Corso di laurea in Ottica e Optometria

Coherently with the didactic goals of the Course of Study planned by the SUA, CdS frames, it is expected that the student acquires familiarity with the basic notions of analytic geometry in the plane and 3D space.

In particular, the student is asked to understand the concept of frame and to be able to use coordinates to describe the basic geometric entities. Furthermore, the student is expected to be able to solve linear systems and to interpret their solutions in the light of the problem at hand.

Knowledge and understanding of basic analytic geometry. Skill in applying knowledge and understanding of the geometry to the use of the most advanced instrumentation in the field of data collection and data processing in optics and optometry.

The course is articulated in 48 hours of formal in-class lecture time, of which 20 dedicated to linear algebra and 28 to analyt ic geometry. A substantial part of the lessons will be reserved to examples and exercises.

THE ORGANIZATION OF THE SCRIPT AND ORAL EXAMS VIA WEBEX, made necessary in view of the Coronavirus emergency, WILL BE DETAILED IN THE MOODLE PLATFORM.

Euclidean Geometry.: the notion of axioms and theorems. Use of coordinates and frames.

Natural, integer, rational and real numbers, and their relation with the points of a line. Frames on the Euclidean line S1, the Euclidean plane S2 and the Euclidean space S3.

The space Rn, linear systems, solutions of a linear system. Matrix of coefficients. Gauss-Jordan method.
Row reduction of a matrix: pivot elements and rank. Rouche'-Capelli's theorem. Homogeneous systems and the structure of the set of solutions.

Matrices: sum, products and their properties. Transpose and inverse of a matrix. Matrix equations and computation of the inverse of a matrix with Gauss-Jordan. Invertibility and rank.

Vector spaces and their properties. The space of vectors of S1, S2 and S3. Vector subspaces. Linear dependence and independence, free sets and generators, bases. Frames in S1,S2 ed S3, II.
Theorems about bases in a vector space. Dimension of a vector space. Space of lines and columns of a matrix: the rank+nullity theorem. Change of bases, and the relation between the induced coordinates

Scalar products and their properties, angle between vectors. Orthonormal sets and bases, components, lengths. Changes of orthonormal frames: orthogonal matrices. Gram-Schmidt orthonormalization method.

Geometry in the plane S2.
Orthonormal frame and coordinates of points. Vector joining two points. Parametric and non-parametric expression of a line, parallelism and orthogonality between two lines. Line passing for two points, middle point of a segment, axis of a segment. Distance between points and lines.

Conic sections:
circle, center and radius. Tangent lines to a circle that pass through a given point. Circle passing through 3 given points;
ellipse and hyperbola in a canonical form: semiaxes, foci and focal property. Equilatera hyperbola referred to the xy axes;
parabola in a canonic form: focus, direttrix and focal property.

Geometry in the space S3.
Orthonormal frames. Parametric and nonparametric equations of planes and lines. Intersection of 3 planes and solutions of a system. Intersection between a line and a plane. Intersection between lines.
Distance point-line and point-plane. Distance line-plane and line-line. Angles between crossing lines and between crossing planes.

The program of the concluding part has changed, to better fit with the remaining hours of this online course. Here are the new contents.

Spheres: Center and radius. Intersections line-sphere, plane-sphere and sphere-sphere. 

Cones over a conic: the cone tangent (circumscribed) to a sphere. Conics as sections of a cone with a plane.

Cylinder: the cylinder tangent (circumscribed) to a sphere. 

Other relevant quadrics (example in canonic form)

Determinants of matrices 2x2 and 3x3: Geometric meaning and computation (Sarrus rule). Bases that are positively oriented with respect to the canonical one.

This was the old program.

Identification of a conic in non-canonical form. The matrix form of a conic.
Determinants of 2x2 and 3x3 matrices, basics about eigenvectors and eigenvalues of a matrix, orthogonality of eigenvectors belonging to different subspaces. Computation of eigenvectors/eigenvalues for 2x2 matrices and determination fo the principal axes of a conic. The conic in the new coordinates referred to the principal frame, and identification (circle/ellipse/hyperbola/parabola or degenerate).

1) E.Abbena, A.M.Fino, G.M.Gianella, Algebra lineare e Geometria Analitica, Vol I, Aracne Ed, 2012

Other texts will be indicated during the lectures. Students are strongly encouraged to take notes.

Some complementary books that may be consulted during the course:

2) David C. Lay, "Linear Algebra and ita applications", 3rd ed. Pearson Education Limited

3) S.Console, M.Roggero, D.Romagnoli, Matematica per le scienze applicate, Levrotto & Bella, 2010

4) Ambrogio, Garrione, Romagnoli, Esercizi di matematica per le scienze applicate, Levrotto & Bella, 2010

Office hours: by appointment