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This book summarizes research carried out in workshops of the SAGA
project, an Initial Training Network exploring the interplay of
Shapes, Algebra, Geometry and Algorithms. Written by a combination
of young and experienced researchers, the book introduces new ideas
in an established context. Among the central topics are approximate
and sparse implicitization and surface parametrization; algebraic
tools for geometric computing; algebraic geometry for computer
aided design applications and problems with industrial
applications. Readers will encounter new methods for the
(approximate) transition between the implicit and parametric
representation; new algebraic tools for geometric computing; new
applications of isogeometric analysis and will gain insight into
the emerging research field situated between algebraic geometry and
computer aided geometric design.
This open access book provides insights into the novel Locally
Refined B-spline (LR B-spline) surface format, which is suited for
representing terrain and seabed data in a compact way. It provides
an alternative to the well know raster and triangulated surface
representations. An LR B-spline surface has an overall smooth
behavior and allows the modeling of local details with only a
limited growth in data volume. In regions where many data points
belong to the same smooth area, LR B-splines allow a very lean
representation of the shape by locally adapting the resolution of
the spline space to the size and local shape variations of the
region. The iterative method can be modified to improve the
accuracy in particular domains of a point cloud. The use of
statistical information criterion can help determining the optimal
threshold, the number of iterations to perform as well as some
parameters of the underlying mathematical functions (degree of the
splines, parameter representation). The resulting surfaces are well
suited for analysis and computing secondary information such as
contour curves and minimum and maximum points. Also deformation
analysis are potential applications of fitting point clouds with LR
B-splines.
This volume contains revised papers that were presented at the
international workshop entitled Computational Methods for Algebraic
Spline Surfaces ("COMPASS"), which was held from September 29 to
October 3, 2003, at Schloss Weinberg, Kefermarkt (A- tria). The
workshop was mainly devoted to approximate algebraic geometry and
its - plications. The organizers wanted to emphasize the novel idea
of approximate implici- zation, that has strengthened the existing
link between CAD / CAGD (Computer Aided Geometric Design) and
classical algebraic geometry. The existing methods for exact
implicitization (i. e., for conversion from the parametric to an
implicit representation of a curve or surface) require exact
arithmetic and are too slow and too expensive for industrial use.
Thus the duality of an implicit representation and a parametric
repres- tation is only used for low degree algebraic surfaces such
as planes, spheres, cylinders, cones and toroidal surfaces. On the
other hand, this duality is a very useful tool for - veloping
ef?cient algorithms. Approximate implicitization makes this duality
available for general curves and surfaces. The traditional exact
implicitization of parametric surfaces produce global rep-
sentations, which are exact everywhere. The surface patches used in
CAD, however, are always de?ned within a small box only; they are
obtained for a bounded parameter domain (typically a rectangle, or
- in the case of "trimmed" surface patches - a subset of a
rectangle). Consequently, a globally exact representation is not
really needed in practice."
This volume contains revised papers that were presented at the
international workshop entitled Computational Methods for Algebraic
Spline Surfaces ("COMPASS"), which was held from September 29 to
October 3, 2003, at Schloss Weinberg, Kefermarkt (A- tria). The
workshop was mainly devoted to approximate algebraic geometry and
its - plications. The organizers wanted to emphasize the novel idea
of approximate implici- zation, that has strengthened the existing
link between CAD / CAGD (Computer Aided Geometric Design) and
classical algebraic geometry. The existing methods for exact
implicitization (i. e., for conversion from the parametric to an
implicit representation of a curve or surface) require exact
arithmetic and are too slow and too expensive for industrial use.
Thus the duality of an implicit representation and a parametric
repres- tation is only used for low degree algebraic surfaces such
as planes, spheres, cylinders, cones and toroidal surfaces. On the
other hand, this duality is a very useful tool for - veloping
ef?cient algorithms. Approximate implicitization makes this duality
available for general curves and surfaces. The traditional exact
implicitization of parametric surfaces produce global rep-
sentations, which are exact everywhere. The surface patches used in
CAD, however, are always de?ned within a small box only; they are
obtained for a bounded parameter domain (typically a rectangle, or
- in the case of "trimmed" surface patches - a subset of a
rectangle). Consequently, a globally exact representation is not
really needed in practice."
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