Much of the interest in cyclidic patches comes from their structural properties. In the in- troduction to this Chapter we discussed some of these benefits. First, as the mappings are conformal, all local angles are preserved, which eliminates shearing. This results in torsion- free nodes which facilitates physical construction [16, 97]. In computer graphics, this same property helps preserve texture and mesh detail under deformation [31]. Second, as patches are built with circles rather than lines, they provide curvature discretizations which converge more quickly to the continuous case. Third, both of the first two properties are shared by offset surfaces along the normals, and are in fact generalizable to an n-D smooth orthogonal coordinate system for working with differential geometry [17].
As a result, parameterizing surfaces and volumes with an orthogonal curvilinear sys- tem can facilitate their deformation and construction. Such designs can come to fruition as freeform architectures covered in flat glass, where offsetting along normals to the surface is critical, and where the ability to manufacture a single 90 degree node reduces costs. The ability to blend between shapes in a precise way is also useful in machine-milling of small parts, where oddly shaped curved pieces are often necessary [52, 86]. Conformal lattices can provide a robust way to animate virtual forms.
C1smooth surfaces are built from a series of patches in Figure 6.11. A new patch is added to an existing edge by specifying an additional tangent sphere. The two additional tangent frames necessary to define the second patch are then found using the procedure outlined in Algorithm 6.1. This process can also occur in a direction orthogonal to the first patch, as shown in Figures 6.12 and 6.13. In this way a 3D volume can be described, as demonstrated by Bobenko and Huhnen-Venedey in [17], where it is shown that three patches are sufficient to define a six-sided volume, orhexahedron. Here we show how such a system can be used to warp a mesh in a way similar to the trilinear interpolation of motors demonstrated in Figure 5.4.
Given a hexahedron of patches arranged as six sides of a warped cube, the technique for specifying a point at coordinate (x,y,z) with x,y,z 2 [0,1] requires transforming the normal directionswcoordinate surfaces on opposite sides of the cube and then finding a new rotor
from their ratio. The complete steps are enumerated in Algorithm 6.2 and its application to finding the transformed positions of a warped sphere in Figure 6.14.
Algorithm 6.2 Calculating A 3D Conformal Coordinate
1. Given a hexahedron (six-sides) of surface patches, pick the two opposing patches at w = 0 and w = 1.
2. Calculate the Cuxrotor of each of the two opposing patches using Equation 6.3.
3. Apply these rotors to the constant coordinate osculating surfacessw0
u0 andsuw01 of each
corresponding patch. Call these transformed two spheressw0
ux andsuwx1.
4. Generate a transformation from the ratio of these two transformed surfaces as Cwz =
e 12zlog([suxw1/suxw0]normalized).
5. Multiply this transformation by Cu,v, the bilinear rotor of the w = 0 patch, found using
Equation 6.5.
6. The result, CwzCux,vy can be applied to the point p(0,0,0)to find its transformed position
in the 3D conformal coordinate grid.
6.5 Discussion and Future Work
We have introduced a new method of rationalizing cyclidic nets by composing conformal transformations from orthogonal coordinate surface pairs. Our technique emerges from the orthogonal decomposition of general conformal transformations uncovered by Dorst and Valkenburg in [46] and detailed further in [43]. We use concircularity as a constraint to ensure our bivector point pairs are well-chosen. Constructing spheres with the outer product of points and bivector tangents, we touch base with contact geometry and the curvilinear co- ordinates explored by Bobenko and Huhnen-Venedy in [17]. To develop continuous blending
Figure 6.11: C1smooth cylidic net surfaces constructed from a series of principal patches.
Figure 6.12: Smooth surfaces can break a corner orthogonally to form nets along a normal. Three sides are sufficient to uniquely define the remaining three necessary to define a hexa- hedron volume.
Figure 6.13: Circular nets can be extended in a third direction to discretize volumes.
techniques we rely on the orthogonal plunge of circle through two coordinate surface spheres. In contrast to matrix-based rationalization techniques, the incorporation of spinor theory into geometric algebra allows us to generate transformations through ratios of the coordinate sur- faces themselves. To help with this, we use a local 6-sphere coordinate system to encode curvature in every possible direction.
By using geometric algebra we are able to discretize smooth shapes in a way that preserves the two goals of discrete differential geometry proposed by Bobenko and Suris in [18]: the transformation group principle (wherein discretizations and their smooth counterparts trans- form the same way) and the consistency principle (whereby constructions can be extended to higher dimensions). Because of the structure-preserving nature of conformal transformations within the CGA mechanism, our composed transformations are as capable of operating on tangents and normals as well as points, greatly simplifying basic calculations in differential geometry. We suspect that approaching differential geometry by careful study of integration of conformal mappings will help in developing the discrete geometric calculus within the conformal model. In our treatment, our frames are already orthogonal – a condition which can be applied to non-orthogonal frames using the reciprocal construction of geometric cal- culus. Hestenes’ writings are, as usual, a good place to start this mapping [68, 72, 75] as is Sobczyk’s simplicial calculus [125] and the rich literature on discrete exterior calculus [31, 32]. We would like to more carefully consider the relationship between the rotors that transforms these spheres across a surface patch and the shape tensor and shape bivector or curl, to better pin-point the pair generators that most clearly and generally match Hestenes’ definition of the shape bivector as the “angular velocity of the pseudoscalar as it slides along the manifold” [75]. Explicating such relationships will give more space for a discrete calculus to form.
A good next step in our formulation will be to analyze an input simplicial surface and try to find its closest conformal representation through piecewise integration of cyclidic patches. It is possible that, in order to fully articulate a discete differential geometry, we decide to
move into the space of R4,2to allow for geometries other than Möbius, in particular those of
Laguerre. In [18], Bobenko and Suris construct discretizations using Lie Sphere Geometry, of which the Möbius geometry of Rn+1,1 is one subset and Laguerre geometry the other. In
[97], Liu et al use a Laguerre-based approach to discretization based on conical rather than circular quad meshes. Recently, Krasauskas has used Laguerre transformations in the geo- metric algebra model of R4,2 to solve hole-filling problems [86]. As opposed to the Möbius transformations which preserve points (spheres of zero radius), transformations Laguerre ge- ometry preserves hyperplanes (spheres of infinite radius). Their combination, Lie Geometry, preserves oriented contact. The surfaces that can be carved out in these other geometries include Darboux cyclides, more general versions of the Dupin cyclides depicted here. Thus having seen the power of rationalization using Möbius transforms, and the demonstrations of Laguerre transformations in the aforementioned texts, we might consider moving to the more encompassing Lie Sphere Geometry. Leo Dorst has also suggested potential advan- tages to working with the Lie Sphere system of oriented contact, for instance for matching broken pieces of pottery. Luckily, with the universal geometric algebra, one could conduct such experiments by extending the constructive methods developed here.
Chapter 7
Conclusion: A Mechanism for Design
7.1 Summary
The current work began as a simple question – how can we design structures with geometric algebra? The investigation of this question has unfolded into an operational strategy for spatial composition on a computer: a synthesis of form through transformation. Felix Klein’s identification of transformation as the central characteristic defining a space is honored by the model of space adopted here. Geometric algebra allows us to practice the processes that constitute spatial systems. With enough practice, these processes can lead to a unified methodology for structural design.
To advance this methodology, we have developed techniques for encoding three proper- ties of spatial structures – symmetric, kinematic, and curvilinear. Examining transformation (reflecting, folding, twisting, bending, and knotting), distance constraints (round incidence relationships), and rationalization (logarithms of ratios) we have implemented a collection of constructive geometric design techniques with a single unifying spatial computing engine. This relationship between technique, parameter, property, and structure is outlined in Table 7.1.
Technique Transformation Incidence Tensor Parameter Groups Constraint Differential
Property Symmetry Kinematics Curvature Structure Balanced Deployable Freeform
Table 7.1: The relations between techniques, parameters, properties, and structures that have been developed in this text.
of thought, first establishing relationship between the geometric product and the associated algebraic concept of inversion and the fundamental spatial concept of reflection. We then explored how – from from that algebraic-spatial connection– the catalysis of mathematical invention generates a coherent system of relationships between geometric numbers. With these relationships thus crystallized, we explored how they can be used to place constraints on each other in order to articulate position and orientation in moving structures. Finally, to demonstrate the elegance achieved, we offered a direct method for rationalizing curved sur- faces via a tensor product of sphere ratios, and illustrated a method for conformally warping a mesh using this discretization of curvature.
The three spatial systems selected were not chosen at random. From start to finish we generated a pedagogy for practice: the notion of a screw displacement as a rotation and a translation discussed at the end of Chapter 3 is used to model kinematics in Chapter 4. The logarithm of such motors are used to introduce a method interpolation in Chapter 4, and the application of such interpolation methods onto general bivectors reveals further forms. In Chapter 6 these are regimented into the deliberate construction of curvilinear surfaces and volumes. A single mathematical model of 3D space has proven adequate for designing a range of spatial configurations. We have not only shown how individual expressions can be used to generate structures, but the processes by which such expressions can be extrapolated. Using a comprehensive language of space, our processes can evolve intuitively; further forms suggest themselves from current manipulations.
Within each Chapter, we argued that the production of geometric expressions enables a novel way to design structures. Applications of our synthesis techniques include machine
parts, tensegrity systems, deployable structures, and freeform architecture, bringing the theo- retical underpinnings of our models one step closer to real-world manufacture. By visualizing this language of space, we have encouraged the adoption of these algebraic methods by dig- ital practitioners. We find the expressive powers of the algebra make it an essential tool for those who seek to develop their spatial thinking.