What Happens When 300 Applied Geometers Meet in Italy?

By Jorg Peters

The 2023 SIAM Conference on Computational Geometric Design (GD23) took place in July and was paired with the International Geometry Summit 2023 (IGS’23) in Genova, Italy — bringing together more than 300 researchers in applied geometry, geometric modeling and processing, solid and physical modeling, computational geometric design, and shape modeling and analysis. The meeting attracted a broad range of attendees and sponsors from industry, research centers, professional societies, and governmental bodies. Sessions and lectures were held in the University of Genoa’s lecture halls, which are housed in a collection of restored ancient monasteries and medieval churches.

GD23 was co-located with several other conferences that focus on similar research areas: artistic geometric tools at Shape Modeling International, manufacturing tools at the Symposium on Solid and Physical Modeling, geometry processing tools and shape collections at the Symposium on Geometry Processing, and simulation tools at the International Conference on Geometric Modeling and Processing. Each conference explored a different approach to enhancing our understanding of the geometric basis for computation. They all promoted the development of geometric tools in applied mathematics and computation.

The SIAM Activity Group on Geometric Design (SIAG/GD) specializes in geometric and topological tools that enable algebraic, differential, and discrete computations, as well as industrial applications in design, analysis, manufacturing, biomedicine, geophysics, and digital entertainment. While talks in the other conference tracks summarized pre-refereed, highly competitive proceedings, GD23 opened the floor for the newest results in the form of minisymposia, contributed talks, and posters. A number of these novel results are currently being collected for a special journal issue.

The invited talks at IGS’23 and GD23 reflected the diversity of both the theory and application of geometric computational tools:

• “The Character of Computer Animation Research,” Theodore Kim (Yale University)
• “From Alpha to Voronoi,” Sara McMains (University of California, Berkeley)
• “Structure Preserving Discretizations: Towards Generalizations of Conformal Maps and Circle Patterns,”  Alexander Bobenko (Technische Universität Berlin)
• “Learned Parametric 3D Shape Models,” Angela Dai (Technische Universität München)
• “Surfaces: Modelling and Analysis,” Jorg Peters (University of Florida)
• “Structured Mesh Generation,” Xianfeng David Gu (State University of New York at Stony Brook)
• “Spatio-semantic Modeling for Urban Digital Twins,” Thomas Kolbe (Technische Universität München)
• “Monte Carlo Geometry Processing,” Keenan Crane (Carnegie Mellon University)
• “Robust Geometry Processing for Physical Simulation,” Daniele Panozzo (New York University)
• “Sorting in Space,” Hanan Samet (University of Maryland).

Martin Skrodzki (Delft University of Technology) received the SIAG/GD Early Career Prize for his work on three-dimensional scanning, point clouds, denoising, and visualization.

Additionally, GD23 featured nine minisymposia with speakers from industry and academia:

• “Paul de Faget de Casteljau, A Pioneer in CAGD: His Work and Some Direct Consequences”
• “Isogeometric Modeling”
• “Subdivision and Refinability”
• “Algebraic Spline Geometry”
• “Applied Geometry, Rational Motions, and Pythagorean-hodograph Curves”
• “Industrial Applications of Robust Geometry Processing”
• “Deep Learning for Geometric Design”
• "Analysis, Design, and Manufacturing Using Microstructures”
  
• "Learning, Processing, and Optimising Shapes.”

In the context of the minisymposium on Paul de Casteljau’s legacy, Figure 1 depicts an example of the efficient and elegant representation that de Casteljau pioneered [2]. His Surfaces à Pôles are widely known as Bézier surfaces: a sibling of the equally ubiquitous B-spline representation [1, 3]. These polynomial representations have a surprising simplicity and elegance, which offers an approach that tackles difficult geometric problems by connecting discrete and continuous computations. For example, the black circles in Figure 1 constitute the four coefficients of a planar curve piece \(p\) of degree 3 in de Casteljau’s representation. Practitioners can interpret the diagram in many ways:

• As an evaluation of \(p\) at \(t\) by repeated linear interpolation, which generates the blue, green, and red points in order
• As a splitting (subdivision) of the curve segment into two pieces; the sequence of black, blue, green, and red points defines the left piece \(p_l\) and the reverse sequence defines the right piece \(p_r\)
• As a proof that this splitting generates the same curve \(p\)
• As an evaluation of a point outside of the domain of \(p_l\), namely to generate the rightmost black point by the same process that was applied to \(p\) in the first bullet
• As the construction of the derivative polynomial, since the blue and green edge vectors are obtained by the same repeated linear interpolation as the points
• As the rules for matching two curve pieces with \(C^3\) continuity (\(p_l\) and \(p_r\)).

That is, one can easily read off the stable evaluation of positions, derivatives, and even differential geometric properties — such as the tangent and osculating plane for spatial curves and normals for surfaces.

One of SIAG/GD’s key contributions to the SIAM community is its attention to overcoming the constraints of the grid structure of tensor-product splines. Modern splines adapt to intricate computational domains and spaces, such as the smooth piecewise polynomial handlebar grip in Figure 2. This mastery of complex shapes complements the ongoing push to tighten the design and analysis loop. Recent progress has allowed researchers to compute directly on planar, volumetric, or even curved surface domains without introducing additional meshing or switching to a different representation than a geometrical one. This technique yields a much tighter and more accurate optimization cycle.

Smart integration with industry standards and finite element tools is an ongoing process. Correspondingly, several GD23 minisymposia featured locally refined splines for localized adaptive refinement and generalized subdivision surfaces. The goal of isogeometric approaches for design and analysis is the creation of generalized splines that both describe the geometry of surfaces or volumes and solve partial differential equations on these irregular geometries. Figure 3 illustrates a polyhedral-net defining smooth geometry with uniform highlight lines across irregularities, as well as the use of the same polyhedral-net spline space for the computation of a heat field on this geometry in real time. The design’s emphasis on suitability for engineering analysis is complemented by novel approaches to structured mesh generation for spline-based analysis that leverage Abel-Jacobi theory and recent efficient techniques to work with “trimmed” tensor-product splines whose domains are restricted by curves. These new tools will help industrial and applied mathematicians reach beyond the unit square or unit cube for modeling and computation.

Johannes Kepler’s dictum “ubi materia, ibi geometria” indicates the breadth and importance of geometric representation. While researchers are currently incorporating learning-based approaches into the field, these are far from dominant when it comes to representing or designing geometry. Machine learning techniques for images most directly carry over to level sets of trivariate distance fields that are described by neural networks. And although classic geometric design encompasses both implicit and parametric representations—indeed, one subfield specializes in exact or approximate conversions between the two—practice has long favored parametric object representations (such as triangulations or higher-order splines) over algebraic, implicit representations. The presentations at GD23 and its co-located meetings addressed these and other trends in the field, and we can look forward to witnessing the continued evolution of computational geometric design.

Acknowledgments: The author thanks all GD23 participants who contributed to this article.

References 
[1] Farin, G. (2001). Curves and surfaces for computer-aided geometric design: A practical guide (5th Ed.). Burlington, MA: Morgan Kaufmann.
[2] Mueller, A. (2023). Paul de Casteljau: The story of my adventure. SSRN. Retrieved from https://ssrn.com/abstract=4497291.
[3] Prautzsch, H., Boehm, W., & Paluszny, M. (2002). Bézier and B-spline techniques (Vol. 6). In Mathematics and visualization. New York, NY: Springer-Verlag.

Jorg Peters is a professor in the Department of Computer and Information Science and Engineering at the University of Florida. He is interested in representing, analyzing, and computing with geometry and has developed new tools for free-form modeling and design in spline, Bézier, subdivision, and implicit representations.