Geometry II
General data
Course ID: | 1000-135GM2 |
Erasmus code / ISCED: |
11.173
|
Course title: | Geometry II |
Name in Polish: | Geometria II |
Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
Course groups: |
Elective courses for 1st degree studies in mathematics |
ECTS credit allocation (and other scores): |
6.00
|
Language: | Polish |
Main fields of studies for MISMaP: | mathematics |
Type of course: | elective courses |
Short description: |
(in Polish) Inwersja, przekształcenia afiniczne oraz stożkowe w ujęciu czysto geometrycznym. Ogniska i kierownice stożkowych, własności izogonalne stożkowych, przekroje stożka obrotowego. Liczne zastosowania i geometryczne dowody najsłynniejszych twierdzeń m.in.: Gaussa-Bodenmillera, Brianchona, o motylku, Ponceleta (dla trójkąta), Feuerbacha, o łańcuchach Steinera, Newtona oraz formuł Kartezjusza, Eulera i Fussa. |
Full description: |
1. Power of a point with respect to a circle - the radical axis of two circles, the radical center - Brianchon's Theorem 2. Inversion - circles and lines under an inversion - inversion as a conformal mapping - constant circles under an inversion - the change of the distances under an inversion, Ptolemy's Theorem - the nine-point circles, Feuerbach Theorem 3. Conics - The focus and the directrix of a parabola - The tangent to the conic. Triangles circumscribing conics: foci as isogonal points. - Eccentricity and the directrix of a parabola. - Flat section of a cone - Brianchon's and Pascal's Theorems for ellipse - The canonical equations of conics 4. Affine mappings - The group of affine mappings - Shear mappings - Darboux Lemma - preseving of the ratio by the affine mapping - Composition of the affine mapping into a similarity and a shear mapping - The main directions of the affine mapping - Preserving of the ratio of areas 5. Elements of projective geometry - Pole and polar with respect to a circle - Projective plane, pencils and chains - The double-ratio - Central projections, projective mapping between planes and lines - Projective involutions - Duality - Cones on the projective planes - Desargue's, Pascal's and Brianchon's Theorems on the plane. |
Bibliography: |
[1] Oswald Veblen, John Wesley Young "Projective geometry" [2] Robin Hartshorne "Foundations of Projective geometry" [3] R. A. Johnson ,,Advanced Euclidean Geometry'' |
Learning outcomes: |
1. Student knows: the power of a point with respect to a circle, radical axis, radical center, Brianchon's Theorem and can apply in selected geometrical problems. 2. Student knows inversion with respect to a circle, can transform selected configurations using inversion, understands the importance of conformal mappings and circles being preserved by inversion, knows the formula for changing the distances and radii of circles under inversion and can apply it in the selected configurations. 3. Student knows the notion of a conic (several equivalent definitions) and related notions: focus, directrix, eccentricity, can construct tangent lines to conics and apply it to solve related problems. 4. Student knows the geometric definiton and the properties of affine mappings, can transform selected configurations using affine mappings and can apply it to solve related problems. 5. Student knows the basic notions of projective geometry: projective plane, double-ratio, pole, polar, projective involution and can apply them in selected geometric problems. |
Assessment methods and assessment criteria: |
(in Polish) Ocena z przedmiotu będzie zależała od wyników pracy na ćwiczeniach, kolokwium w trakcie semestru, egzaminu pisemnego i ewentualnego egzaminu ustnego. |
Classes in period "Summer semester 2023/24" (in progress)
Time span: | 2024-02-19 - 2024-06-16 |
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MO WYK
CW
TU W TH FR |
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Waldemar Pompe | |
Group instructors: | Waldemar Pompe | |
Students list: | (inaccessible to you) | |
Examination: |
Course -
Examination
Lecture - Examination |
Classes in period "Summer semester 2024/25" (future)
Time span: | 2025-02-17 - 2025-06-08 |
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MO TU W TH FR |
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Joanna Jaszuńska | |
Group instructors: | Joanna Jaszuńska | |
Students list: | (inaccessible to you) | |
Examination: |
Course -
Examination
Lecture - Examination |
Copyright by University of Warsaw.