Bicubic Splines for Fast-Contracting Control Nets

Abstract

Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for control nets that merge parameter lines. Additionally, untilrecently, none provided a good shape for fast-contracting polyhedral control nets. This work improves the state-of-the-art of piecewise polynomial spline surfaces accommodating fast-contracting control nets. The new fast-contracting (FC) surface algorithm yields the industry-preferred uniform degree bi-3 (bi-cubic). The surfaces are by default differentiable, have an improved shape, measured empirically as to highlight the line distribution, and require fewer pieces compared to existing methods.

1. Introduction

To accommodate narrower surface passages or account for less shape fluctuation, a designer can merge parallel parameter lines as illustrated in Figure 1a,b (see the gray regions). Aggressive merging in quad-meshing algorithms such as [1,2] packs the gray contraction regions too close to each other as shown in Figure 1c,d. Existing algorithms require these faces to be separated by a frame of quadrilaterals. Mitigation strategies range from ad hoc designer intervention, to an improved Doo-Sabin refinement step [3,4], to special re-meshing rules for $T_0$- and $T_1$-locations, [5] (see Figure 1e). The drawbacks of these mitigations are both an increase in the number of patches and a decrease in the surface quality. The surface quality suffers because to obtain the required combinatorial structure, the natural cross field (flow) of the geometry is disturbed.

The recent publication [6] presented two new fast-contracting (FC) spline constructions: FC⁴ and FC³. FC⁴ generates bi-degree surfaces (2, 4) or (2, 3), and FC³ generates bi-degree 3. Both assume that the regular quad-grid of the control net define bi-2 $C^1$ splines, and both use the un-isotropic ${▵}^2$ configuration of Figure 1f as the control net that retains the two preferred directions of the tensor-product splines. The split of the gray core is ignored and no re-meshing or refinement is required to guarantee geometric $G^1$ continuity of the resulting surfaces.

Unlike FC³, the new bi-3 construction ${\mathrm{FC}}_8^3$ manages the transition from two (top) to four (bottom) bi-3 pieces via two internal T-junctions, and requires only the two T-junctions of the center line (see Figure 2e vs. Figure 2f). Remarkably,

• ${\mathrm{FC}}_8^3$ is a 8-piece bi-3 (bi-cubic) construction.
• ${\mathrm{FC}}_8^3$ yields improved shape compared to FC³, measured empirically with a more uniformly highlighted line distribution, and has fewer polynomial pieces, the minimal number required for a good shape.
• The ${\mathrm{FC}}_8^3$ formulas for generating bi-3 patches are linear in the input control net, and hence can be collected into a matrix.
• The implementation of ${\mathrm{FC}}_8^3$ can so be reduced to gathering the control net in a vector of points and multiplying the vector by a matrix.

The patch count can be further reduced to 7 by merging the two middle patches of ${\mathrm{FC}}_8^3$; however, as will be demonstrated, this diminishes the resulting surface quality significantly.

We note that tensor product splines (NURBS) are the preferred representation in many modeling packages and degree bi-3 (bi-cubic) is the preferred degree. Catmull-Clark subdivision surfaces [7], popular in computer graphics, are also of degree bi-3 but consist of an infinite sequence of nested rings generated by recursion. More importantly, however, Catmull-Clark and bi-2 Doo-Sabin [3] subdivision rules are used for treating control net configurations with n directions equally, i.e., isoptropically, whereas FC splines retain the two preferred direction of the bivariate tensor-product splines. That is, FC splines are more general than tensor-product splines in that they allow for merging of quad strips.

After a brief literature review, Section 2 introduces the technical nomenclature and reviews the existing constructions FC³ and FC⁴. Section 3 introduces and derives the new ${▵}^2$-net construction with explicit tables for implementation. Section 4 provides example-driven critical assessment and discussion of variants as well a comparison to FC³. For rotationally symmetric scenarios that permit regular layout or less contraction, ${\mathrm{FC}}_8^3$ is shown to be at least as well-shaped as regular bi-2 splines and the surfaces presented in [8].

Figure 1. Contracting control net configurations: one strip contraction in (a) ${\tau }_0$-net and (b) ${\tau }_1$-net from [8,9] and two-strip contraction via (c) cascading triangles, (d) $T_0$-gon + $T_1$-gon and (e) refinement according to [5]; (f) ${▵}^2$-net with triangulated gray core as a generalization of (c,d), see [6]. Removing from the core one inner bottom edge yields (c), removing both yields (d).

Figure 1. Contracting control net configurations: one strip contraction in (a) ${\tau }_0$-net and (b) ${\tau }_1$-net from [8,9] and two-strip contraction via (c) cascading triangles, (d) $T_0$-gon + $T_1$-gon and (e) refinement according to [5]; (f) ${▵}^2$-net with triangulated gray core as a generalization of (c,d), see [6]. Removing from the core one inner bottom edge yields (c), removing both yields (d).

Figure 2. Rapid contraction: (a) ${▵}^2$-net with 20 labels of its nodes ${\mathbf{d}}_{ij}$. (b) Extended ${▵}^2$-net allows to produce one bi-2 frame via B-to-BB conversion as shown in (c–f). Bottom row: layouts of FC⁴ and FC³ from [6] and ${\mathrm{FC}}_8^3$.

Figure 2. Rapid contraction: (a) ${▵}^2$-net with 20 labels of its nodes ${\mathbf{d}}_{ij}$. (b) Extended ${▵}^2$-net allows to produce one bi-2 frame via B-to-BB conversion as shown in (c–f). Bottom row: layouts of FC⁴ and FC³ from [6] and ${\mathrm{FC}}_8^3$.

Related Work

FC constructions assemble a finite number of polynomial pieces to join smoothly after a change of variables. Such $G^k$ constructions complement constructions for isotropic configurations such as rational multi-sided surfaces [10,11,12], and singularly parameterized surfaces. The sub-genres of singular constructions are subdivision surfaces [3,7,13,14,15], edge collapse, polar surfaces [16,17,18], and vertex singular surfaces [19,20,21,22] or rational singular constructions [23,24].

The shapes of $G^2$ constructions of degree bi-7 [25] or degree bi-6 [26], and lower-degree tangent-continuous splines [27,28,29,30,31,32,33] are empirically measured via highlighted line distribution [34]. FC surfaces fill irregularities in a $C^1$ bi-quadratic (bi-2) tensor-product surface, which is attractive since bi-2 splines have minimal bi-degree for smoothing quadrilateral meshes. Subdivision generalizations of bi-2 splines consist of an infinite sequence of nested (contracting) bi-2 polynomial surface rings. Ref. [3] has visible artifacts already in the first ring, Augmented Subdivision presented in [4] improves the shape by following a carefully chosen central guide point and polyhedral-net splines [35] combine algorithms from [8,18,36] to generalize tensor-product bi-quadratic (bi-2) splines, filling in finitely many polynomial pieces of degree at most bi-3. T-splines [37] address the merging parallel quad strips but typically serve only to refine an existing quad partition; due to their global parameterization requirement, they may not be well-defined for a given T-configuration, (see Figure 2 in [38] and Figure 6 in [39]). Alternatively, T-junctions in the control net can be associated with smooth surfaces of bi-degree $(2,4)$ [9] or bi-3 [8] that result in smooth surfaces of good quality. ${\mathrm{FC}}_8^3$ is partly motivated by the output of quad-dominant meshing algorithms such as [1,2], that introduce (fast) mesh contractions. We note that the present paper does not touch on re-meshing [40,41,42], but focuses on frequently occurring contracting configurations.

2. Control Nets, Macro-Patches, FC³ and FC⁴

${\mathrm{FC}}_8^3$ is an improvement of FC³. We therefore use the notation of [6]. Similarly to tensor-product spline control nets, the ${▵}^2$ nets have two distinguished directions that we refer to as ‘vertical’ and ‘horizontal’ due to their layout in Figure 2. The number of mesh lines is reduced or expanded only in the vertical direction.

2.1. Control Nets

Figure 2a displays the labels of ${▵}^2$-net used for the derivation and presentation of pre-calculated data. The bottom row of Figure 2 compares the patchworks of three FC surfaces. FC⁴, see Figure 2d, consists of pieces of bi-degree $(2,4)$ (i.e., 2 in the horizontal direction) except for a middle row of bi-degree $(2,3)$. The central, light-red patch serves as a model for the bi-3 constructions. For FC³, see Figure 2e; this central patch is degree-raised to bi-3 and split into two to keep a tensor-product structure of red macro-patch. The layout of the bi-3 pieces of ${\mathrm{FC}}_8^3$ is shown in Figure 2f. The circles in Figure 2d–f mark the locations of T-junctions: small ∘ points to a single T-junction, ◯ to multiple T-junctions. The two T-junctions in (d) merge three strips into one. The additional T-junctions in (e) follow from the fact, proven in [6], that any $C^1$ bi-3 construction requires an even number of pieces, both at the bottom and the top.

2.2. Polynomial Pieces

${\mathrm{FC}}_8^3$ consists of tensor-product pieces of polynomial bi-degree $(d,d^{\prime })$ in Bernstein-Bézier form (BB-form, [43]). That is, for Bernstein polynomials $B_k^d\left(t\right):=\left(\genfrac{}{}{0pt}{}{d}{k}\right){(1-t)}^{d-k}{t}^k$, the bi-dgree 3 (bi-3) patch $\mathbf{p}$ is defined as

$$\mathbf{p}(u,v):=\sum _{i=0}^3\sum _{j=0}^3{\mathbf{p}}_{ij}{B}_i^3\left(u\right)B_j^3\left(v\right),0\le u,v\le 1.$$

With the convention that u is the parameter tracing out the horizontal direction.

Connecting the BB-coefficients ${\mathbf{p}}_{ij}\in {\mathbb{R}}^3$ to ${\mathbf{p}}_{i+1,j}$ and ${\mathbf{p}}_{i,j+1}$ wherever well-defined yields the BB-net, see Figure 3. Any $3\times 3$ grid can be interpreted as the control net of a uniform bi-2 spline in uniform knot B-spline form. In Figure 3, the B-spline control points are marked ∘. The B-to-BB conversion (e.g., by knot insertion) expresses the spline in bi-2 BB-form illustrated by the green BB-nets in Figure 3. Conversion of a partial sub-grid yields a partial BB-net $\mathbf{t}$, called tensor-border, that defines the position and first derivatives across an edge.

The changing number of mesh lines forces a change of parameterization and hence the introduction of geometric continuity: Two polynomial pieces $\mathbf{p},\mathbf{q}:{\mathbb{R}}^2\to {\mathbb{R}}^3$ join $G^1$ along the common curve with BB-coefficients ${\mathbf{p}}_{i0}={\mathbf{q}}_{i0}$ if there exists a reparameterization $\rho :{\mathbb{R}}^2\to {\mathbb{R}}^2$, see, e.g., [44],

$$\begin{array}{cc}\hfill \mathbf{p}(u,v)& :=\mathbf{q}\circ \rho (u,v),\rho (u,v):=(u+b(u)v,a(u\left)v\right)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\partial }_v\mathbf{q}(u,0)& =a\left(u\right){\partial }_v\mathbf{p}(u,0)+b\left(u\right){\partial }_u\mathbf{p}(u,0)\hfill \end{array}$$

(2)

for scalar-valued, univariate functions a and b. In addition to the shared BB-coefficients of the common boundary, only the layers of BB-coefficients ${\mathbf{p}}_{i1}$ and ${\mathbf{q}}_{i1}$ of adjacent patches enter the $G^1$ continuity constraints. In the derivation, u-, v-directions can be assigned as convenient, but typically u is used to parameterize along the boundary and v in the orthogonal direction of the tensor-border, towards the interior or core.

2.3. Summary of FC⁴ and FC³

The bi-2 tensor-border frame (dark green in Figure 4a) represents first-order Hermite data. This input for all FC constructions stems from the ${▵}^2$-net by B-to-BB conversion.

Figure 4 summarizes the construction of FC⁴. The khaki-colored bottom layer in (a) is degree-raised to bi-degree $(2,4)$ and the yellow top layer is uniformly split into 3 pieces and degree-raised. The light-green left and right layers are the result of reparameterizing the left pieces ${\mathbf{t}}^0$, ${\mathbf{t}}^1$, and ${\mathbf{t}}^2$, and the right pieces ${\mathbf{t}}^3$, ${\mathbf{t}}^4$, ${\mathbf{t}}^5$ of input tensor-border frame, respectively. Unconstrained BB-coefficients are marked •, and the BB-coefficients marked × are defined by $C^1$ extension of the central (light-red) patch. The remaining BB-coefficients are the averages of their horizontal (black and light-green) neighbor BB-coefficients.

With Figure 5 providing tensor-border labels, the construction of FC³ is summarized in Figure 6. In (a), leaving ${\mathbf{t}}_i$, $i=0,\dots ,5$, from Figure 4a and ${\underline{\mathbf{t}}}_0$ and ${\underline{\mathbf{t}}}_3$ unchanged. ${\underline{\mathbf{t}}}_1$ and ${\underline{\mathbf{t}}}_2$ represent a uniformly split piece at the middle of the bottom frame of Figure 4a. Mimicking the bottom split, ${\overline{\mathbf{t}}}_i$, $i=0,\dots ,3$, top split with ratio $1:\frac{1}{2}:\frac{1}{2}:1$. In (b), the reparameterized pieces ${\mathbf{t}}_i$, $i=0,\dots ,5$, ${\underline{\mathbf{t}}}_i$, $i=0,\dots ,3$ and ${\overline{\mathbf{t}}}_i$, $i=0,\dots ,3$ (light-green) are consistent at all four corners and the light-red bi-3 patches 6 and 7 represent the central light-red bi-degree $(2,3)$ patch from Figure 4b, after degree-raising and split. The reparameterizations ${\rho }^s$, $s=0,1,2$, ${\underline{\rho }}^s$, $s=0,1,2,3$ and ${\overline{\rho }}^s$, $s=0,1$ are also used in ${\mathrm{FC}}_8^3$ construction, see Figure 5 for the labeling. Formulas (3) and (4) define the bi-3 tensor-border frame (light-green in Figure 6b) due to the symmetries ${\rho }^{3+s}:={\rho }^s$, $s=0,1,2$.

$$\begin{array}{cc}\hfill {\rho }^s(u,v):=& (u,a^s\left(u\right)),a^s\left(u\right):=a_0^s(1-u)+a_1^{s}u;{\tilde{\mathbf{t}}}^s:={\mathbf{t}}^s\circ {\rho }^s;\hfill \\ \hfill [a_0^0,a_1^0]:=& [1,\frac{7}{9}],[a_0^1,a_1^1]:=[\frac{7}{9},\frac{5}{9}],[a_0^2,a_1^2]:=[\frac{5}{9},\frac{1}{3}];\hfill \\ \hfill {\tilde{\mathbf{t}}}_{01}^s:=& \left(1-\frac{2}{3}{a}_0^s\right){\mathbf{t}}_{00}^s+\frac{2}{3}{a}_0^s{\mathbf{t}}_{01}^s,\hfill \\ \hfill {\tilde{\mathbf{t}}}_{11}^s:=& \left(\frac{1}{3}-\frac{2}{9}{a}_1^s\right){\mathbf{t}}_{00}^s+\left(\frac{2}{3}-\frac{4}{9}{a}_0^s\right){\mathbf{t}}_{10}^s+\frac{2}{9}{a}_1^s{\mathbf{t}}_{01}^s+\frac{4}{9}{a}_0^s{\mathbf{t}}_{11}^s.\hfill \end{array}$$

(3)

The boundary BB-coefficients ${\tilde{\mathbf{t}}}_{i0}^s$, $i=0,\dots ,3$, are obtained from ${\mathbf{t}}_i^s$, $i=0,1,2$ by degree-raising. The remaining BB-coefficients ${\mathbf{t}}_{21}^s$ and ${\mathbf{t}}_{31}$ are defined by the symmetry ${\mathbf{t}}_{ij}^s$↔${\mathbf{t}}_{2-i,j}^s$, $i=0,1,2$, $j=0,1$; $a_i^s$↔$a_{1-i}^s$, $i=0,1$.

The bottom reparameterizations ${\underline{\rho }}^s$ and top reparameterizations ${\tilde{\rho }}^s$, $s=0,1,2,3$, are defined by formulas

$${\underline{\rho }}^s:=(u+{\underline{\gamma }}^s{B}_1^2\left(u\right)v,v),{\underline{\gamma }}^s:=\frac{{(-1)}^{s+1}}{9};{\overline{\rho }}^s:=(u+{\overline{\gamma }}^s{B}_1^2\left(u\right)v,v),{\overline{\gamma }}^s:=\frac{{(-1)}^s}{3}.$$

Setting $\gamma :={\underline{\gamma }}^s$, $\mathbf{p}:={\underline{\mathbf{t}}}^s$, $\mathbf{q}:={\widetilde{\underline{\mathbf{t}}}}^s$ in (4) yields explicit formulas for the BB-coefficients of the reparameterized bi-3 tensor-border ${\widetilde{\underline{\mathbf{t}}}}^s:={\underline{\mathbf{t}}}^s\circ {\underline{\rho }}^s$. Setting $\gamma :={\overline{\gamma }}^s$, $\mathbf{p}:={\overline{\mathbf{t}}}^s$, $\mathbf{q}:={\widetilde{\overline{\mathbf{t}}}}^s$ in (4) yields ${\widetilde{\overline{\mathbf{t}}}}^s:={\overline{\mathbf{t}}}^s\circ {\overline{\rho }}^s$.

$$\begin{array}{cc}\hfill {\mathbf{q}}_{01}:=& \frac{1}{3}{\mathbf{p}}_{00}+\frac{2}{3}{\mathbf{p}}_{01},{\mathbf{q}}_{31}:=\frac{1}{3}{\mathbf{p}}_{20}+\frac{2}{3}{\mathbf{p}}_{21},\hfill \\ \hfill {\mathbf{q}}_{11}:=& \left(\frac{1}{9}-\frac{4}{9}\gamma \right){\mathbf{p}}_{00}+\left(\frac{2}{9}+\frac{4}{9}\gamma \right){\mathbf{p}}_{10}+\frac{2}{9}{\mathbf{t}}_{01}+\frac{4}{9}{\mathbf{p}}_{11},\hfill \\ \hfill {\mathbf{q}}_{21}:=& \left(\frac{2}{9}-\frac{4}{9}\gamma \right){\mathbf{p}}_{10}+\left(\frac{1}{9}+\frac{4}{9}\gamma \right){\mathbf{p}}_{20}+\frac{4}{9}{\mathbf{p}}_{11}+\frac{2}{9}{\mathbf{p}}_{21},\hfill \end{array}$$

(4)

the boundary BB-coefficients ${\mathbf{q}}_{i0}$, $i=0,1,2,3$, are obtained from ${\mathbf{p}}_{i0}$, $i=0,1,2$, by degree-raising.

3. The ${\mathrm{FC}}_8^3$ Construction

Starting with the bi-2 tensor-border frame, see Figure 7a, of FC³, cf. Figure 6a; however, anticipating the layout displayed in Figure 7b, uniformly splitting the top, we construct the bi-3 tensor-border frame (light-green in (a)) according to Section 2.3, Figure 6, and the reparameterized tensor-borders ${\tilde{\mathbf{t}}}^1$, ${\tilde{\mathbf{t}}}^2$ (and analogously ${\tilde{\mathbf{t}}}^4$, ${\tilde{\mathbf{t}}}^5$) in Figure 8a re-scaled as $•:=\frac{3}{2}$•− $\frac{1}{2}$• (Hermite data: • on the boundary remain unchanged). All data of ${\mathbf{p}}^5$ in Figure 8b is displayed in Figure 8a, middle-left and again, enlarged, in Figure 8c,d. The light-red underlain piece in Figure 8c represents the bi-3 patch in Figure 6b with label 6. Leaving × unchanged, we set the BB-coefficients marked $\times :=3$×$-2$×, to form the layer ${\mathbf{p}}_{2j}^5$, $j=0,\dots ,3$ in Figure 8b. The BB-coefficients ${\mathbf{p}}_{1j}^6$, $j=0,\dots ,3$ are constructed analogously. By construction, ${\mathbf{p}}_{3j}^5={\mathbf{p}}_{0j}^6$, $j=0,\dots ,3$ and ${\mathbf{p}}_{3j}^5:=\frac{1}{2}{\mathbf{p}}_{2j}^5+\frac{1}{2}{\mathbf{p}}_{1j}^6$. On top, the re-scaled tensor-border is reparameterized by ${\overline{\rho }}_0$, ${\overline{\rho }}_1$ defined in Section 2.3 yielding ${\overline{\mathbf{t}}}_0$, ${\overline{\mathbf{t}}}_1$, see Figure 7a. By inspection, the parameterization is consistent at the top two corners in Figure 8b. The patches ${\mathbf{p}}^7$ and ${\mathbf{p}}^8$ are completed by light-gray extension of patches ${\mathbf{p}}^5$, ${\mathbf{p}}^6$, see Figure 8b. The patches ${\mathbf{p}}^s$, $s=1,\dots ,4$, are completed by dark-gray extension of patches ${\mathbf{p}}^5$, ${\mathbf{p}}^6$ and a subsequent split.

${\mathrm{FC}}_8^3$ expresses the BB-coefficients as affine combinations of the ${▵}^2$-net nodes $\mathbf{d}$. That is, with the formulas of Section 2.3, explicit formulas are available once the BB-coefficients of the central patches are known (light-red in Figure 6b). These can be gleaned from [6]; however, here, we present the explicit weights $\mu$ of formulas formulas for $j=0,\dots ,3$, $s=0,\dots ,3$ (see Figure 8b).

$$\begin{array}{cc}\hfill {\mathbf{p}}_{2s}^5:=& \sum _{i=1}^5\sum _{j=1}^2{\mu }_{ij}^s{\mathbf{d}}_{ij}+\sum _{i=1}^4{\mu }_{i3}^s{\mathbf{d}}_{i3}+\sum _{i=1}^3{\mu }_{i4}^s{\mathbf{d}}_{i4},\hfill \\ \hfill {\mathbf{p}}_{1s}^6:=& \sum _{i=1}^5\sum _{j=1}^2{\mu }_{6-i,j}^s{\mathbf{d}}_{ij}+\sum _{i=1}^4{\mu }_{5-i,3}^s{\mathbf{d}}_{i3}+\sum _{i=1}^3{\mu }_{4-i,4}^s{\mathbf{d}}_{i4}.\hfill \end{array}$$

(5)

Without the loss of quality, the coefficients $\mu$ can be stated with 5 decimal accuracy and corrected by less than 0.00009 to form a partition of 1. That is, the weights ${\mu }_{ij}^s$ listed below are exact, not approximations of the implementation weights.

$$\mathrm{Table}{M}^s\mathrm{lists}{10}^5\left(\begin{array}{ccccc}{\mu }_{14}^s& {\mu }_{24}^s& {\mu }_{34}^s& \times & \times \\ {\mu }_{13}^s& {\mu }_{23}^s& {\mu }_{33}^s& {\mu }_{43}^s& \times \\ {\mu }_{12}^s& {\mu }_{22}^s& {\mu }_{32}^s& {\mu }_{42}^s& {\mu }_{52}^s\\ {\mu }_{11}^s& {\mu }_{21}^s& {\mu }_{31}^s& {\mu }_{41}^s& {\mu }_{51}^s\end{array}\right).$$

$$M^0:=\left(\begin{array}{ccccc}1287& -2571& 1287& \times & \times \\ -896& 38395& 15479& -2979& \times \\ 4441& 9437& 47248& -13479& 2358\\ -428& 1705& -2561& 1705& -428\end{array}\right),M^1:=\left(\begin{array}{ccccc}1952& -3904& 1952& \times & \times \\ 496& 62003& 25198& -4364& \times \\ 6080& -9877& 32592& -16127& 3997\\ -934& 3739& -5608& 3739& -934\end{array}\right)$$

$$M^2:=\left(\begin{array}{ccccc}2448& 12463& 1753& \times & \times \\ 6449& 56051& 23412& -2578& \times \\ 2892& -12303& 18821& -12303& 2892\\ -828& 3315& -4971& 3315& -828\end{array}\right),M^3:=\left(\begin{array}{ccccc}8725& 38797& 2475& \times & \times \\ 5059& 32439& 13689& -1190& \times \\ 1674& -7179& 11010& -7179& 1674\\ -531& 2128& -3188& 2128& -531\end{array}\right)$$

4. Assessments and Comparisons

One motivation of ${\mathrm{FC}}_8^3$ was to significantly reduce the number of bi-3 patches in FC³ without overtly harming surface quality. For completeness, we also compare a FC construction composed of 7 patches with FP7, and with FF4, of the same layout as ${\mathrm{FC}}_8^3$, but pinning down degrees of freedom via the functional ${\mathcal{F}}_4:={\int }_0^1{\int }_0^1{\sum }_{i+j=4,i,j\ge 0}\frac{4!}{i!j!}{\left({\partial }_s^i{\partial }_t^{j}f(s,t)\right)}^{2}dsdt.$

4.1. Small Local Challenge Nets

Figure 2b already introduced the ${▵}^2$-net (nodes marked by •) extended by a quad frame whose outermost nodes can be of any valence. While a FC surface is fully defined by ${▵}^2$-net, the extended ${▵}^2$-net yields a regular B-spline bi-2 frame as in Figure 2c–f, that reveals any problems in the transition from regular to FC surface. Conversely, the restriction to the extended ${▵}^2$-net avoids the need to zoom-in into area of irregularity on a large model, see, e.g., Figure 9a and emphasize problematic nets.

The nodes of the ${▵}^2$-net shown Figure 10a are obtained from the planar layout of Figure 2b by projection onto an elliptic paraboloid. This type of control net is key when testing surfaces obtained from nets contracting as in Figure 1: a high quality surface, similar to an elliptic paraboloid within the region of Figure 10b is expected. FP7 displays undesired strong oscillation of the highlighted lines, and FF4, as well as ${\mathrm{FC}}_8^3$ reveals no artifacts under scrutiny, justifying the subtle construction in [6]. Improving on FC³ is not expected in this specific hard configuration. However, Figure 11, Figure 12 and Figure 13 show an improvement of ${\mathrm{FC}}_8^3$ over FC³ in the highlighted line distribution across the transition between patches 5 and 6, and 7 and 8, respectively (see Figure 6b for labels), likely due to the increased smoothness when combining the patches into one.

The extended ${▵}^2$-net of Figure 11a is derived from Figure 10a by lifting a ‘horizontal’ row of nodes (labels $i3$, $i=0,\dots 5$) and a ‘vertical’ triple ($3j$, $j=0,1,2$) to 32: (b) displays the region shown in (c,d). Red arrows in Figure 11c point to abrupt changes in highlighted lines for FC³ that disappear in ${\mathrm{FC}}_8^3$, while the ↓ in Figure 11d points to sharper turns in the ${\mathrm{FC}}_8^3$ highlighted lines than in same location for FC³. The bottom row compares side-views revealing unexpected dips indicated by ↓ of FF4 and FP7.

The extended ${▵}^2$-net of Figure 12 is derived from Figure 10a by pushing down the nodes with labels $ij$$i,j=0,1,2$. Red arrows in Figure 12c point to abrupt changes in highlighted lines for FC³ that disappear in ${\mathrm{FC}}_8^3$, while the ← in Figure 12d points to slightly sharper turns in the ${\mathrm{FC}}_8^3$ highlighted lines than in the same location for FC³. The bottom row compares side-views for ${\mathrm{FC}}_8^3$ and FP7, revealing an unwanted dip, indicated by ↓, in FP7. Figure 13 adds a wavy ${▵}^2$-net commonly occurring in automatically generated meshes. Again, ${\mathrm{FC}}_8^3$ has a slight shape advantage over FC³ indicated by ←.

In summary, empirically, we see only a slight (if any) degradation of the highlighted line distribution by using fewer pieces in ${\mathrm{FC}}_8^3$ over FC³ and an improvement of ${\mathrm{FC}}_8^3$ over FC³ in the highlighted line distribution across the transition between the merged patches. That is, sharp turns either become considerably milder or disappear.

4.2. Large Hand-Crafted Models

The original motivation of FC³ (FC⁴) was to address local challenges in re-meshing for spline surfaces. However, the solution lends itself to direct design of larger objects, e.g., to design surfaces that start with the ubiquitous shape from revolution, such as the examples in Figure 14.

Figure 15 demonstrates the design with ${▵}^2$-nets. The input (a) is a mesh of revolution: the nodes in the 10 horizontal loops lie on co-axially stacked circles. The bottom 4 layers consist of 36 nodes each, the 5th has 24, the 6th and 7th have 12, the 8th layer has 8, and the top two layers have 4 nodes each. The mesh is capped by a quad face; (b) shows the surface layout: the bottom 4 layers become bi-2 patches; the 12 lower and the 4 upper ${▵}^2$-nets become ${\mathrm{FC}}_8^3$ macro-patches; the four 3-valent vertex neighborhoods at the top are covered by bi-3 patches according to [36]. While the ${▵}^2$-nets overlap, their cores are sufficiently separated to build ${\mathrm{FC}}_8^3$ macro-patches. Note that (c) contains no submesh defining a bi-2 patch. The second row zooms in on and displays the highlighted line distribution of three challenging neighborhoods: (e,f) focuses on the transition between the 12 bottom and 4 top ${\mathrm{FC}}_8^3$ surfaces; (g,h) on the transition between the 4 top ${\mathrm{FC}}_8^3$ and the valence 3 neighborhoods; (i,j) on the transition between the regular bi-2 patches and the 12 bottom ${\mathrm{FC}}_8^3$. The most noticeable changes in the highlighted lines occur where regular bi-2 $C^1$ splines are used, as further discussed later with respect to Figure 16c. The bottom row illustrates hand-crafted designs, whose highlighted line distributions are on par with regular bi-2 splines.

In Figure 17a, the rapid merging part of the quad strips with ${▵}^2$-nets is taken from Figure 15a, while the bottom part (now four quad strips) is modified anticipating a ‘connecting tube’. The valence 5 vertices are treated with [36] to yield the smooth transitions in Figure 17b. Figure 17c,d show a design modification involving the ${▵}^2$-nets.

Figure 16. Detailed comparison of surfaces from Figure 15l and Figure 18j. (a,b) Figure 15l bottom; (c) Figure 18j bottom; (d,e) Figure 15l middle; (f) Figure 18j middle.

Figure 16. Detailed comparison of surfaces from Figure 15l and Figure 18j. (a,b) Figure 15l bottom; (c) Figure 18j bottom; (d,e) Figure 15l middle; (f) Figure 18j middle.

The goal of Figure 18 is to compare ${\mathrm{FC}}_8^3$ for ${▵}^2$-nets to the less rapidly contracting bi-3 surfaces of [8] and to regular $C^1$ bi-2 splines for designs where any of the three options can be chosen. That is, if ${\mathrm{FC}}_8^3$ looks no worse than its two alternatives, despite supporting rapid contraction, ${\mathrm{FC}}_8^3$ passes the test. We recall that [8] takes as input ${\tau }_0$-nets, namely triangles with two vertices of valence 4 and one with valence 5, surrounded by one layer of quad facets. Row 1 shows surfaces created using [8], and Row 2 regular bi-2 spline surfaces. To make the corresponding surfaces as similar as possible, the 10 layers of (a) and (g) in Figure 18 lie on the same circles as for Figure 15. The main change is the number of nodes in Figure 18: in (a), layers $1,2,3,4$ have 32, layers 5, 6 have 16, layers 7, 8 have 8 and layers 9, 10 have 4 nodes; in (g), all layers have 36 nodes. Unlike (a), in the regular mesh (g) the top is not closed: the best option in terms of simplicity and quality is to cap with 36 triangles sharing a common central vertex, i.e., a polar configuration; see [18] for the details. We omit this cap to focus on the comparison of regular splines with their counterparts based on ${▵}^2$- and ${\tau }_0$-nets. (Analogously, the cap from Figure 15b and Figure 18b is omitted displaying the corresponding surfaces in Figure 18n,o.) All surfaces in Figure 15b and Figure 18b,h have the same axial symmetries and similar highlighted line distributions; see Figure 18m–o. That is, the highlighted line distributions depend on the geometry (large-scale flow) of input meshes more so than on the choice of construction. Modified surfaces can look alike as in Figure 17d and Figure 18f, or they can differ due to a different layout of ${\tau }_0$ vs ${▵}^2$ cores (gray in Figure 1a,f); see the bottom of Figure 18d vs. Figure 15l. By contrast, the bottom of the regular bi-2 surface Figure 18j is very similar to Figure 15l; at the top the surface is less of a global shape modification than a local embossing since the corresponding layer in Figure 15a has 12, whereas Figure 18g has 36 nodes and in each case four nodes are displaced outwards (Figure 18k,l displacing by a smaller amount makes a local embossing milder; however, an effect of global modification is not achieved). Achieving the effect of ${\mathrm{FC}}_8^3$ with the un-contracted net is a tricky challenge for a designer as is also clear if the connecting un-contracted tube ends at a single quad on the outer disk of Figure 17.

Figure 17. Design inspired by Figure 14b.

Figure 17. Design inspired by Figure 14b.

Figure 18. Designs mimicking Figure 15 but using ${\tau }_0$-nets or regular bi-2 splines. (m–o) visual equivalence of un-contracted, moderate-speed-contracted and fast-contracted surfaces. (o) corresponds to Figure 15b.

Figure 18. Designs mimicking Figure 15 but using ${\tau }_0$-nets or regular bi-2 splines. (m–o) visual equivalence of un-contracted, moderate-speed-contracted and fast-contracted surfaces. (o) corresponds to Figure 15b.

Figure 16top compares the base of Figure 15l to the base of Figure 18j. The highlighted line distributions in Figure 16b,c indicate similar surface quality. Also comparing the highlighted line distribution of the mid-section, Figure 16e,f, show no clear winner. Indeed, we observe in numerous tests that the quality of ${\mathrm{FC}}_8^3$ surfaces is no worse than the quality of regular $C^1$ bi-2 splines, despite the reduction in degrees of freedom.

5. Conclusions

While very specialized, the optimal treatment of rapid-contraction ${▵}^2$-nets by ${\mathrm{FC}}_8^3$ surfaces is an important building block that allows for properly adjusting the parameterization of free-form surfaces when accommodating narrow passages or a decrease in detail. Artificially comparing ${\tau }_0$-surfaces and regular bi-2 splines where such rapid contraction is not needed shows the ${\mathrm{FC}}_8^3$ surfaces to be of equal quality and therefore to fit nicely into the polyhedral-net spline framework [35].