Geometry plays an essential role in the fields of physics and engineering. Its importance extends beyond theoretical studies to shape the design, simulation, and manufacturing of products. As such, the way geometry is represented has profound implications on how we approach and solve engineering problems. Today, this representation is no longer confined to traditional methods; it is driven by powerful computational tools that enable greater precision, speed, and versatility in design and simulation processes.
In this article, we explore the evolution and significance of geometry representation in modern engineering, highlighting key technologies, applications, and the challenges ahead.
The Importance of Geometry Representation Today
Geometry is integral to every phase of engineering, from concept to design, testing, and production. As technology continues to advance, new methods for representing geometry are constantly emerging, reshaping engineering practices. The ability to represent and manipulate geometry with high precision is crucial not only for product development but also for optimizing engineering designs, ensuring efficiency, and accelerating innovation.
Applications of Geometry Representation in the Computer Age
With the advent of computers, the representation of geometry has moved beyond the limitations of hand-drawn sketches and manual calculations. Today, various applications leverage computational tools to create, manipulate, and simulate geometry, pushing the boundaries of what’s possible in engineering.
Engineering
Before the advent of CAD, engineers relied on pencil and paper, along with specialized drafting tools, to create designs. Early CAD systems in the 1970s and 1980s aimed to digitize these manual processes. By the 1990s, the concept of parametric CAD was introduced, transforming design practices. Parametric modeling, which is now the standard in nearly all engineering design software, allows engineers to create models that are based on parameters that can be adjusted to produce different design variations.
Computer-Aided Design (CAD)
CAD, which emerged in the 1960s, revolutionized engineering and architecture by replacing manual drafting with digital tools. Over time, its use has expanded beyond architecture and engineering to other domains like visual effects (FX) and gaming. The increasing variety of fields employing CAD tools has resulted in the development of specialized features tailored to specific applications. Consequently, new types of geometric representations have emerged to meet the unique needs of each domain.
Parametric 3D Modeling
Parametric 3D modeling, a key aspect of modern CAD software, uses a feature-based design approach. Engineers start with a basic sketch and apply features such as extrudes, lofts, and sweeps to gradually build up a 3D model. The beauty of parametric design lies in its use of constraints and relationships connected through a dependency tree, allowing designers to create models that are flexible and modifiable. By organizing these geometric operations into a dependency tree, parametric design enables a seamless flow of edits from one design iteration to another. This step-by-step approach ensures that every change is recorded and can be easily adjusted.
Parametric modeling also introduces significant computational benefits. For example, a simple geometry like a cylinder can be fully described by just two parameters — diameter and length — making it an efficient representation. More complex geometries are similarly reduced to a set of parameters and relationships. Interestingly, this type of representation has even found applications in machine learning (ML). For example, Gaussian process models can predict quantities, such as the drag coefficient of an object, based on a set of design parameters.
Internally, parametric design software typically uses boundary representation (B-rep) to describe geometries. B-rep represents shapes through their boundaries in 3D space, which consist of faces, vertices, and edges. The relationship between these entities, encoded in mathematical formulas, enables a software kernel to transform user input into the 3D models that are displayed on the screen.
The major players in parametric design — Siemens, Autodesk, PTC and Dassault — rely on powerful CAD kernels like Parasolid and ACIS to handle these geometric representations, offering robust solutions for a range of industries.
Limitations of Parametric Design
While parametric design is a powerful tool, it is not without its limitations. Its precision-based approach works well for engineering geometries, where features can be broken down into mathematically defined elements. However, this method struggles with modeling more organic shapes, such as those found in nature. For instance, modeling complex curves and irregular surfaces can be difficult with parametric constraints alone.
To overcome these challenges, many CAD tools have combined parametric design with other modeling techniques. Programs like Rhino, Alias, Blender, and Maya, for example, use non-uniform rational B-splines (NURBS) and polygonal modeling, which offer greater flexibility for capturing freeform shapes.
Alternatives to Parametric Design
While parametric design is a dominant paradigm in engineering, it is not tied to a single modeling approach. Many CAD systems combine parametric workflows with different geometric representations, each offering distinct advantages. For example, NURBS and CSG models can be part of parametric systems, and even polygonal or implicit models can be manipulated with parametric tools to some extent. CAD software is often categorized by the core technology it uses to represent geometry, which influences how models are built, edited, and constrained.
- Constructive Solid Geometry (CSG): CSG is a method that combines simple shapes using Boolean operations (e.g., union, difference, intersection). It’s used in tools like OpenSCAD and AutoCAD.
- NURBS: These mathematical representations of curves and surfaces are widely used in programs like Rhino and Blender. While NURBS can be used alongside BREPs, they are distinct in their ability to handle complex curves smoothly.
- Polygonal Modeling: Used in applications like Maya and Blender, polygonal modeling relies on mesh representations made up of vertices, edges, and faces, which can be combined to form 3D shapes. Here, the modeling tools resemble a sculpting workbench, and shape creation is much more freeform, unconstrained, and dynamic, which allows users to morph and deform meshes.
- Implicit Representations: Technologies like signed distance functions (SDFs) and level set methods use mathematical functions to describe shapes implicitly, providing another layer of flexibility for complex designs. Here, users can create structures like gyroid lattices where organic patterns can be created across large volumes.
Beyond these approaches, the choice of modeling method often depends on the specific needs of the industry, whether it's engineering, industrial design, or entertainment.
Beyond Design: Geometry Representation in Simulation
While geometry representation plays a central role in design, its importance extends into simulation, where engineers assess the viability of their designs. Simulation allows engineers to test how a design will behave under real-world conditions, such as stress, heat, and motion.
One key aspect of simulation is discretization, a process where continuous geometry is converted into discrete elements for computational purposes. This process, known as meshing, is essential because many physical problems cannot be solved analytically. Instead, numerical methods such as finite element analysis (FEA) approximate the solution by dividing the geometry into smaller, manageable pieces. Meshing enables the simulation of complex behaviors that would be difficult to model with analytical methods alone.
Looking Ahead: Geometry Representation at the Intersection of Engineering and AI
As artificial intelligence (AI) continues to evolve, it will undoubtedly play a significant role in how geometry is represented and manipulated in engineering. AI can potentially optimize engineering workflows by reducing the time and cost associated with simulation and geometry creation.
The AI Challenge
AI presents a unique opportunity to augment traditional engineering design processes. By improving simulation speed and geometry generation, AI can help engineers explore more design variations in less time, leading to more efficient optimization strategies. However, the traditional CAD and simulation systems were not designed with machine learning (ML) in mind. Most of the tools used for digital geometry representation are proprietary and closed-source, which limits the extent to which AI can be integrated into existing systems. Overcoming these barriers is key to unlocking AI’s full potential in engineering.
Innovation vs Intuition vs Integration
Integrating AI into engineering workflows involves balancing three key factors:
- Innovation: AI can generate novel designs that push the boundaries of traditional methods, but it needs to do so without violating established engineering principles.
- Intuition: Engineers must trust the results of AI models. Although innovation is critical, AI-generated designs must also align with expert intuition and existing knowledge in the field.
- Integration: AI-driven designs must seamlessly integrate into existing workflows, including CAD, simulation, and manufacturing. The complexity of geometry representation may hinder AI’s ability to work effectively unless it can operate within these established frameworks.
For AI to truly drive innovation, it must enable the exploration of novel designs — even those that push or outright violate traditional constraints. This kind of boundary-pushing is where the most groundbreaking ideas often emerge. However, it’s also where trust becomes critical. Navigating this space requires a delicate balance: challenging conventional limits while still earning the confidence of domain experts who must evaluate and ultimately adopt these results.
Our latest work with the Large Geometry Model (LGM) demonstrates what’s possible when that balance begins to take shape. By stepping beyond the confines of parametric CAD, LGM breaks free from the rigidity of design specificity, opening up a far broader and less biased design space. This allows users to explore geometry in ways unshackled from traditional, incremental methodologies. Furthermore, LGM sidesteps the limitations of mesh-based representations — namely their high dimensionality, which inflates compute cost — by leveraging signed distance functions (SDFs) to encode geometry within a latent space. This shift enables more powerful optimization strategies, capable of escaping local optima and unlocking designs that are not only novel but previously unreachable using conventional tools.
Parameters Still Matter
While the LGM approach offers exciting freedom from the limitations of traditional parametric design and shows great promise in driving innovation, its real-world adoption hinges on more than just performance. Like any AI tool in a high-stakes domain, success depends on explainability, interpretability, and manufacturability. A model that produces breakthrough designs and outperforms existing ones across key metrics is undoubtedly impressive, but if that design is too complex to manufacture or fails to meet essential engineering constraints, its practical value quickly diminishes.
The real challenge — and opportunity — lies in maintaining the innovative power of models like LGM while staying grounded in the realities of engineering. Bridging this gap means aligning AI-generated outputs with the constraints engineers are accustomed to working with. Research efforts such as BrepGen point in this direction, suggesting that decoding LGM outputs into boundary representations (B-reps) structured similarly to parametric models could provide a vital link. Achieving this would bring us significantly closer to integrating cutting-edge generative approaches with the structured, constraint-driven design methods that underpin real-world engineering workflows.
Conclusion
Geometry representation is at the heart of modern engineering and continues to evolve with the advancement of computational tools. From traditional CAD systems to AI-powered innovations, the way we represent and manipulate geometry is undergoing a transformative shift. While the future holds immense promise, the integration of AI and machine learning into engineering design will require careful consideration of existing workflows, design constraints, and manufacturing capabilities. As we move toward more intelligent and automated systems, the role of geometry representation will be pivotal in shaping the future of engineering and design.