Course contents
I) Geometry of curves in three-dimensional space:
1) Curves parametrized by arc-length: arc-length; the Frenet
trihedron; curvature and torsion; Frenet's formulae; rectifying,
normal and osculating planes; osculating circle; Frenet's
Theorem;
2) Frenet's formulae, curvature and torsion for curves not
necessarily parametrized by arc-length;
3) Main geometric properties of special curves.
II) Geometry of surfaces in three-dimensional space:
1) Definition of parametrized surface; tangent space and tangent
plane; normal vector field; the Gauss map;
2) The First Fundamental Form;
3) Normal curvature and geodesic curvature of curves on a
surface;
4) The Second Fundamental Form; Meusnier's Theorem; the Weingarten
map; Rodriguez' Theorem;
5) Gauss curvature and Mean curvature; curvature-based
classification of points on a surface;
6) Rotationally invariant surfaces; ruled surfaces; developable
surfaces;
III) Elements of Matlab programming for computer modeling.
Readings/Bibliography
1) A. Parmeggiani, "Il concetto di Forma in Matematica: il corso di
Matematica Applicata", Architettura 3, Facolta' di Architettura
dell'Universita` di Bologna (2002);
2) E. Cohen, R. F. Riesenfeld and G. Elber, "Geometric
Modeling with Splines - An Introduction", A. K. Peters (2001)