Prof. Zeev Rudnick
Fall 2004-2005, Tel Aviv
The course is an introductory course in the geometry of two and three
It assumes knowledge of the courses calculus 1, linear algebra 1.
It is preferable that the students have either taken calculus 2 and
linear algebra 2 or are studying them in parallel with the course, but
in practice this was not universally true.
- Background from linear algebra:
Inner product spaces, orthogonality, triangle inequality and the
Projections, reflections, isometries.
eigenvalues and eigenvectors
- Orientation, the cross product and the vector triple
product. Determinants as volumes.
- Affine geometry. metric characterization of lines and segments
- Conic sections - synthetic definition: An ellipse is
the locus of points whose sum of distances from two given points (the
foci) is constant. A hyperbola is
the locus of points so that the absolute value of the difference
of the distances from two given points (the foci) is constant.
A parabola is the locus of points equidistant from a point (the focus)
and a line (the directrix). Eccentricity.
Canonical equations for these conic
The "Boscovitch property" - existence
for directrices for an ellipse (not a circle) and a hyperbola.
- The classification of plane quadrics
Q(x,y)=ax2 +bxy +cy2 +dx +ey+f=0 .
- Curves in the plane: Tangents, normals, characterization of the
tangent as the closest line to the curve which passes through a given
point. The tangent and normal to curves defined implicitly as
(background: The chain rule and the gradient).
- Tangents to conic sections: A line intersects an ellipse at
exactly one point iff it is tangent to the ellipse.
The optical properties of ellipses, hyperbolas and parabolas.
Given a line and two points on the same side of the line, there is a
unique ellipse whose foci are the two points and to which the line is
- Billiards: the shortest path between two points and hitting a
line, the law of reflection, the billiard map and phase space.
- Billiards in a circle: the angle with the tangent is constant, an
orbit is periodic iff this angle is a rational multiple of π,
and otherwise the impact points are dense on the boundary of the
Kroneckers' theorem on density of orbits of an irrational rotation.
- Billiards in an ellipse: If a segment of a
billiard trajectory intersects the the major axis segment then all
segments do so. If a trajectory segment does not intersect the major
axis segment then is tangent to an ellipse confocal with the original
one and all other segments of the trajectory remain tangent to that
ellipse. For a trajectory not intersection the major axis segment,
there is a hyperbola confocal with the original ellipse to which all
trajectory segments are tangent (did not prove in class).
- Arc length and natural parametrization
- curvature and the differential geometry of curves.
Definition of curvature as the length of the curvature vector k(s) =
||R''(s)||. (we did not discuss signed curvature).
Then k(s)=d θ/ds where dθ is the angle
between the tangents R' (s) and R' (s+ds).
A formula for arbitrary parametrization:
k(t) = ||r' X r''||/||r'||3.
If k(s) is nonzero then the curve lies on one side of the tangent,
together with the curvature vector R''(s).
- The radius, center and circle of curvature (osculating circle),
the evolute of a curve. Characterization of the osculating circle as
the closest circle to a curve passing through the given point on the
curve (3-rd order contact with the curve).
- Frenet's equation.
- Uniqueness and existence of curves with given nowhere zero