Affine and Projective Geometry
Section
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Course description
REQUIREMENTS AND PRIOR KNOWLEDGE
To follow this course properly only an elementary mathematical level of education is needed (highschool level in Spain, or equivalent in other countries). Some material to be highlighted: Elementary theory of discussion and resolution of systems of linear equations. Basic matrix algebra.
GENERAL DESCRIPTION OF THE SUBJECT
This course starts with an introduction to Linear Algebra. These concepts will be used to study the affine euclidean space and its transformations. In particular, we will focus on the study of isometries in the affine plane and space. The third part of the course is an affine and projective study of conics and quadrics.
OBJECTIVES: KNOWLEDGE AND SKILLS
General goals:
1. Development of a geometrical way of thinking, both in the qualitative and quantitative sense.
2. To provide a rigurous introduction to Linear Algebra, Affine Geometry and the study of conics and quadrics.
Specific goals:
1. To achieve a Basic knowledge of the euclidean affine space.
2. Classify and determine vector and affine isometries.
3. Work with homogeneous coordinates in the projective space.
4. Classify affine conics and quadrics. Learn how to obtain their more important elements.
TEACHING MATERIAL
• Class material
• Working plan
• Vocabulary English/Spanish
EVALUATION ACTIVITIES OR PRACTICAL TASKS
Lab assignments and partial exams.
Syllabus
CHAPTER I: LINEAR ALGEBRA
Part 1: Matrices
Part 2: Linear systems of equations
Part 3: Vector spaces and subspaces
Part 4: Bases and dimension
Part 5: Equations of subspaces
Part 6: Operations with subspaces
Part 7: Linear maps
Part 8: Diagonalization
CHAPTER II: AFFINE AND EUCLIDEAN GEOMETRY
Part 1: The affine space
Part 2: Affine transformations
Part 3: Afiine Euclidean space
Part 4: Isometries
CHAPTER III: CONICS AND QUADRICS
Part 1: Introduction to the projective space
Part 2: Conics
Part 3: Pencil of conic
Part 4: Quadrics
Bibliography
- B-B-001. J. de Burgos, Curso de Álgebra y Geometría, Ed. Alhambra, 1980.
- B-B-002. M. Castellet, I. Llerena, Álgebra lineal y Geometría, Ed. Reverté, 2009.
- B-B-002. D.C. Lay. Linear Algebra and Its Applications. Third Edition Update. Addison Wesley, 2006.
- B-B-002. J.R. Sendra, S. Pérez-Díaz, J. Sendra y C. Villarino. Introducción a la Computación Simbólica y Facilidades Maple. Addlink Media, 2009.
- B-B-002. J.L. Pinilla. Cónicas, cuádricas, curvas y superficies. Varicop, 1970.
- B-B-002. H. Pottmann, A. Asperl, M. Hofer and A. Kilian. Architectural Geometry. Bentley Institute Press, 2007.
- B-B-002. A. de la Villa, Problemas de Álgebra con esquemas teóricos, Ed. CLAGSA, 1994.
Class material
CHAPTER I: LINEAR ALGEBRA
Week 1 MC-F-001. (PDF)
Week 2 MC-F-002. (PDF)
Week 3 MC-F-003. (PDF)
Week 4 MC-F-004. (PDF)
CHAPTER II: AFFINE AND EUCLIDEAN GEOMETRY
Week 5 MC-F-005. (PDF)
Week 6 MC-F-006. (PDF)
Week 7 MC-F-007. (PDF)
Week 8 MC-F-008. (PDF)
CHAPTER III: CONICS AND QUADRICS.
Week 9 MC-F-009. (PDF)
Week 10 MC-F-010. (PDF)
Week 11 MC-F-011. (PDF)
Week 12 MC-F-012. (PDF)
Week 13 MC-F-013. (PDF)
Glossary English/Spanish
ENGLISH/SPANISH GLOSSARY
Affine and Projective Geometry
A
– affine conic cónica afín.
– affine space espacio afín.
– affine subspace subespacio afín.
– affine transformation transformación afín.
– asymptote asíntota.
– autoconjugated autoconjugado.
– axis (pl axes) eje.
B
– basis (pl bases) base.
– bijective biyectiva.
– bilinear bilineal.
– bilinear form forma bilineal.
C
– charasteristic polynomial polinomio característico.
– cartesian equations ecuaciones cartesianas.
– conjugated points puntos conjugados.
– conic pencil haz de cónicas.
– coordinates coordenadas.
– coordinate system sistema de referencia.
– cross product producto vectorial.
D
– degenerate conic cónica degenerada.
– determinant determinante.
– diagonalize diagonalizar.
– diameter diámetro.
– dimension dimensión.
– direct sum suma directa.
– discriminant discriminante.
– dot product (ver scalar product) producto escalar.
E
– echelon form forma escalonada.
– eigenspace subespacio propio.
– eigenvalue valor propio, autovalor.
– eigenvector vector propio, autovector.
– elliptic type tipo elíptico.
– endomorphism endomorfismo.
– euclidean space espacio euclídeo.
F
– finitely generated finitamente generado.
– fixed point punto fijo.
G
– generating set conjunto generador.
–
H
– helical movement movimiento helicoidal.
– homogeneous coordinates coordenadas homogéneas.
– homotethy homotecia.
– hyperplane hiperplano.
I
– improper line recta impropia, del infinito.
– improper point (ver point at infinity) punto impropio, del infinito.
– injective (ver one-to-one) inyectiva.
– intersection intersección.
– invariant subspaces subespacios invariantes.
– inverse image imagen inversa.
– isometry isometría.
– isomorphism isomorfismo.
J
K
– kernel núcleo.
L
– line bundle haz de rectas.
– linear algebra álgebra lineal.
– linear combination combinación lineal.
– linear equation ecuación lineal.
– linear transformation aplicación lineal.
– linearly independent linealmente independiente.
– linearly dependent linealmente dependiente.
M
– map aplicación, función.
– matrix (pl matrices) matriz.
– metric invariants invariantes métricos.
– minor menor.
N
O
– one-to-one (ver injective) inyectiva.
– origin origen.
– orthogonal ortogonal.
– orthonormal ortonormal.
P
– parallelepiped paralelepípedo.
– parametric equations ecuaciones paramétricas.
– point at infinity (ver improper point) punto del infinito, impropio.
– polar form forma polar.
– polar line recta polar.
– polar variety variedad polar.
– polarity polaridad.
– pole polo.
– projection proyección.
– projective conic cónica proyectiva.
– projective space espacio proyectivo.
– projectively independent proyectivamente independientes.
– projectivized affine space espacio afín proyectivizado.
– proper line recta propia.
– proper point punto propio.
Q
– quadratic form forma cuadrática.
R
– rank rango.
– real vector space espacio vectorial real.
– reflection (ver specular symmetry) simetría especular.
S
– scalar product (ver dot product) producto escalar.
– set conjunto.
– signature signatura.
– singular point punto singular.
– skew lines rectas que se cruzan.
– specular symmetry (ver reflection) simetría especular.
– submatrix submatriz.
– subset subconjunto.
– surjective sobreyectiva.
T
– tangent variety variedad tangente.
– translation traslación.
– translational symmetry simetría deslizante.
– triple terna.
– triple product producto mixto.
U
– unknown variable indeterminada.
V
– vector subspace subespacio vectorial.
– vertex (pl vertices) vértice.
W
Working plan
Week 3. Linear transformations
Week 6. Affine transformations I
Week 7. Affine transformations II
Week 8. Euclidean affine space
Week 10. Introduction to projective space
Table of contents Class material Time Methodology Self-testing • Definition.
• Subspaces.
• Linear combination of vectors.
• Generated subspaces.
• Linear dependency and independency.
• Base and dimension.
Week 1 3 Theory Sheet 1 1 Exercises • Overview of rank, determinant, and their application to the study of linear dependency of vectors.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Vector coordinates.
• Intersection of subspaces.
• Sum of subspaces.
• Equations of subspaces.
Week 2 3 Theory Sheet 2 1 Exercises • Overview of methods to resolve systems of equations.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Linear transformation.
• Matrix expression.
• Kernel and image.
• Operations with linear transformations.
• Change of base.
Week 3 3 Theory Sheet 3 1 Exercises
• Exercises to determinate linear transformations, images.
• Discussion and obtainment of the origin of a vector.
2 MAPLE lab
Table of contents Class material Time Methodology Self-testing • Eigenvalue and eigenvector.
• Eigenspaces.
• Obtainment of an eigenvector base.
• Diagonalization.
Week 4 3 Theory Sheet 4 1 Exercises • Exercises about diagonalization.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Affine space.
• Affine subspace.
• Dimension.
• Coordinate systems, change of coordinate systems.
Week 5 3 Theory Sheet 5 1 Exercises • Exercises about equations of subspaces.
• Equations of subspaces in different coordinate systems, in dimensions 2 and 3
• Examples of sum and intersection of subspaces.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Affine transformation.
• Matrix expression.
• Subspaces of fixed points. .
Week 6 3 Theory Sheet 6 1 Exercises • Exercises to determine affine transformations.
• Exercises about homotheties, oblique symmetries and projections.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Invariant subspaces in affine transformations.
Week 7 3 Theory Sheet 7 2 Exercises Mid-term exam 2 Problem solving and practice • Exercises to determine and obtain invariant subspaces.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Affine euclidean space.
• Orthogonal and orthonormal coordinate systems.
• Orthogonal matrices.
Week 8 3 Theory Sheet 8 1 Exercises • Exercises about changing from an orthonormal base to an orthogonal one.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Isometry.
• Matrix expression.
• Classification
• Determination.
Week 9 3 Theory Sheet 9 1 Exercises • Exercises to determinate and obtain invariant subspaces.
• To sum up affine transformations and isometries, examples of similarities.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Projective plane and space.
• Projectivized affine space.
• Homogeneous coordinates.
• Conics will be introduced as plane sections of a cone.
Week 10 3 Theory Sheet 10 1 Exercises • Exercises about equations of lines and planes in the projective space.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Definition as locus.
• Optic propieties.
• Reduced equations.
• Equation in the projective space.
• Matrix expression.
• Intersection with a line, tangents.
• Intersection with the line at infinity. Affine classification.
Week 11 3 Theory Sheet 11 1 Exercises • Exercises with equations of conics in different coordinate systems.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Harmonic quatern.
• Pairs of conjugated points.
• Polarity.
• Singular points, degenerated conics.
• Notable elements.
Week 12 3 Theory Sheet 12 1 Exercises • Exercises about polarity, obtainment of notable elements, center, diameters, aymptotes, axes.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Determination of conics, bundles.
Week 13 3 Theory Sheet 13 1 Exercises • Exercises to determine conics with bundles.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Geometric introduction.
• Reduced equations.
• Equation in the projective space.
• Matrix expression.
• Intersection with a line, tangent lines and planes.
• Intersection with the point at infinity, affine classification.
• Singular points, degenerate conics.
• Notable elements
Week 14 3 Theory Sheet 14 1 Exercises • Exercises with quadrics in different coordinate systems.
2 MAPLE lab Table of contents Class material Time Methodology Self-testing • Polarity.
• Singular points, degenerate quadrics.
• Notable elements.
Week 15 1.5 Clase teoríca Método Expositivo Sheet 15 5 Explicación de contenidos
• Exercises about polarity, obtainment of notable elements, center, diameters, diametral planes, axis.
Authors of material
Matemática Aplicada a la Edificación, al Urbanismo y al Medio Ambiente
Mª Eugenia Rosado
Profesora Titular Interina
Departamento de Matemática Aplicada de la ETSAM
Sonia Luisa Rueda
Profesora Titular Interina
Departamento de Matemática Aplicada de la ETSAM
María Copado
Becaria Colaboración
Departamento de Matemática Aplicada de la ETSAM