### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
IRENE IGNAZIA ONNIS (Tit.)
Period
Second Semester
Teaching style
Convenzionale
Lingua Insegnamento

Informazioni aggiuntive

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

### Objectives

KNOWLEDGE AND UNDERSTANDING ABILITIES
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.

APPLIED KNOWLEDGE AND UNDERSTANDING ABILITIES
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.

JUDGEMENT
Learn the standard linear algebra proof techniques.

COMMUNICATION SKILLS
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.

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

### Prerequisites

Ability to manipulate and simplify rational expressions even in more variables. Knowing how to solve equations and inequalities in one variable of 1° and 2° degree. Familiarity with the geometric properties of elementary plane figures. Knowing how to calculate the length of a circle, the area of the circle, the volumes of cube, cuboid, pyramid, cone, cylinder, and sphere. Familiarity with the theorems of Thales, Pythagoras and Euclid and ability to use them to solve problems of elementary geometry. Familiarity with the geometric meaning of sine, cosine and tangent, and the main trigonometric formulas.

In addition to these elementary notions students must have acquired the knowledge and skills of Algebra 1 and Geometria 1.

### Contents

Affine spaces. Affine space and subspace. Affine coordinates. Geometric transformations. The Group of geometric transformations. Affine transformations.

Euclidean spaces. Euclidean vector spaces. Orthogonal complement of a subspace. Orthogonal projection. Minimum distance properties of the orthogonal projection and applications. Cartesian coordinates. Euclidean transformations. Linear isometries. Isometries. The Group of isometries.

Geometry of the plane and space. Cartesian coordinates. 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. Symmetric of a point relative to a line or a plane. Half-planes (half-spaces) determined by a straight line (a plan). Classification of orthogonal transformation and isometries in dimension two and three.
Circles, spheres, cones and cylinders. The circle and the sphere: cartesian and parametric equation. Cones and cylinders: Cartesian and parametric equations. Cone tangent to a sphere. Plane tangent to a sphere. Polarity. Cylinder tangent to a sphere. Power of a point with respect to a sphere. Radical plan. Pencil of circles.

Geometry of conics and quadrics. Conics as geometric loci. Eccentricity. Quadrics of revolution. Non-degenerate quadrics in canonical form. General equation of a quadric in space or a conic in the plane. The intersection of a line with a quadric: asymptotic direction and asymptote. Tangency of a straight line to a quadric in a point. Polar line of a point with respect to a conic and polar plan of a point with respect to a quadric. The reciprocity theorem. Center of symmetry of a quadric. Definition of a conjugate diameter to a direction. Conjugate directions. Principal directions. Axis and plane of symmetry. Orthogonal classification of conics. Invariants of a quadric. Homographic hyperbole. Change of Cartesian coordiantes and canonical form. The method of invariants to determine the Euclidean canonical form of a conic. Affine Canonical equation of a conic. The method of Gauss. Pencil of conics. Classification of quadrics. Reduction of a quadric in canonical form.

### Teaching Methods

Blackboard and slides during the lectures.
The teacher updates a website dedicated to students
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 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.

### Texts

Textbook

E. Sernesi, Geometria 1, Bollati Boringhieri.

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

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

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

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