Mathematician from the FAV UWB seeks faster and more cost-effective methods of CNC machining

Accurate CNC machining depends on matching surface curvature, avoiding tool-material collisions, and controlling tool paths. Alena Lindauerová from the Department of Mathematics at the Faculty of Applied Sciences develops methods to improve machining precision while reducing costs.

In her research, the doctoral student combines mathematics and engineering. CNC machines are used to produce a wide variety of components made of metal, plastic, wood, and other materials. These components are then used, for example, in the automotive and aerospace industries or in healthcare as medical instruments or implants. Using mathematics, Alena Lindauerová is investigating how to manufacture such parts as precisely as possible—without errors and with the highest surface quality. For their published results, a panel of experts at the GMP 2026 conference awarded the entire research team second place in the Best Paper Award competition.

As a mathematician, what inspired you to focus on CNC machining?

My dad studied CNC machining, so I’ve been familiar with it since I was a child. My advisor, Michal Bizzarri, introduced me to the specific problem we addressed in the article during my final semester of my bachelor’s program. My bachelor’s thesis didn’t have a topic that could be built upon directly, and Michal offered me what was, at the time, a very appealing area for my master’s thesis. The opportunity to apply advanced mathematics to a real-world problem was a huge bonus for me.

What exactly do you focus on in your research?

Simply put, we’re investigating the best way to position a cutting tool on the material so that it conforms as precisely as possible to the target surface. In our award-winning paper, we focused on 5-axis CNC machining of a specific group of surfaces known as Dupin cycloids. These have the enormous advantage that we know their key geometric properties in advance. For general surfaces, however, we have to determine these properties through complex, computationally expensive calculations to set up the tool optimally.

Where are the weak points in current CNC machining technologies?

There are several challenges we’re addressing. The first is the tool’s tilt itself. In practice, only a constant tilt angle is often used, but this limits the resulting surface quality. We’re striving for a tilt that changes dynamically and perfectly follows the surface’s curvature. Another major challenge is tool-to-material collisions. We need the tool to reach even complex areas, but at the same time, the holder to which it is clamped must not collide with the material or cut off anything it isn’t supposed to. The directions of the tool’s movement themselves pose a challenge. Using mathematical calculations, we can find paths that more closely respect the geometry of a given product, thereby improving its quality and, in some cases, reducing production time.

Was the research also conducted under actual operating conditions?

In the award-winning paper, everything was carried out using computer simulations and purely mathematical calculations. However, we have recently applied for a joint grant with mechanical engineers from the Czech Technical University in Prague (ČVUT). While this topic is mathematically beautiful, real-world applications must take into account many more factors than just pure geometry. In actual production, the materials of the tool and workpiece, cutting speed, cooling, the machine’s behavior under vibration, and other physical quantities—which we as mathematicians do not primarily investigate—all play a role. It is precisely the combination of our theory with their practical perspective that could help us collaborate with the company.