approximation theory and computational geometry

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"

New Allies: Approximation Theory & Computational Geometry


The Problem

Computational Geometry often operates on straight lines, piecewise linear surfaces, …

Approximation often better with non-linear objects (circular arcs, B-splines, NURBS, …)



Intersection of 2 curvesin the plane

1. Approximate curve e.g. with polygonal chain

2. Intersect approximated curves

a) find critical segments

b) intersect them



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



Example

Compare two different levels of approximations

Chain of: straight lines vs. circular arcs

SL: d=2 CA: d=3



m

O{mlog(m)}

O{m3/2+}

O{mlog(m)}

Algorithms

straight lines

n

O{nlog(n)}

O{n3/2log(n)}

O{n}

chain of:

#elements

1 intersection

count inters.

convex hull

circular arcs

n2/3

O{n2/3log(n)}

O{n1+}

O{n2/3log(n)}



real world input

Algorithm withcomplexity c(n)

Question/Problem

Complexity of approachfor 0 [c*(n)=c*(,d)]

Image of real world withcomplexity n = f(,d,…)

preprocessing structural intelligence

real

wor

dap

prox

imat

ion

theo

ryco

mpu

tatio

nal

geom

etry



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



Suitable tasks

Intersection of curves

Minkowski-sum

Minimum distance between surfaces

Convex hull of (plane) curves

Calculation of Medial Axis



Thank you for your attention

approximation theory and computational geometry

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