|
" Geometric Problems that take possesion of me "
The lecture deals with my way of doing research: It often starts with getting stimulated by an article or a lecture at a conference. Putting one of its details in another context, mostly a projective geometric one, and then, by looking for meaningful generalisations, it is finally possible to extract a general principle which brings together divergent and often well-known facts. The resulting aha moments are almost addictive. Examples of such topics will be “polyhedrons made by setsquares”, “C^r-biarcs of circles and conics”, “Stellae octangulae”, “Frégier’s theorem”, “Minkowski Geometry resp. metric planes”. There remain still many open questions, which could be perhaps answered by researchers having another scientific background than mine. Unfortunately the mentioned topics are not within the actual scientific mainstream, but their inner beauty justifies to be curious and to struggle for getting answers to open questions.
|
|
" Perception-Aware Computational Fabrication "
Haptic and visual feedback are important for assessing objects' quality and affordance. One of the benefits of additive manufacturing is that it enables the creation of objects with personalized tactile and visual properties. This personalization is realized by the ability to deposit functionally graded materials at microscopic resolution. However, faithfully reproducing real-world objects on a 3D printer is a challenging endeavor. A large number of available materials and freedom in material deposition make exploring the space of printable objects difficult. Furthermore, current 3D printers can perfectly capture only a small amount of objects from the real world which makes high-quality reproductions challenging. Interestingly, similar to the manufacturing hardware, our senses of touch and sight have inborn limitations given by biological constraints. In this talk, I will demonstrate how we can leverage the limitations of the Human Sensorial System to increase the apparent gamut of a 3D printer by combining numerical optimization with perceptual insights. Instead of optimizing for exact replicas, we search for perceptually equivalent solutions. I will show applications of the proposed methodology to designing objects with prescribed compliance, mimicking the haptics of drawing tools, and manufacturing objects with spatially varying gloss.
|
|
" Geometry and Tool Motion Planning for Curvature Adapted CNC Machining "
CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between “dual” classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software.
|
|
" Aktuálny odkaz Euklidových Základov "
Príspevok uvádza krátky prehľad obsahu trinástich kníh kardinálneho Euklidovho diela Základy (Stoicheia) z hľadiska dnešnej klasifikácie matematických odborov. Dielo je po prvý raz v úplnosti pripravené do tlače v slovenskom preklade autora príspevku a je opatrené stručným komentárom približujúcim obsah diela v jazyku dnešnej matematiky.
Dôležitým zámerom príspevku je poukázať na niekoľkovrstvový význam Euklidovej odbornej terminológie, ktorý z pohľadu dnešných zásad logiky a presnosti odborného jazyka vyžaduje hlbšie zásahy do tradičnej rutiny v používaní matematickej terminológie v písomných prejavoch, najmä v tvorbe učebníc, i v dennej didaktickej praxi.
Druhým významným trvalo aktuálnym odkazom Euklida, skryto prestupu-júcim celým jeho dielom, no osobitne Základmi, je zdôrazňovanie sústavnej primeranej didaktickej axiomaticko-deduktívnej metódy vo vyučovaní matematiky (a nielen matematiky).
|