# GAIA II Sintesi della relazione

Project ID:
IST-2001-35512

Finanziato nell'ambito di:
FP5-IST

Paese:
Norway

## Classification of curves and surface

To use algebraic curves and surfaces in CAGD one needs to know about their shape: topology, singularities, selfintersections. Most of this kind of classification theory is performed for algebraic curves and surfaces defined over the complex numbers, i.e., one considers complex (instead of only real) solutions to polynomial equations in two or three variables (or in three or four homogeneous variables, if the curves and surfaces are considered in projective space). Complete classification results exist only for low degree varities and mostly only in the complex case. A simple example, the classification of conic sections, illustrates well that the classification over the real numbers is much more complicated than over the real numbers.

We have collected known results about such classifications, especially what is known for real curves and surfaces of low degree. Of particular interest in CAGD are parameterizable (i.e. socalled rational) curves and surfaces, and we have made explicit studies of various such objects. These objects, or patches of these objects, are potential candidates for approximate implicitization problems. For example, when the rough shape of a patch to be approximated is known, one can choose from a "catalogue" what kind of parameterized patch that is suitable - this eliminates many unknowns in the process of finding an equation for the approcximating object and will therefore speed up the application.

In addition to the survey of known results, particular objects that have been studied are:

- Monoid curves and surfaces, especially quartic monoid surfaces

- Tangent developables

- Triangle and tensor surfaces of low degree of low (bi)degrees

We have collected known results about such classifications, especially what is known for real curves and surfaces of low degree. Of particular interest in CAGD are parameterizable (i.e. socalled rational) curves and surfaces, and we have made explicit studies of various such objects. These objects, or patches of these objects, are potential candidates for approximate implicitization problems. For example, when the rough shape of a patch to be approximated is known, one can choose from a "catalogue" what kind of parameterized patch that is suitable - this eliminates many unknowns in the process of finding an equation for the approximating object and will therefore speed up the application.

In addition to the survey of known results, particular objects that have been studied are:

- Monoid curves and surfaces, especially quartic monoid surfaces.

- Tangent developables.

- Triangle and tensor surfaces of low degree of low (bi)degrees.

We have collected known results about such classifications, especially what is known for real curves and surfaces of low degree. Of particular interest in CAGD are parameterizable (i.e. socalled rational) curves and surfaces, and we have made explicit studies of various such objects. These objects, or patches of these objects, are potential candidates for approximate implicitization problems. For example, when the rough shape of a patch to be approximated is known, one can choose from a "catalogue" what kind of parameterized patch that is suitable - this eliminates many unknowns in the process of finding an equation for the approcximating object and will therefore speed up the application.

In addition to the survey of known results, particular objects that have been studied are:

- Monoid curves and surfaces, especially quartic monoid surfaces

- Tangent developables

- Triangle and tensor surfaces of low degree of low (bi)degrees

We have collected known results about such classifications, especially what is known for real curves and surfaces of low degree. Of particular interest in CAGD are parameterizable (i.e. socalled rational) curves and surfaces, and we have made explicit studies of various such objects. These objects, or patches of these objects, are potential candidates for approximate implicitization problems. For example, when the rough shape of a patch to be approximated is known, one can choose from a "catalogue" what kind of parameterized patch that is suitable - this eliminates many unknowns in the process of finding an equation for the approximating object and will therefore speed up the application.

In addition to the survey of known results, particular objects that have been studied are:

- Monoid curves and surfaces, especially quartic monoid surfaces.

- Tangent developables.

- Triangle and tensor surfaces of low degree of low (bi)degrees.