Keywords

1 Introduction

In this work we present a design model for a (grid)shell composed from an assembly of 3D components fabricated from planar and/or developable faces. Our approach rests fundamentally on the so-called Triply Orthogonal system of Surfaces (TOS) giving rise to explicit parameterizations of three families of surfaces by principal curves. These principal-curves parameterizations are orthogonal conjugate networks, which make them optimal in the discretization of doubly-curved surfaces using planar and developable pieces. Using a TOS thus provide us a three-dimensional principal network in the sense of a curvilinear coordinate system which will be used to define the 3D component (or voxel) of the assembly forming the (grid)shell. The six sides of a voxel are given (by pieces of) surfaces from the three families in the TOS, hence are principal patches. There follows that we can transform our voxel into a rational-voxel whose sides are either planar or developable, allowing for a feasible construction by simple methods. In Sect. 2, we present the geometry of principal networks and the TOS, in Sect. 3, we present the design model and degrees of design freedom, in Sect. 4, we present a realized (grid)shell based on our design model.

2 Geometry

We will now briefly recall few facts from differential geometry (cf. [4] and [5]).

2.1 Principal Patches

Consider a surface S given by a smooth parameterization X(u,v) where the domain of (u,v) is an open set in the plane and the coordinate lines form a network on S. We will thus refer to X(u,v) as a patch or as a network. We denote by Xu, Xv the partial derivatives generating the tangent plane TpS and by N the normal vector. The first and second fundamental forms are denoted by I, II, the shape operator is given by S = II/I.

Let a,b be tangent vectors in TpS then their directions are called

$$ \left\{ {\begin{array}{*{20}l} {{\text{Orthogonal}}\;{\text{if}}\;\langle a,b\rangle = 0\;{\text{that}}\;{\text{is}}\;a^T \cdot I \cdot b = 0} \hfill \\ {{\text{Conjugate}}\;{\text{if}}\;\langle a,S\left( b \right)\rangle = 0\;{\text{that}}\;{\text{is}}\;a^T \cdot II \cdot b = 0.} \hfill \\ \end{array} } \right. $$

The eigenvalues of the shape operator are the principal curvatures k1, k2 and the corresponding (necessarily orthogonal) eigenvectors define the principal directions.

  • A curve in S is a principal curve if at every point it is tangent to a principal direction that is, it is an integral curve for that principal direction field.

  • A patch X(u,v) on S is a principal patch (network) if its coordinate lines are principal curves, that is the directions of Xu, Xv are both orthogonal and conjugate.

  • A mesh on S is a principal mesh if it is constituted of quadrilateral faces obtained by joining the intersection points of a principal network.

Fig. 1.
figure 1

a) Principal network. (b) Principal mesh.

Full size image

2.2 Geometric Properties of Principal Networks

Let X(U,v) be a principal patch (network), that is Xu, Xv are principal directions (orthogonal and conjugate). Orthogonality is clear as seen in Fig. 1(a), as for conjugation, we recall that conjugate directions a and b are parallel to the edges of a parallelogram tangent to the DUpin's indicatrix, as seen in Fig. 1(B). Naturally, there is an infinity of such parallelograms and if we impose orthogonality between a, b then we have a rectangle with edges parallel to the principal directions (Fig. 2).

Fig. 2.
figure 2

A) Orthogonality property of Xu, Xv. (b) Conjugation property of Xu, X v.

Full size image

We now define two important types of ruled surfaces or ‘strips’ that will be useful (Fig. 3).

The Normal strips for a fixed uo, vo given by:

$$ \left\{ {\begin{array}{*{20}l} {N_1 \left( {s,u} \right) = X\left( {u,v_o } \right) + sN\left( {u,v_o } \right)} \hfill \\ {N_2 \left( {s,v} \right) = X\left( {u_o ,v} \right) + sN\left( {u_o ,v} \right).} \hfill \\ \end{array} } \right. $$

The ‘Tangent’ strips for a fixed uo, vo given by:

$$ \begin{array}{*{20}l} {T_1 \left( {s,u} \right) = X\left( {u,v_o } \right) + s\left( {X\left( {u,v_o + \varepsilon } \right) - X\left( {u,v_o } \right)} \right)} \hfill \\ {T_2 \left( {s,v} \right) = X\left( {u_o ,v} \right) + s\left( {X\left( {u_o + \varepsilon ,v} \right) - X\left( {u_o ,v} \right)} \right).} \hfill \\ \end{array} $$
Fig. 3.
figure 3

a) Normal strips. (b) Tangent strips.

Full size image

2.3 Continuous Principal Networks and Discrete Circular PQ Meshes

Deviation from Planarity and Developability

It is important to point out that our principal meshes (arising from generic principal networks) are not technically circular PQ meshes. However, they are very close to being so, in the sense that if Q is a quad from a principal mesh it deviates from planarity by a very small error (see [3]) as seen in Fig. 4.

Fig. 4.
figure 4

Q deviate from planarity by O (ε4) as opposed to O (ε3) in other networks.

Full size image

Hence, non-linear optimization problems of approximating surfaces by circular PQ meshes, see [2, 7] and [9] are much more likely to work if initiated from principal networks. Another (somehow inverted way) of looking at the deviation from planarity of principal meshes, which also shows the deviation from developability of the tangent strips T1(s,u), T2(s,v) is that:

$$ \begin{array}{*{20}l} {{\text{Circular}}\;{\text{PQ}}\;{\text{meshes}}} \hfill \\ {{\text{Sequence}}\;{\text{of}}\;{\text{PQ}}\;{\text{faces}}} \hfill \\ \end{array} \mathop{\longrightarrow}\limits_{{{\text{refinement}}}}^{{{\text{at}}\;{\text{the}}\;{\text{limit}}\;{\text{of}}}}\begin{array}{*{20}l} {{\text{Principal}}\;{\text{networks}}} \hfill \\ {{\text{Developable}}\;{\text{strips}}} \hfill \\ \end{array} $$

Error Tolerance in our 3D Models

In our approach we start by principal networks hence the normal strips N1(s,u), N2(s,v) are truly developable. Next, we produce discrete principal meshes (hence almost planar) and semi-discrete tangent strips (hence almost developable), as explained above. However, the errors are so small, that if we consider the realization of the model using Rhinoceros with its tolerance set to 0.001. Then, planarity of the principal meshes and the developability of the tangent strips in question are completely within this tolerance. Thus for all purposes of our architectural application:

  • The principal meshes will be considered planar

  • The tangent strips T1(s,u), T2(s,v) will be considered developable.

2.4 Triply Orthogonal System of Surfaces

As mentioned above the Triply Orthogonal system of Surfaces (TOS) are systems consisting of three families of surfaces that are mutually orthogonal. A theorem of Dupin asserts that given such system then the curves of intersection of the surfaces are principal curves on each surface, see [6]. Similar to discrete principal networks, there is a discrete version of TOS, see [1]. However, we will focus on smooth TOS giving rise to elliptic curvilinear coordinates, as seen in Fig. 5.

Fig. 5.
figure 5

a) Ellipsoids R. (b) One-sheet hyperboloids G. (c) Two-sheet hyperboloids B.

Full size image

Explicit Parametric Definition of the TOS

For a, b, c real suitably bounding the parameters u, v, w we obtain the patches:

$$ \begin{gathered} X\left( {u,v,w} \right) \hfill \\ \; = \left\{ { \pm \sqrt {\frac{{\left( {a - u} \right)\left( {a - v} \right)\left( {a - w} \right)}}{{\left( {a - b} \right)\left( {a - c} \right)}}} , \pm \sqrt {\frac{{\left( {b - u} \right)\left( {b - v} \right)\left( {b - w} \right)}}{{\left( {b - a} \right)\left( {b - c} \right)}}} , \pm \sqrt {\frac{{\left( {c - u} \right)\left( {c - v} \right)\left( {c - w} \right)}}{{\left( {b - c} \right)\left( {a - c} \right)}}} } \right\} \hfill \\ \end{gathered} $$

parameterizing the elliptic TOS (R, G, B) where R are ellipsoids, G are one-sheeted hyperboloids and B are two-sheeted hyperboloids respectively, by principal patches:

$$ \left\{ {\begin{array}{*{20}l} {R\left( {v,w} \right) = X\left( {u_o ,v,w} \right)} \hfill \\ {G\left( {w,u} \right) = X\left( {u,v_o ,w} \right)} \hfill \\ {B\left( {u,v} \right) = X\left( {u,v,w_o } \right)} \hfill \\ \end{array} } \right. $$

where u in [−c,c], v is in [c,b] and w in [b,a].

Important Advantages of the TOS

  1. (1)

    We obtain an explicit expression for the parameterizations of the surfaces in question therefore, bypassing the need for solving the differential equation of principal curves in order to obtain the principal networks.

  2. (2)

    Exploring the third dimension through the elliptic curvilinear coordinates and the generation of the 3D component (voxel), as seen in Fig. 6.

  3. (3)

    The TOS gives us a way to simultaneously have caps-surfaces that are principal patches with varying thickness all-the-while maintaining that the lateral-surfaces are developable. This is done by having the cap-surfaces given by two successive surfaces from one family in the TOS and the lateral-surfaces developable tangent strips on surfaces from the remaining two families, as seen in Fig. 7(b). This gives a larger variety of shell designs (of varying thickness) with the geometric advantages of constant-thickness shells. Note that, the principal patches cap-surfaces used here are not parallel neither are their principal meshes, by contrast to the parallel meshes addressed in [8].

Fig. 6.
figure 6

a) Principal patches R, G, B. (b) Elliptic curvilinear coordinate lines u, v, w.

Full size image
Fig. 7.
figure 7

a) Constant thickness using offset. (b) Varying thickness using TOS.

Full size image

3 Design Model

Let us outline the five degrees of design freedom constituting the design model.

3.1 Choice of Shell-Slice Type

The first degree of design-freedom is in the choice of the leaves determining the caps-surfaces of the shell-slice. More precisely, by fixing one curvilinear direction and considering the continuum of leaves given by the family of surfaces orthogonal to that direction as seen in the Fig. 8.

Fig. 8.
figure 8

a) R-leaves. (b) G-leaves. (c) B-leaves.

Full size image

3.2 Choice of Shell-Slice Thickness

The second degree of design-freedom is in the choice of the specific two leaves determining the caps-surfaces of the shell-slice, as seen in Fig. 9.

Fig. 9.
figure 9

a) Highly curved thick. (b) Medium curved thin. (c) Highly curved very thick.

Full size image

3.3 Choice of Voxel Assembly

Consider the voxel bounded by two R-leaves, two G-leaves and two B-leaves as seen in Fig. 10(a). This voxel has (doubly-curved) faces which are principal patches and edges which are principal curves. The third degree of design-freedom is in the way we can choose a configuration of these voxels, for example, the chosen shell-slice can be seen as a two-dimensional assembly of these voxels, as seen in Fig. 10(b). This generalizes the assemblies on based revolution surfaces and normal offsets, seen in [10].

Fig. 10.
figure 10

a) One voxel. (b) Two-dimensional assembly. (c) One-dimensional assembly.

Full size image

Three-Dimensional Assembly of Voxels

It is important to note that by construction of the TOS, this degree of design-freedom can be extended in three directions in the sense of three-dimensional assemblies, as seen in Fig. 11(b).

Fig. 11.
figure 11

a) Discrete number of (R, G, B)-sheets. (b) Three-dimensional assembly of voxels.

Full size image

3.4 Choice of Rational-Voxel Type

Since a voxel has six sides all of which are principal patches then from discussion above, we can then rationalize the voxel into one whose faces are either planar or developable. The resulting rationalization of the voxel is a what we call the ‘rational-voxel’ and the choice of planar and/or developable faces in the voxel is the fourth degree of design-freedom, as seen in Fig. 12. We will call a rational-voxel with:

  1. (1)

    Only planar faces a P-voxel, its assembly is called a P-assembly.

  2. (2)

    Only developable faces a D-voxel, its assembly is called a D-assembly.

  3. (3)

    Both planar and developable faces a PD-voxel, its assembly is a PD-assembly.

Fig. 12.
figure 12

Rationalization of the voxel into the rational-voxel.

Full size image

Note that the choice of planarity or developability of the faces of the rational-voxel determines its continuity with its neighboring voxel, as seen in Fig. 13. In particular, the case of a D-assembly the results in series of developable strips amenable to feasible fabrication techniques seen for instance in [11].

Fig. 13.
figure 13

P, PD and D-assemblies with the continuity with neighbors indicated.

Full size image

3.5 Choice of Solid or Hollow Voxel-Assembly

The fifth degree of design-freedom is the choice of solid (shell) or hollow (grid-shell), as seen in Fig. 14. Illustrating the some of the design possibilities arising from the five degrees of design-freedom.

Fig. 14.
figure 14

Matrix of some of the prototypes based on the five degrees of design-freedom.

Full size image

4 Realized Prototype

Next, we present a realized architectural prototype based on the design model above (Fig. 15).

Fig. 15.
figure 15

The final realized realized pavilion.

Full size image

A. Form:

We started by manipulating the five degrees of design-freedom. This has allowed us to explore different architectural interpretations. Next we narrowed them down to the following proposals seen in Fig. 16 (P-assembly Hollow).

Fig. 16.
figure 16

(1) G, B, R-P-assembly solid. (2–3) G, B, R-P-assembly hollow.

Full size image

From these proposals we decided on proposal (2B): the B-P-assembly hollow (version 1) as a base form upon which some adjustments will be made taking into account the size of the object, its spatial quality, and the needs of the project, as seen in Fig. 17.

Fig. 17.
figure 17

Adjustments made to the B-P-assembly hollow version 01.

Full size image

Once the exact final form was finalized, we made a 1:10 physical model in cardboard to test the form and the box-folding technique (Fig. 18).

Fig. 18.
figure 18

1:10 cardboard model of the final form.

Full size image

B. Material:

As we mentioned above, the choice of the P-assembly was fixed earlier on given the material we chose (27 sheets of plywood of 2.5 m length x 1.2 m width x 5 mm thickness) where we considered it to be mostly adapted to realizing planar surfaces given its limited bending capacity.

C. Fabrication and assembly:

The advantage in working with P-assembly is that we were able to realize all the boxes (P-voxels) simply by folding flat pieces of plywood. These pieces were cut using the 3-axis milling and then folded as shown in Fig. 19.

Fig. 19.
figure 19

The folding technique employed for the assembly of the boxes.

Full size image

Once the boxes were assembled, they were then grouped into partitions according to their position in the structure (Fig. 20).

Fig. 20.
figure 20

The grouping of the assembled boxes per partition.

Full size image

The boxes of each partition were bolted together, then the partitions themselves were bolted together to form the final assembled structure as shown in Fig. 21. Notice the use formwork to maintains the levels of the partitions while they’re being joined and the footings fixed to the blocks.

Fig. 21.
figure 21

The assembling of the partitions into the fully assembled model.

Full size image

5 Conclusion

In this paper, we gave an exposition of a TOS showing its potential as a mathematical basis for generating design models. We showed that the geometric properties of the principal patches constituting the TOS can be used as tools to generate a large variety of architectural prototypes realized as assemblies of rational-voxels made from planar and developable surfaces. To end the paper, we would like to mention two ways for furthering this work; the first being, exploring the three-dimensional assemblies, in the sense of going into ‘volumes’ and not just shells. Next, it could be seen that the TOS presented here is to certain extent limiting in terms of architectural form, because of its symmetry and geometric logic. Thus, an interesting continuation for us, would be to explore certain deformations of TOS (and principal patches) that preserve the principal networks. Hence, allowing for more freeform shapes while maintaining the geometric properties presented here and their advantages for fabrication.