Complements of Geometry

Code:

M3004

Acronym:

M3004

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	48	162
L:CC	1	study plan from 2021/22	2	-	6	48	162
			3
L:F	0	Official Study Plan	2	-	6	48	162
			3
L:G	0	study plan from 2017/18	2	-	6	48	162
			3
L:M	23	Official Study Plan	2	-	6	48	162
			3
L:Q	0	study plan from 2016/17	3	-	6	48	162

Teaching language

Portuguese

Objectives

The aim of this course is to relate elementary geometry to its "modern" developments (i.e., affine and projective geometry) by using an algebraic approach.

Learning outcomes and competences

As described in the above section.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

- Linear Algbra
- elementary Geometry

Program

1) Real (or complex) affine spaces: affine bases; baricentric coordinates; affine transformations; ratio of three collinear points; use of affine methods to prove elementary geometric results.

2) Circles in the Euclidean plane (an algebraic approach): power of a point with respect to a circle, orthogonal circles, pencils of circles, inversion with respect to a circle.

3) Real (or complex) projective lines: Möbius transformations; cross-ratio of four real (or complex) numbers; cross-ratio of four collinear points or four concurrent lines; harmonic division of four points; homographies between projective lines. Real (or complex) projective plane: homographies; line at infinity; the affine transformations of R^2 into itself are precisely those homographies of the projective real plane that leave the line at infinity invariant.

4) Conics in the Euclidean plane - a (nearly) synthetic approach: definition of conic by focus and directrix; equation in Cartesian coordinates; conic defined by its foci; conics as plane sections of cones of revolution (Dandelin's theorem); intersection of a conic with a straight line; tangent lines to conics; special properties (including optical properties) of conics.

5) Conics in the complex projective plane: second degree algebraic curves; intersection of a regular conic with a line; a conic fails to be regular if and only if it can be decomposed in two straight lines; conjugate points with respect to a conic; polar line of a point and pole of a straight line; construction of the tangent lines to a conic through a given point using only an unmarked ruler. Conic defined by five points. Theorems of Pascal and Brianchon (about hexagons inscribed or circumscribed on a conic).

Mandatory literature

André Gramain; Géométrie élémentaire. ISBN: 2-7056-6333-9

Teaching methods and learning activities

Formal lectures, complemented by exercises for the students to solve.

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

Class attendance is not compulsory.

Calculation formula of final grade

Final examination only.

Special assessment (TE, DA, ...)

In some special circumstances, assessment of the students may consist of an oral examination only.