1st Meeting Industrial Geometry Approximation Theory and Computational Geometry 1st IG-Meeting New Allies Joint research with project S09202 "Coupling evolving level sets with curves and surfaces"

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 1 The Problem  Computational Geometry often operates on straight lines, piecewise linear surfaces, …  Approximation often better with non- linear objects (circular arcs, B-splines, NURBS, …)

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 2 Intersection of 2 curves in the plane 1.Approximate curve e.g. with polygonal chain 2.Intersect approximated curves a)find critical segments b)intersect them

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 3 Error  If   0 then n   !  CG-algorithm: c(n) … n = f( ,d,…)  But: original problem consists of only 2 curves NOT n elements ! with and

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 4 Example  Compare two different levels of approximations  Chain of: straight lines vs. circular arcs SL: d=2CA: d=3

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 5 m O{mlog(m)} O{m 3/2+  } O{mlog(m)} Algorithms straight lines n O{nlog(n)} O{n 3/2 log(n)} O{n} chain of: #elements 1 intersection count inters. convex hull circular arcs n 2/3 O{n 2/3 log(n)} O{n 1+  } O{n 2/3 log(n)}

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 6 real world input Algorithm with complexity c(n) Question/Problem Complexity of approach for  0 [c*(n)=c*( ,d)] Image of real world with complexity n = f( ,d,…)  preprocessingstructural intelligence real word approximation theory computational geometry

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 7 Synergy of AT and CG  Combine both concepts to get overall theory and global insights  Leads to new approach ?  Maybe even establish a new paradigm  Propose new notion for complexity of CG-algorithms

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 8 Suitable tasks  Intersection of curves  Minkowski-sum  Minimum distance between surfaces  Convex hull of (plane) curves  Calculation of Medial Axis

1st Meeting Industrial Geometry New Allies: Approximation Theory & Computational Geometry 9 Thank you for your attention