Algorithmic Algebraic Geometry for Geometric Modeling
{'Code': '05020000', 'Directorate': {'Abbreviation': 'CSE', 'LongName': 'Direct For Computer & Info Scie & Enginr'}, 'Division': {'Abbreviation': 'IIS', 'LongName': 'Div Of Information & Intelligent Systems'}}
{'SignBlockName': 'Joyce'}
Geometric modeling is a relatively new discipline that focuses on               the problem of constructing and manipulating representations of                 geometric objects.  One of the fundamental problems is choosing                 an appropriate representation for surfaces.  In one popular                     approach, surfaces are represented as zero sets of polynomials.                 Such surfaces, referred to as algebraic surfaces, include almost                all of the surfaces currently used in geometric modeling (e.g.,                 Bezier surfaces, B-spline surfaces, quadric surfaces).                                                                                                          A benefit of the use of these surfaces is the existence of an                   extensive mathematical theory of algebraic surfaces known as                    algebraic geometry.  We propose to apply theory from algebraic                  geometry to three current problems in geometric modeling:                                                                                                       1.  Symbolic manipulation of systems of polynomial equations:  We               propose to investigate symbolic manipulation methods involving                  elimination theory (e.g., resultants).  Specifically, we intend                 to explore new methods for eliminating  several variables from a                system of polynomials simultaneously.                                                                                                                           2.  Computing the intersection of algebraic surfaces:  we propose               to apply the theory of birational maps to the problem of                        constructing an algorithm for computing the intersection of                     algebraic surfaces.                                                                                                                                             3.  Construction of piecewise smooth algebraic surfaces:  we                    propose to use the theory of polynomial ideals to aid in the                    design and construction of a system of piecewise continuous                     algebraic surfaces.  In particular, we hope to develop methods                  for constructing free-form algebraic surfaces.
{'FirstName': 'Joe', 'LastName': 'Warren', 'EmailAddress': 'jwarren@rice.edu', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'William Marsh Rice University', 'CityName': 'Houston', 'ZipCode': '770051827', 'PhoneNumber': '7133484820', 'StreetAddress': '6100 MAIN ST', 'CountryName': 'United States', 'StateName': 'Texas', 'StateCode': 'TX'}
{'Code': '6840', 'Text': 'ROBOTICS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8810747.xml'}