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Second Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2017]  PERCORSO COMUNE 8 64


One among the aims of the course is to provide students with the basic elements of analytical geometry, which will then be used in most of their subsequent studies.
The theoretical structure of the course consists of the development of the themes of the program, through the introduction of fundamental concepts and development of a series of theorems together with the correspondent proofs, accompanied by significant examples, exercises and applications.
In particular, the course aims to develop a rigorous mathematical language; assimilation of abstract concepts, algebraic structures, theorems and their proofs, related to analytical geometry.

Learning of proof techniques; ability to solve new problems and standard exercises, in which you one has to develope a strategy and apply the concepts learned, or process a little proof similar to those seen in class. Using linear algebra tools in other contexts.

Learn the standard linear algebra proof techniques.

The student will be able to present and argue the solution of problems; he will also be able to discuss and properly show the most relevant results related to analytical geometry.

Ability to learn to independently solve complex problems and exercises. Ability to read and understand advanced mathematical textbooks.


The students must have acquired the knowledge and skills of Algebra 1 and Geometria 1.


Bilinear forms. Matrix associated to a bilinear form. Congruent matrices. Duality and rank of a bilinear form. Quadratic forms. Polar bilinear form. Canonical form of a quadratic form and the method of completing squares.

Scalar product and euclidean spaces. Norm and properties. Euclidean vector spaces. Orthogonal basis and the Gram-Schmidt algorithm. Orthogonal complement of a subspace. Orthogonal projection. Minimum distance properties of the orthogonal projection and applications.

Symmetric endomorphisms. Properties and Spectral Theorem. Real symmetric matrices. Orthogonal transformations.

Affine spaces, definition and examples. Affine coordinates. Affine subspaces, dimension and codimension. Intersection of affine subspaces. Parallelism. Representation of affine subspaces: parametric and cartesian equations. Affine transformations. Translations. Euclidean affine space and Cartesian coordinates. Isometries and geometric characterization. Orthogonal affine subspaces.

Geometry of the plane and space. Affine line in the plane. Mutual position of two lines. Straight line for two points. Alignment condition for three points. Proper and improper pencil of lines. Distance from a point to a line. Parametric and cartesian equation of the affine plane. Plane for three points. Coplanarity condition of four points. Cartesian equation of the straight line in space. Pencil of planes. Mutual position of a line and a plane. Mutual position of two lines in space. Distance from a point to a plane, distance of a point from a straight line. Distance of two points on a straight line. Central symmetry. Orthogonal symmetry with respect to a hyperplane in a n-dimensional euclidean space.

The circle and the sphere: cartesian and parametric equation. Plane tangent to a sphere. Power of a point with respect to a sphere. Radical plan. Pencil of circles.

Cones and cylinders. Cartesian and parametric equations. Homogeneous functions of degree k. Cone and cylinders tangent to a sphere.

Classification of orthogonal transformations and isometries in dimension two and three.

Conics as geometric loci. Canonic equations and eccentricity. Equilateral hyperbola. Equilateral hyperbola with the coordinate axes as its asymptotes.

Revolution surfaces: meridians, parallels, profile curve and rotation axis. Examples. Quadrics of revolution.

Geometry of conics and quadrics. General equation of a quadric in space or a conic in the plane. Euclidean invariants and rank. Center of symmetry. Axis and plane of symmetry. Orthogonal classification and reduction of conics and quadrics in canonical form.

Teaching Methods

Compatibly with the teaching methods provided for in the in the Manifesto Accademico 2021-22, as a consequence of the COVID-19 emergency, the tools used for the frontal lessons will mainly be the blackboard with, possibly, a synchronous streaming system via internet using the Microsoft Teams platform

The teacher updates a website dedicated to students
(https://www.unica.it/unica/page/it/ireneionnis) where students can find any notes of the teacher and the weekly exercises to do at home and
corrected by the tutor during the tutor's class (once a week).

Verification of learning

Written and oral. Students who pass the written test (3 hours) are admitted to the oral test. The oral examination is considered to be exceeded if the student answer correctly at least three questions on different topics of the syllabus.

In any case a reply too insufficient can impair the entire oral test. The final grade is determined by the vote in the written test and by the evaluation of the oral test.



E. Sernesi, Geometria 1, Bollati Boringhieri.

M.R. Casali, C. Gagliardi, L. Grasselli - Geometria - Esculapio Editore.

I. Vaisman, Analytical Geometry, World Scientific, 1997.

A. Sanini, Lezioni di Geometria, Levrotto & Bella.

A. Sanini, Esercizi di Geometria, Levrotto & Bella.

More Information

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