Fundamental research is key
Indeed, many technologies we now take for granted are made possible because of fundamental research on methods for curves and surfaces. This later served as the foundation for applied research and development that has benefitted our societies.
For a general audience without in-depth knowledge of mathematical methods and its cutting-edge research, these questions might not be something we consider in our daily lives. We rather enjoy the technologies that come out at the other end and enrich our lives.
Gathering leading researchers
However, there are also people that are deeply invested and advancing this research. This summer, leading researchers are gathering at the International Conference on Mathematical Methods for Curves and Surfaces (MMCS) at the University of Oslo, June 26-28th.
This is not the first time many of them have met.
The Norwegian mathematician Tom Lyche from University of Oslo is part of the team organising the conference. Lyche in fact also organised the first conference in 1988 together with the American mathematician Larry L. Schumaker.
The International Conference on Mathematical Methods for Curves and Surfaces have historic roots, but fundamental research on curves and surfaces dates back much further.
Historic background – from ancient shipbuilding to computer-aided fingerprint processing
The history of curves and surfaces has evolved through ancient shipbuilding techniques to modern computer-aided design methods. Early developments in the field trace back to the Venetians, who perfected the use of curves in ship hull design between the 13th and 16th centuries.
Aeronautics also played a crucial role, with pioneers like Robert Liming establishing numerical methods for defining aircraft shapes in the 1940s. ?
Around the same time Iso Schoenberg and other Romanian and Bulgarian mathematicians developed the theory of B-splines. As the field progressed, spline curves evolved into B-splines and Non-Uniform Rational B-Splines (NURBS), providing a unified approach to curve and surface representation based on (piecewise) polynomials. ?
Innovations from Paul de Casteljau and Pierre Bézier at French automotive companies Citro?n and Renault led to the development of Bézier curves, which revolutionised how designers approached free-form curves and surfaces.?
At the same time, work by Even Mehlum and others at the Center for Industrial Research (now SINTEF) in Oslo led to a software system for the mathematical modelling of ship hulls. It was based on nonlinear spline curves consisting of pieces of Cornu spirals and straight lines tied together smoothly.
Parametric surface patches, both rectangular and triangular, enabled designers to define complex surfaces with precision. The introduction of subdivision surfaces further advanced the field, with methods like the Catmull-Clark and Doo-Sabin algorithms enhancing surface modeling. ?
Computer-Aided Geometric Design’s (CAGD) influence extended into scientific applications such as weather mapping and ship design, necessitating robust algorithms for data interpolation and approximation. The field also borrowed techniques from computational geometry and computer graphics, with triangulation algorithms becoming essential for data interpolation. Shape analysis, through techniques like curvature plots and fairing algorithms, became a vital aspect of design optimisation, reflecting the field's interdisciplinary growth and technological advancements.?
The field has had a major influence on a wide spectrum of applied science, ranging from data science to animation and robotic planning.
Indeed, police agencies around the world now use some of this technology to process fingerprints.?
Fingerprints fall into three main patterns: loops, whorls, and arches. Analysing these patterns, along with the ridge line patterns, requires significant computational effort. When experts analyse sets of fingerprints, they don't rely solely on visual inspection to spot similarities. Instead, they apply rigorous statistical analysis, grounded in mathematics, to draw their conclusions. ?
Conference series
The term computer-aided geometric design was established at a landmark conference in 1974 at the University of Utah. See photos from the conference here. It was organised by Robert Barnhill and Richard Riesenfeld. Many of the delegates from the US and Europe, including Even Mehlum, have later been participants at the MMCS meetings in Norway.
Much of the development of computer-aided geometric design took place in the 1980's, and the first journal devoted to the subject began publishing in July of 1984.
While the subject was discussed in several invitation only workshops and conferences in the fields of Numerical Analysis and Approximation Theory, during this period there was not yet an international conference devoted exclusively to the field.
This was the motivation for Tom Lyche and Larry Schumaker, Tom’s PhD advisor at the University of Texas in Austin, to organise the first of this series of meetings. It was held in 1988 in Oslo and attracted over 100 participants from around the world.
Due to its success, it was decided to hold the meeting on a regular basis – at first every three years, then later every four years at various venues in Norway.
Meetings were held in Biri 1991, Ulvik 1994, Lillehammer 1997, Oslo 2000, Troms? 2004, T?nsberg 2008, Oslo 2012, and T?nsberg 2016.
This year's meeting will be the tenth in the series. Over the years the list of topics covered by the meeting has broadened to reflect current activity in the field, which now includes the numerical solution of partial differential equations.
Tom Lyche and the organising team are gathering leading researchers yet again at University of Oslo, June 26-28th, 2024.
Click here for more information and full programme.
See more historic images from the conference series and other related meetings
