Computational Geometry
A.Y. 2024/2025
Learning objectives
The course aims to provide the indispensable mathematical bases of analytical, differential and projective geometry for the use and study of computer graphics.
Expected learning outcomes
At the end of the course students are able to use analytical, differential or projective geometry to set up a computer graphics project. This project can be a video game, or have a didactic character, or represent natural phenomena.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1) Euclidean and affine Geometry: transformations; affine maps;
2) Curves: differential curves; geometric continuity; Bezier curves; spline curves; conics; interpolation.
3) Surfaces: differential surfaces; geometric continuity; Bezier surfaces; Coons patches; interpolation.
4) Projective Geometry: projective transformations; multiple view Geometry; clipping; reconstruction and camera calibration.
2) Curves: differential curves; geometric continuity; Bezier curves; spline curves; conics; interpolation.
3) Surfaces: differential surfaces; geometric continuity; Bezier surfaces; Coons patches; interpolation.
4) Projective Geometry: projective transformations; multiple view Geometry; clipping; reconstruction and camera calibration.
Prerequisites for admission
Prerequisites: knowledge of the contents of the Discrete Mathematics and Continuum Mathematics courses. It is strongly recommended to pass both these exams.
Ability to set up web pages, possibly with the use of the Java
Ability to set up web pages, possibly with the use of the Java
Teaching methods
Traditional teaching on the blackboard.
Teaching Resources
A) Basic references:
G. Farin: "Curves and surfaces for computer aided geometric design" ed. Academic Press, 1990 (o edizioni successive).
G. Farin-D. Hansford: "The essentials of CAGD" ed. A. K. Peters, Wellesly Mass. U. S.A. 2000.
R. Hartley-A. Zisserman: "Multiple view Geometry in computer vision" ed. Cambridge Univ. Press, 2002.
J. J. Risler: "Methodes mathematiques pour la C. A. O." Recherches en Mathematiques Appliquees, 18, ed. Masson, 1991.
B) Additional references:
W. Boehm-H. Prautzsch: "Geometric concepts for Geometric Design" ed. A. K. Peters, Wellesly Mass. U.S.A., 1994.
J. C. Fiorot-P. Jeannin: "Corbes splines rationelles, applications a la C.A.O." Recherches en Mathematiques Appliquees, 24, ed. Masson, 1992.
M. M. Mortenson: "Computer Graphics: an introduction to the Mathematics and Geometry" ed. Hainemann Newnes, 1989.
M. M. Mortenson: "Modelli geometrici in computer graphics" ed. Mc Graw-Hill, 1989.
A.W. Nutbourne-R. R. Martin: "Differential Geometry applied to curve and surface design" Vol 1: Foundations; ed. Ellis Norwood Limited, 1988.
H. O. Peitgen-P. H. Richter: "La bellezza dei frattali" ed. Bollati Boringhieri, Torino, 1987.
M. A. Penna-R. R. Patterson: "Projective Geometry and its applications to Computational Geometry" ed. Prentice Hall, 1986.
F. Yamaguchi: "Curves and Surfaces in Computer Aided Geometric Design" ed. Springer Verlag, Berlin, 1988.
C) Ariel site of the course or course site
http://www.mat.unimi.it/~alzati/Geometria_Computazionale_98-99/
G. Farin: "Curves and surfaces for computer aided geometric design" ed. Academic Press, 1990 (o edizioni successive).
G. Farin-D. Hansford: "The essentials of CAGD" ed. A. K. Peters, Wellesly Mass. U. S.A. 2000.
R. Hartley-A. Zisserman: "Multiple view Geometry in computer vision" ed. Cambridge Univ. Press, 2002.
J. J. Risler: "Methodes mathematiques pour la C. A. O." Recherches en Mathematiques Appliquees, 18, ed. Masson, 1991.
B) Additional references:
W. Boehm-H. Prautzsch: "Geometric concepts for Geometric Design" ed. A. K. Peters, Wellesly Mass. U.S.A., 1994.
J. C. Fiorot-P. Jeannin: "Corbes splines rationelles, applications a la C.A.O." Recherches en Mathematiques Appliquees, 24, ed. Masson, 1992.
M. M. Mortenson: "Computer Graphics: an introduction to the Mathematics and Geometry" ed. Hainemann Newnes, 1989.
M. M. Mortenson: "Modelli geometrici in computer graphics" ed. Mc Graw-Hill, 1989.
A.W. Nutbourne-R. R. Martin: "Differential Geometry applied to curve and surface design" Vol 1: Foundations; ed. Ellis Norwood Limited, 1988.
H. O. Peitgen-P. H. Richter: "La bellezza dei frattali" ed. Bollati Boringhieri, Torino, 1987.
M. A. Penna-R. R. Patterson: "Projective Geometry and its applications to Computational Geometry" ed. Prentice Hall, 1986.
F. Yamaguchi: "Curves and Surfaces in Computer Aided Geometric Design" ed. Springer Verlag, Berlin, 1988.
C) Ariel site of the course or course site
http://www.mat.unimi.it/~alzati/Geometria_Computazionale_98-99/
Assessment methods and Criteria
Type of exam: oral.
The exam consists of a compulsory oral discussion that focuses on the topics covered in the course during which the candidate must demonstrate that he can master them.
The oral exam can be replaced by the discussion of a project elaborated on computer by the candidate and previously agreed with the teacher. In this case, others to the project itself (completeness, proper functioning, clarity of use, etc.) will also be assessed the display capacity.
The project will then be included in the course site, which can be consulted to get an idea of the projects presented so far.
Exams are held by appointment. The final evaluation is expressed in thirtieths.
The exam consists of a compulsory oral discussion that focuses on the topics covered in the course during which the candidate must demonstrate that he can master them.
The oral exam can be replaced by the discussion of a project elaborated on computer by the candidate and previously agreed with the teacher. In this case, others to the project itself (completeness, proper functioning, clarity of use, etc.) will also be assessed the display capacity.
The project will then be included in the course site, which can be consulted to get an idea of the projects presented so far.
Exams are held by appointment. The final evaluation is expressed in thirtieths.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 48 hours
Professor:
Alzati Alberto
Shifts:
Turno
Professor:
Alzati AlbertoProfessor(s)
Reception:
Monday 14.00-16.00
Office n° 2103, II floor, c/o Dip. Mat., via Saldini 50