=ADD=
=reftype=
14
=number=
97-21
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1997/97-21.ps.gz
=note=
Diploma Thesis
=year=
1997
=author=
Aigner; Klaus
=title=
Symbolic Computation of and with Offset Curves
=month=
07
=abstract=
In geometry and its applications we are sometimes interested in computing the
offset curve to a given generator curve. That is, given a curve C, we wish
to determine a curve O(C,d) such that, for every point p of C, there is a
point q on O(C,d) such that the distance between p and q is exactly d, and 
the line determined by p and q is perpendicular to C at p and vice versa. If
the generator curve is an algebraic curve in implicit or parametric
representation, we describe procedures to compute its implicit offset curve
by using methods of elimination theory (Groebner bases, resultants). 

However, in applications many algorithms implicitly assume that the objects are 
given parametrically. The main difficulty is that the rationality of the generator
curve is not preserved (in general). For deciding the rationality of an offset
curve we introduce the notion of Rational Pythagorean Hodographs (i.e. rationally
parametrized curves whose normal vector has rational lengths). For generator
curves which are RPH we describe an algorithm to compute the offset curve in
parametric representation by using the concept of reparametrizing curves.

By considering higher dimensional spaces and applying a direct isometry to the
normal vector one can extend the classical notion of offsets to the concept of
generalized offsets to hypersurfaces. In this case we also describe criteria for
the rationality of such hypersurfaces.

We have implemented serveral algorithms to compute offsets in MAPLE V and by using
the software package CASA. CASA offers data structures to represent algebraic
sets and to compute with them (parametrization, implicitation, decomposition,...).
The package for computing offset curves consists of the procedures mkImplOffset,
mkParaOffset and RPHcurve. Using mkImplOffset or mkParaOffset we can compute
the offset curve in implicit or parametric representation, respectively, to
a given generator curve (in implicit or parametric representation) at a specified
or unspecified distance. RPHcurve checks if there exists an RPH parametrization
of a rational algebraic curve and, if there exists one, it computes the necessary
reparametrizing function.
=location=
2
=owner=
2
=source=
3