Rony Goldenthal and Michel Bercovier
Curves and surface design using optimization is very common nowadays.
The most obvious example is, perhaps, fitting and fairing.Hence a curve (or
surface) is created by minimizing the approximation error from a given set of
points while minimizing
some fairness function in order to guaranty a certain quality for the resulting
curve, thus resulting in an optimization problem with two objective functions.
Additionaly, constrains are imposed in order to improve the robustness of the
algorithm; constrains
are often modeled as penalty functions.Construction of class A surfaces, based
on given isophotes or reflection lines together with interpolation constrains
is an example of problems that include several goals:fairness,imposed “a
posteriori”patterns, interpolation points, etc. The resulting optimization
problem is now a multi-objective (MO) problem. Classical optimization methods
such as Newton-Raphson or Conjugate Gradient are not designed to handle MO problems.
In this work multi objective optimization using a multi objective
genetic algorithm is applied. This approach has the advantage of avoiding the
need for selecting arbitrary weights. Additionally,the output of the algorithm
is a set of compromised solutions.
Each solution is optimal with regard to a certain selection of priorities among
the different, possibly conflicting,optimization goals.Moving from gradient
based optimization to genetic algorithm has additional advantages:enabling usage
of highly non-linear
and non-differentiable cost functions and robust handling of constrains.The
optimization variables used in this work are the curve’s (surface’s)
parameterization, knot vector and the NURBS’weights. Several applications
of this approach to curve and surface design are given.
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