Viktor Abramov

ENET

Analytical Geometry (Part II)

LTMS.00.051 · 3 ECTS

About the Course

The course begins with the classification of second-order curves. It then considers the most important second-order curves, namely the ellipse, hyperbola, and parabola, which are also called conic sections.

The course introduces the tangent to a parametrized curve; the approach is based on the equation of the tangent to the graph of a function. Geometric properties of tangents to conic sections are proved, with important applications in engineering.

The equation of conic sections in polar coordinates is derived. The final part of the course is devoted to second-order surfaces, such as ellipsoids, hyperboloids, and paraboloids.

Schedule

Lectures: Mondays, 10:15–12:00, room 1007, Delta building.

Practical classes: Fridays, 14:15–16:00, room 1020, Delta building.

I teach the lectures. The practical classes are taught by doctoral student Nikolai Sovetnikov. Questions concerning the practical classes and tests should be addressed directly to him.

Duration and Examinations

The course lasts 8 weeks, beginning in the first week of September.

The first examination will take place in the first week of November. The second examination will take place either at the end of November or in December.

The exact examination dates will be announced later, after discussion with the students.

The examination is written.

Admission to the Examination

Tests are held during the practical classes. A student must obtain at least 10 points in order to be admitted to the examination.

Lecture Topics

  1. Plane coordinate transformations. Translation, rotation and reflection in an axis. Transformation of a second-degree polynomial.
  2. The concept of a second-order curve. The equation of a second-order curve and its investigation.
  3. Investigation of the canonical equation of the ellipse. The focal-radius property of the ellipse. The focal parameter and eccentricity of the ellipse. The directrices of the ellipse and their property.
  4. Investigation of the canonical equation of the hyperbola. Foci, centre, semi-axes and focal radii. The focal-radius property. Asymptotes of the hyperbola. Equations of the directrices of the hyperbola. The directrix property.
  5. Investigation of the canonical equation of the parabola. Focus and vertex of the parabola. Equation of the directrix. Distance of a point of the parabola from the focus and directrix. Polar coordinates. Pole and polar axis. Equations of the ellipse, hyperbola and parabola in polar coordinates.
  6. Equation of the tangent to the ellipse, hyperbola and parabola. Tangent properties of the ellipse, hyperbola and parabola.
  7. The concept of a second-order surface. The equation of a second-order surface. The concept and equation of a surface of revolution. Ellipsoid of revolution and its equation. Equation of the ellipsoid. Second-order cone and its equation. One-sheeted hyperboloid of revolution and one-sheeted hyperboloid.
  8. Two-sheeted hyperboloid. Elliptic paraboloid. Hyperbolic paraboloid. Hyperbolic paraboloid as a ruled surface. Generatrices.

Course Notes

For this course, Chapter 8 of Part I of the course notes and all of Part II are used.

Course notes — PDF

Announcements

Current announcements for students will be posted here.