Geometry II

mathematics

elective courses

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.

[1] Oswald Veblen, John Wesley Young "Projective geometry"

[2] Robin Hartshorne "Foundations of Projective geometry"

[3] R. A. Johnson ,,Advanced Euclidean Geometry''

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.