Approximation Theory and Computational Geometry

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

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

3. Intersection of 2 curvesin the plane • Approximate curve e.g. with polygonal chain • Intersect approximated curves • find critical segments • intersect them New Allies: Approximation Theory & Computational Geometry

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

5. Example • Compare two different levels of approximations • Chain of: straight lines vs. circular arcs SL: d=2 CA: d=3 New Allies: Approximation Theory & Computational Geometry

6. Algorithms straight lines n O{nlog(n)} O{n3/2log(n)} O{n} m O{mlog(m)} O{m3/2+} O{mlog(m)} chain of: #elements 1 intersection count inters. convex hull circular arcs n2/3 O{n2/3log(n)} O{n1+} O{n2/3log(n)} New Allies: Approximation Theory & Computational Geometry

7. real world input  real word preprocessing structural intelligence approximationtheory Image of real world withcomplexity n = f(,d,…) Algorithm withcomplexity c(n) Question/Problem computationalgeometry Complexity of approachfor 0 [c*(n)=c*(,d)] New Allies: Approximation Theory & Computational Geometry

8. 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 New Allies: Approximation Theory & Computational Geometry

9. Suitable tasks • Intersection of curves • Minkowski-sum • Minimum distance between surfaces • Convex hull of (plane) curves • Calculation of Medial Axis New Allies: Approximation Theory & Computational Geometry

10. Thank you for your attention New Allies: Approximation Theory & Computational Geometry