differentiable geometry, which rely on the independence of geometric properties of the chosen parametrization.. On one hand, the parametric lines should follow the shape characteristics (e.g., ridges or curvature lines) of the desired surface to obtain convincing results. On the other hand, a ‘natural’ arrangement of control meshes greatly favors the intuitive character of interactive modeling techniques. Thus, there is little flexibility in the choice of an adequate parametrization, and the corresponding patches must be trimmed at places where neighbors meet. To sum up, the advantages of a well-adapted parametrization seem to be rated higher than the notorious trouble with the trimming issue.

In CAD/CAM applications, the usual solution to the problem of nonmatching trimming curves is to simply accept little gaps in surface representations, and to make them so small that they do not impair the quality of the product to be manufactured. This allows modeling surfaces of adequate quality, but the price to pay is a certain inconvenience of the design process. The challenge is to design patches that are not only fair by themselves but join with neighbors within a typically very small margin³³3Customary tolerances for traditional production techniques have to be decreased even further for modern 3d printing systems with their incredible attention to detail.. Further, not only positions must be matched, but possibly also surface normals and curvature tensors in the sense of an approximate $G^{1}$- or $G^{2}$-continuity of the composite surface. For high quality surface design, as requested for instance in the automotive industry, the only way to achieve this is a time-consuming manual process of iterated boundary adaption and fairing.

Besides these practical considerations, there is another issue with the current state of technology. Surface models with disconnected pieces are not well suited for simulation purposes. No matter how small the gaps are, they appear insurmountable⁴⁴4Advanced methods like discontinuous Galerkin may offer a remedy. for heat conduction, transmission of forces and bending moments, or other physical processes. For this reason, multi-patch CAD models are usually converted to watertight meshes by elaborate procedures to prepare them for simulation. Volumetric modeling techniques as requested for Isogeometric Analysis with its potential to save this effort are currently studied. However, they seem to be quite far from a breakthrough for freeform surface design in industrial applications.

From the abundance of material on the design of surfaces of arbitrary topology in general and trimmed NURBS in particular, we want to briefly mention only two recent approaches. In [MH17], a procedure for converting an assembly of trimmed NURBS patches to a watertight model is suggested. It is based on local remodeling such that trimming curves become parametric lines. Without doubt, this is a reasonable practical procedure, even though some issues with a special kind of singularities appearing there (cf. [MR]) and also general fairness aspects are still waiting for a detailed assessment and analysis. Another line of research that has to be discussed in this context is documented in [VSVS20, VVSS20]. It is based on ideas similar those presented here, but there are also significant differences. Before we compare the methods, we briefly explain the basic principles of our approach.

Let $\mathbf{b}$ be a spline surface, called base, and $\mathbf{r}_{1},\dots,\mathbf{r}_{L}$ be a sequence of other spline surfaces, called ribbons. Then we define the ABC-surface

with suitable reparametrizations $\kappa_{\ell}$ and weight functions $w$ and $w_{\ell}$. Figure 1 illustrates the setting. It is instructive, but not necessary, to think of $\mathbf{b}$ as a standard spline surface describing the overall shape to be designed. When trimmed, its boundaries are close to the desired behavior, but do not match exactly. The ribbons $\mathbf{r}_{\ell}$ are employed to adjust that deviation.

The domain $\Gamma\subset\mathbb{R}^{2}$ of $\mathbf{a}$ is assumed to be bounded by $L$ smooth segments $\Gamma_{1},\dots,\Gamma_{L}$ meeting at corners $\sigma_{1},\dots,\sigma_{L}$. The base $\mathbf{b}$ describes the intended shape globally on $\Gamma$, and each ribbon $\mathbf{r}_{\ell}$ describes that shape locally in a vicinity of the segment $\Gamma_{\ell}$ of the boundary. This is done in such a way that the spline curve $\mathbf{c}_{\ell}:=\mathbf{r}_{\ell}(\cdot,0)$ parametrizes the segment $\mathbf{a}(\Gamma_{\ell})$ of the boundary of the trace of $\mathbf{a}$. The reparametrizations $\kappa_{\ell}$ are used to reconcile the otherwise unrelated surface representations, and the weights are used for blending. When approaching the boundary, $w$ vanishes and makes the base disappear. Further, near the boundary segment $\Gamma_{j}\subset\mathbb{R}^{2}$ corresponding to $\mathbf{c}_{j}$ all weights $w_{\ell}$ except for $w_{j}$ vanish so that also all ribbons but $\mathbf{r}_{j}$ are faded out. Hence, this surface dominates the behavior of $\mathbf{a}$ near the boundary in a way that will be made precise below. The accurate boundary control of the surface $\mathbf{a}$ by means of ribbons explains the acronym ABC-surface.

The basic principle and some technical details of ABC-surfaces were first presented in [Rei]. Roughly at the same time and independent of our research, another group developed similar ideas [VVSS20], which were recently brought to our attention. The formal similarities are obvious, but there are also differences, and one of them is significant.

First, the weighted base $w\mathbf{b}$ does not appear in that form⁵⁵5It does appear in a similar form in [VSR12] for the special case of straight boundaries and polynomial ribbons. in [VVSS20]. Instead, a linear combination $\sum_{j}b_{j}\mathbf{p}_{j}$ of control points $\mathbf{p}_{j}\in\mathbb{R}^{3}$ and B-splines $b_{j}$ is used, where only B-splines with support completely contained in the domain of the surface are permitted so that there is no influence on the shape near the boundary. Accordingly, the term $w$ in the denominator is replaced by $\sum_{j}b_{j}$. This can be considered a minor technical detail, but it also reveals a slightly different, though equally valid philosophy behind the construction. We think of the base as a relatively accurate description of the entire surface, and of the ribbons as small, local correctives, used only to adapt $\mathbf{b}$ to the desired boundary behavior. By contrast, the ribbons in [VVSS20] are the major building blocks for the shape to be modeled. The summands $b_{j}\mathbf{p}_{j}$, if included at all, are only used for variations inside if the shape defined by the ribbons alone does not yet meet expectations.

Second, and this difference is much more important, the surfaces constructed in [VSVS20, VVSS20] are transfinite in the sense that they do not possess a closed-form representation by means of elementary functions. In particular, these surfaces are not piecewise polynomial or rational and thus cannot be stored in standard file formats, as requested in many applications. The point is that the curves $\gamma_{\ell}$ bounding the domain $\Gamma$ are defined explicitly as splines. Once this is done, there is no way to continue the construction within the spline framework. For instance, it is in general impossible to find splines $w_{\ell}$ that vanish on all but one of these boundary curves⁶⁶6 The special case of straight boundaries, which admits rational constructions, was discussed in [VSR12, SVR14, VSK16, VSK17].. In [VSVS20, VVSS20], it is suggested to construct reparametrizations and weights as solutions of variational problems with the curves $\gamma_{\ell}$ providing the boundary conditions. This requests a certain computational effort, but visually, the results appear to be excellent. By contrast, in our construction, the boundary curves $\gamma_{\ell}$ are defined implicitly as zero levelsets of spline functions. That way, the problem described above disappears. It may be objected that explicit evaluation of the boundary of the domain is now much more expensive than it was before, but as a matter of fact, this is hardly ever requested in practice as long as the spatial boundary curves $\mathbf{c}_{\ell}$ are available.

The paper is organized as follows: In the next section, we introduce and analyze the basic principle for the construction of ABC-surfaces in a very general setting. Three theorems concerning the behavior of positions, normals, and curvature tensors at the boundary are provided. They establish the validity of our construction, but also that of the methods presented in the more application-oriented papers [SVR14, VSVS20, VVSS20]. In Section 3, we specialize the principle to the spline setting and describe in detail how to construct adequate reparametrizations and weights from given bases and ribbons. Section 4 addresses some important technical issues related to the limitations of current standard formats for file exchange with respect to trimming. Finally, in Section 5, we present a few examples of ABC-spline surfaces to demonstrate the potential of our method.

Throughout, boldface letters denote points and functions with values in $\mathbb{R}^{3}$, while greek letters denote points and functions with values in $\mathbb{R}^{2}$. Except for the surface $\mathbf{a}$ to be constructed, all functions appearing below are assumed to be sufficiently smooth. In particular, their derivatives mentioned anywhere (explicitly or implicitly through notions like normals or curvatures) are assumed to exist and to be continuous. Further, to avoid a discussion of degenerate cases, surfaces are always assumed to be immersed, i.e., the partial derivatives are linearly independent everywhere. The following definitions are crucial for the construction of ABC-surfaces:

Now, we describe the construction of a surface $\mathbf{a}$, whose domain $\Gamma\subset\mathbb{R}^{2}$ of definition is bounded by a sequence $\gamma_{1},\dots,\gamma_{L}$ of planar trimming curves $\gamma_{\ell}:[0,1]\to\mathbb{R}^{2}$. When concatenated according to

these curves form a closed loop without self-intersections. Here and below, the index $\ell$ always runs from $1$ through $L$ and is understood modulo $L$. Let

denote the corners and boundary segments of $\Gamma$, respectively. Formally, the endpoint $\gamma_{\ell}(1)$ is missing in the definition of $\Gamma_{\ell}$ to obtain a disjoint union $\partial\Gamma=\bigcup_{\ell}\Gamma_{\ell}$ for the boundary of $\Gamma$. Accordingly, for each boundary point $\beta\in\partial\Gamma$, there exist unique values $\ell$ and $u\in[0,1)$ such $\beta=\gamma_{\ell}(u)$.

The building blocks for the construction of the surface $\mathbf{a}$ are the following:

• •

  Let $\kappa_{\ell}:=[p_{\ell},q_{\ell}]:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be functions that define trimming curves $\gamma_{\ell}$ in the sense of Definition 2.1 according to the setting described above.

• •

  Let $w:\mathbb{R}^{2}\to\mathbb{R}$ be an implicit representation of all trimming curves. That is,

  $$w\circ\gamma_{j}=0,\quad j=1,\dots,L.$$

  Further, it is assumed that $w$ is positive on the interior of $\Gamma$, see Figure 3 (left).

• •

  For each $\ell$, let $w_{\ell}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be an implicit representation of all trimming curves but $\gamma_{\ell}$. That is,

  $$w_{\ell}\circ\gamma_{j}=0,\quad j=1,\dots,\ell-1,\ell+1,\dots,L.$$

  Further, it is assumed that $w_{\ell}$ is non-negative on $\Gamma$, and that the only zeros of the sum $\sum_{\ell}w_{\ell}$ in $\Gamma$ are the corners, see Figure 3 (right).

• •

  Let $\mathbf{b}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ be a surface, called the base. Unit normal and curvature tensor of $\mathbf{b}$ are denoted by $\mathbf{n}_{\mathbf{b}}$ and $E_{\mathbf{b}}$, respectively.

• •

  For $\ell=1,\dots,L$, let $\mathbf{r}_{\ell}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ be surfaces, called ribbons. Unit normal and curvature tensor of $\mathbf{r}_{\ell}$ are denoted by $\mathbf{n}_{\ell}$ and $E_{\ell}$, respectively.

With these ingredients, we construct a new surface that combines base and ribbons in a specific way:

For the sake of simplicity, this definition is restricted to the case that the domain $\Gamma$ is simply connected and has always $L$ corners and boundary segments. Generalizations to multiply-connected domains and to disc-like surfaces bounded by one smooth, periodic curve will be discussed in a forthcoming report.

Two basic elements characterize the construction in the definition above. First, the functions $\kappa_{\ell}$, which define the trimming curves, are used as reparametrizations for the ribbons $\mathbf{r}_{\ell}$ to adapt them to the otherwise unrelated base $\mathbf{b}$. Second, near the boundary segment $\Gamma_{j}$ the various implicit representations $w$ and $w_{\ell}$ are used as weights to fade out the influence of the base $\mathbf{b}$ and all ribbons $\mathbf{r}_{\ell}$ except for $\mathbf{r}_{j}$. The ribbon $\mathbf{r}_{j}$ becomes dominant and determines the shape of $\mathbf{a}$. Thus, the construction provides accurate boundary control for $\mathbf{a}$, giving rise to the acronym ABC-surface. For a more detailed assessment of this raw idea, we need a further definition:

Further introducing the notation

for the reparametrized ribbons, $\mathbf{a}$ can be written in the form

Here and below, the variable $\sigma$ is always assumed to be fixed and mostly omitted in the notation for the sake of simplicity. Unit normal $\mathbf{\bar{n}}_{\ell}$ and curvature tensor $\bar{E}_{\ell}$ of $\mathbf{\bar{r}}_{\ell}$ coincide with the respective quantities of $\mathbf{r}_{\ell}$ since these surfaces are related by reparametrization,

In the interior of $\Gamma$, $\mathbf{a}$ is a blend of smooth surfaces and thus can be expected to be smooth, too. As always, exceptions may occur in case of an incidental loss of regularity. We will not discuss such degenerate cases, but focus on the behavior of $\mathbf{a}$ at the boundary. The following three theorems clarify the situation:

In the following proofs, it is convenient to use the abbreviations

to expand $\mathbf{a}$ in a vicinity of the corner $\sigma_{j}$. Since weights are assumed to be nonnegative, it is $h(\tau)\in[0,1]$. For contact order $k=r-1$, we obtain

In the following, $\partial_{1}f,\partial_{2}f$ denote the partial derivatives of the bivariate function $f:\mathbb{R}^{2}\to\mathbb{R}^{n}$, respectively. $Df:=[\partial_{1}f\ \partial_{2}f]:\mathbb{R}^{2}\to\mathbb{R}^{n\times 2}$ denotes the Jacobian.

We note that the regularity condition (7) is purely technical and typically satisfied. It rules out a degenerate situation, which can occur in a similar way anywhere in any common surface modeling system for an unfortune choice of control data.

The crucial conditions for the existence of normal vectors at the boundary are algebraic (proper decay of the functions $\bar{w}$ and $\bar{w}_{\ell}$) and geometric (consistency of ribbons and normals). Similar conditions appear in the following theorem concerning curvature tensors. However, and this may be unexpected, also a parametric condition (consistency of derivatives of ribbons) has to be met to guarantee existence of a curvature tensor at corners.

The proofs of the three theorems show that $G^{k}$-properties on the interior part of the boundary segments $(0<u<1)$ are easily achieved by requesting proper decay of the functions $\bar{w}$ and $\bar{w}_{\ell}$. Consistency and regularity conditions are requested only to guarantee good behavior at the corners. Remarkably, condition (9) concerning consistency of the derivatives of reparametrized ribbons shows that the functions $\kappa_{\ell}$ cannot be chosen independently of the ribbons $\mathbf{r}_{\ell}$. More precisely, the interrelation

must be observed. Assuming that the ribbons are given and the reparametrizations are still unknown, this equation yields a linear constraint on the pair $D\kappa_{j-1},D\kappa_{j}$ at the corner $\sigma_{j}$. Vice versa, if the reparametrizations are given, the space of admissible ribbons is restricted to a linear subspace.

The following example shows that this condition is in fact crucial. Let

Near the origin, the traces of both surfaces coincide so that consistency of point, normal, and curvature tensor is guaranteed. However,

The weight functions $w_{j-1}(x,y)=x^{3},w_{j}(x,y)=y^{3}$ have proper decay at the boundary curves $\gamma_{j-1}(u)=[u,0,0]$ and $\gamma_{j}(u)=[0,u,0]$. Assuming for simplicity that the base and all other ribbons vanish, we obtain

Skipping the technical details, we note that

and

However, the curvature tensor $E_{\mathbf{a}}$ is discontinuous at the origin. Figure 4 shows the surface $\mathbf{a}$ (left) and the Gaussian curvature $K(r\cos\varphi,r\sin\varphi)$ (right) as a function of $r$. The individual lines correspond to $30$ equally spaced angles $\varphi$ in the interval $[0,\pi/2]$. For $\varphi=0$ and $\varphi=\pi/2$, corresponding to the trimming curves $\gamma_{j-1}$ and $\gamma_{j}$, it is $\lim_{r\to 0}K(r,0)=\lim_{r=0}K(0,r)=0$, as suggested by the cylindrical shape of $\mathbf{\bar{r}}_{j-1}$ and $\mathbf{\bar{r}}_{j}$. However, other angles yield non-zero limits.

In the previous section, the ABC-principle was introduced in great generality. Our main goal, however, is to find a special embodiment of the setup that is fully compatible with the predominant standard of industrial surface modeling: trimmed NURBS.

To this end, we assume that all building blocks of the ABC-surface $\mathbf{a}$ are tensor product NURBS. Even more, for the sake of simplicity, we consider only purely polynomial, i.e., non-rational splines. Further, it is practical (but not necessary) to use a common set of knot vectors for the base $\mathbf{b}$, the weights $w,w_{\ell}$ and the reparametrizations $\kappa_{\ell}$; degrees may be different. By contrast, the knots of the ribbons $\mathbf{r}_{\ell}$ are completely independent. It is immediately clear that the concatenation, multiplication, summation and division of piecewise polynomial functions yields a piecewise rational function $\mathbf{a}$. Denoting the coordinate degree of the bivariate spline $s$ by $\operatorname{deg}s\in\mathbb{N}_{0}^{2}$ and its total degree by $|\operatorname{deg}s|\in\mathbb{N}_{0}$, we obtain

Together, the maximal degree of the rational pieces of $\mathbf{a}$ equals that of the corresponding numerators, and is given by

where the maximum is taken component-wise. More specific results will be presented later on.

The crucial question that has to be addressed now concerns the construction of suitable splines $w,w_{\ell}$, and $\kappa_{\ell}$. In recent works [VSVS20, VVSS20], reparametrizations for domains bounded by curves are obtained as solutions to variational problems with the given trimming curves $\gamma_{\ell}$ as boundary conditions. It can be expected that this yields good visual results, but of course, there does not exist a closed-form solution, and much less a spline fulfilling the condition $\kappa_{\ell}(\gamma(u))=[u,0]$. Similar problems arise when seeking weights vanishing at the boundary or parts thereof.

The solution of the problem to find weights and reparametrizations compatible with the boundary is amazingly simple: There is no need to define the boundary curves $\gamma_{\ell}$ explicitly. It suffices to define them implicitly by the functions $\kappa_{\ell}$ according to Definition 2.1,

This setting is justified by the fact that standard CAD systems and also standard file exchange formats like STEP or IGES allow trimming curves to be specified explicitly as a closed loop of spline curves in the parametric domain, or alternatively–and this is the option used here–as a closed loop of spline curves in $\mathbb{R}^{3}$ bounding the trace of the given NURBS surface. By Theorem 2.4, the curves

can serve exactly that purpose. Actually, the planar trimming curves $\gamma_{\ell}$ need not be computed at any processing step of ABC-spline surfaces as long as geometric trimming by spatial curves, as explaind in Section 4, is available.

Now, we elaborate on methods how to determine suitable reparametrizations $\kappa_{\ell}$ and weights $w,w_{\ell}$ under the assumption that the base $\mathbf{b}$ and the ribbons $\mathbf{r}_{\ell}$ are given.