Geometrical models play a critical role in diverse fields such as engineering design and analysis, manufacturing automation, computer graphics and animation, architectural and environmental planning, computational fabrication, and medicine and biomedical technology. The field of computer aided geometric design (CAGD) arose in the 1960s from a desire to supplant two-dimensional drawings and three-dimensional (3D) sculpted models—which had previously sufficed to specify complex shapes—with precise digital representations based on novel mathematical formulations.
Explosive growth in the speed and memory of digital computers has fueled substantial progress regarding basic mathematics, efficient algorithms, and adoption of CAGD methods in the aforementioned applications through both commercial software packages and open source algorithm libraries. Nevertheless, fundamental mathematical challenges inhibit further progress in CAGD.
The SIAM Activity Group on Geometric Design was founded in 1989 to tackle these challenges. It provides an interactive environment that brings together researchers and practitioners from academia and industry at the biennial SIAM Conference on Computational Geometric Design.
In CAGD, a very high premium is placed on the accuracy and robustness of geometrical models, as they are typically employed as input to subsequent applications (finite element analysis, surgical planning, manufacturing processes, etc.), and model inaccuracies or inconsistencies generally incur failure of the applications. As in other fields of applied mathematics, the range of mathematical problems of practical interest that admit exact solutions is quite limited in scope, necessitating the use of approximate and computational solutions.
There have been two independent evolutionary threads in the specification of geometrical models, motivated primarily by distinct application domains. Engineering design places great emphasis on the use of analytic or piecewise-analytic (spline) geometries, which satisfy demanding functional requirements like surface smoothness and precision tolerances of mating parts. On the other hand, mesh models (i.e., surfaces tessellated by triangular facets) are preferred in representing natural or “amorphous” shapes, such as computer-animated figures or human organ models employed in surgical or radiation therapy. The mesh model approach offers great robustness and topological flexibility but often at the expense of huge file sizes, and in most instances does not satisfy critical engineering precision requirements (e.g., for aerodynamic surfaces).
Accurate and efficient porting of geometrical models to applications remains a key challenge in their optimal utilization. Over the past decade, the field of isogeometric analysis (IgA) has generated considerable interest and activity. One of the original goals of IgA was to directly exploit 3D geometrical models in finite element analyses of stress, fluid flow, heat transfer, etc., rather than immediately replacing them with 3D mesh approximations. Due to the complexities associated with “trimmed” surfaces—parametric surfaces with complicated parameter domains defined by their intersections with other surfaces—the focus has instead drifted to adoption of the Bézier/B-spline bases employed in geometric modeling as “shape functions” for finite element analyses.
Similar limitations prevail in manufacturing. In computer numerical control (CNC) machining, the precise part model is typically replaced by voluminous piecewise-linear/circular (G code) toolpaths that limit the accuracy, speed, and smoothness of the tool relative to the workpiece (see Figure 1). Likewise, in 3D printing, the precise part model is first replaced by a surface tessellation known as an STL file (the acronym is derived from stereolithography, the original 3D printing process). The printer must slice the STL model into hundreds or thousands of parallel planes and generate area-covering paths for each slice. G-code part programs and STL files are universally applicable but can incur huge data volumes and loss of accuracy, efficiency, and reliability. Although originally intended as stop-gap measures to facilitate CNC machining and 3D printing technology, they have become entrenched “industry standards” that are extremely difficult to dislodge.
Figure 1. Depiction of the toolpaths for computer numerical control machining of a swept surface (left), and the swept surface machined in aluminum directly from its exact procedural definition (right). Figure courtesy of .
Geometrical 3D models are also of paramount importance in physical simulations, since the physical behavior of an object is intimately tied to its geometry. Although discrete differential geometry helps to elegantly discretize physical effects, it is hard for computational tools to take problem- and material-specific constraints into account and correctly handle the highly nonlinear and global deformations that models experience during simulation (see Figure 2). While engineering applications emphasize physical accuracy, the key aspect in computer graphics is speed. During his keynote presentation at the 2017 SIAM Conference on Industrial and Applied Geometry in Pittsburgh, Pa., Tony DeRose of Pixar Animation Studios entertained attendees by sharing secrets of Pixar’s solution to this problem.
Such problems, especially at the interface of geometrical models and their intended applications, continue to make CAGD research an active and fruitful endeavor. The development of novel mathematical representations and algorithms—rather than just increased computing power—will be imperative to addressing these challenging issues.
Figure 2. Example of a nonlinear and physically realistic interpolation (yellow) between two given meshes (blue). Figure courtesy of .
The upcoming SIAM Conference on Computational Geometric Design (GD19) is part of the International Geometry Summit, which will feature four geometry-related conferences from June 17th-21st in Vancouver, Canada. This summit is a prime opportunity to interact with the CAGD community and learn about recent advances in the field. In particular, GD19 will feature exciting keynote presentations by Jessica Zhang (Carnegie Mellon University) on incorporating isogeometric analysis into existing software, such as Abaqus and LS-DYNA; Michael Bronstein (Università della Svizzera italiana (USI), Imperial College London, and Intel Perceptual Computing, Israel) on debunking “folklore” of spectral geometry regarding the behavior of manifold Laplacian eigenvalues and eigenfunctions; and Shahram Izadi (perceptiveIO) on the latest advances in virtual and augmented reality.
 Marras, S., Cashman, T.J., & Hormann, K. (2013). Efficient interpolation of articulated shapes using mixed shape spaces. Comp. Graph. Forum, 32(8), 258-270.
 Nittler, K.M., & Farouki, R.T. (2016). A real-time surface interpolator methodology for precision CNC machining of swept surfaces. Int. J. Adv. Manufact. Tech., 83, 561-574.