All the material which is written in small scripts, in footnotes, in Appendices ,which is marked by the symbol $^*$ or $\dagger$, and which is written in handwritten notes using red pen (or is put in the red frames) is not compulsory.
The same is for Homeworks:
All exercises marked by the symbol
$\dagger$ are not compulsory.
Here is the pdf file
of the lecture notes.
These lecture notes cover the lectures
of first 7 weeks.
See below the pdf files of scanned
lectures for last four weeks.
This is lectures on conic
sections and basic notions of projective geometry.
Every `lecture' contains 4-5 pages.
It normally
covers material of 2/3-rd
of usual 50 minutes lecture
(The content of last 12 lectures cover the
8 50 minutes lectures for last four weeks.)
These lectures are so `small' because of
some technical reasons related with scanning of
the files.
Here it is the pdf file
of the lecture C1 of second part of the course.
Here
we give the geometrical definition of
ellipse, hyperbola and parabola.
(This lecture began at 22-nd March and will
end at 23-March)
Here is
the pdf file
of the lecture C2 of second part of
the course. Here
we will recall Cartesian coordinates,
and transformation
from Cartesian coordinates to another
Cartesian coordinates.
At the end very briefly we consider general
affine transformations of plane.
(This lecture began and ended
at 23 march.)
Here is
the pdf file
of the lecture C3 of second part of the course.
Here
we give analytical definition of
conic sections---ellipse, hyperbola and parabola.
We show that
analytical definition of conic sections is
equivalent
to their geometrical definition, which we considered
in the lecture C1)
(This lecture began at 24-th March and it ended
at 29 March.)
Here it is the pdf file of the lecture C4 of second part of the course. Here we will formulate and prove the fundamental Theorem that intersection of plane with surface of cone is an ellipse, hyperbola or parbola (except degenerate cases). (This justifies the termin `conic section' for these curves.)
We consider the orthogonal projection of a conic section (intersection of plane with conic surface $k^2x^2+k^2y^2=z^2$) on the horizontal plane $z=0$. The orthogonal projection is also a conic section, and vertex of the cone iz one of the foci of this conic section.
The very last page of this lecture is
not compulsory.
(This lecture began at 29-th March and it ended
at 30 March.)
Here is
the pdf file of the very short lecture,
the lecture C5 of second part of the course. Here
we show that affine transformations
(not orthogonal!)
transform ellipse to circle, and
using this fact we come to
the formula for area of an ellipse.
(This lecture began and ended
at 30 March.)
Here is the pdf file of the lecture C6 of second part of the course. In this lecture we introduce the concept of projective line. Projective line= Line with a point at infinity. It can be defined as a set of lines in $R^2$ which pass through origin. It is useful to consider the usual line $y=1$ in $R^2$. Then points of projective line $RP^1$ can be viewed as points of usual line $y=1$ with coordinate $u$, and the point at infinity. Every point $u$ of the line $y=1$ can be viewed as the intersection of the line $x=uy$ passing through origin with the line $y=1$. The point at infinity is represented by the line $y=0$, which is parallel to the line $y=1$. We introduce homogeneous coordinates on projective line $RP^1$. (This lecture began and ended at 26 April-the first lecture after Easter break.)
Attention: the third line of the first
page of this lecture
is written `projected line'. This has to be written
`projective line'.
Here is the pdf file of the lecture C7 of second part of the course. In this lecture we consider projective transformations of projective line $RP^1$. The transformations of Euclidean line which preserve the Euclidean structure (distance)--the isometry of Euclidean line--- are translations (and reflection). The transformations of of affine line $R$ which preserve affine structure are enlargements (dilations) and translations.
Projective transformations of projective
line $RP^1$ are described
by linear transformation in $R^2$:
to linear transformation
$x\to x'=ax+by, y\to y'=cx+dy$
we assign transformation
$[x:y]\to [x':y']=[ax+by:cx+dy]$
in homogeneous coordinates.
(We suppose that linear transformation
is non-degenerate, i.e. $ad-bc\not=0$.)
Respectively in affine coordinate
(u=x/y) we come to
linear fractional transformations:
$u'=x'/y'=(au+b)/(cu+d)$. They
include affine trasnformation of affine line.
(This lecture began
at 26 April and it ended at 27 April.)
Here is
the pdf file of the
lecture C8 of second part of the course.
In this lecture we
defined the cross-ratio $(A,B,C,D)$
of four distinct points
$A,B,C,D$ on the porjective line.
If $u_A$ is affine coordinate of a point $A$,
$u_B$ is affine coordinate of a point $B$,
$u_C$ is affine coordinate of a point $C$,
$u_D$ is affine coordinate of a point $D$
then the cross ratio of these points is
equal to
$(A,B,C,D)=
{(u_A-u_C)(u_B-u_D)}
/
{(u_A-u_D)(u_B-u_C)}
$.
One can define also cross-ratio in
homogeneous coordinates.
The cross-ratio is the invariant of
projective
transformations. We will give the
proof of this result
(not compulsory) and show that the central
projection preserves the cross-ration, and it
is the special case of projective transformations.
(One can show that every projective
transformation is composition
of central projections.)
(This lecture began and ended at
at 27 April.)
Here it is
the pdf file of the
lecture C9 of second part of the course.
In this lecture we
define the projective plane $RP^2$.
Projective plane $RP^2$ is the set of lines
in $R^3$ passing through origin.
Projective plane may be viewed
as a set which possesses the usual
affine plane $R^2$
and set of points at infinity.
Indeed consider the usual plane $z=1$ in $R^3$.
Then every point $(u,v)$ of this plane
can be viewed as the intersection of the line
$x=uz,y=vz$ passing through origin with this plane.
The lines passing through origin which are parallel
to the plane $z=1$ represent points at infinity.
(We can think on $RP^2$ as union of plane
$R^2$ and the projective line $RP^1$.)
(Compare with definition of
projective line.)
(This lecture was on 3-rd May).
Here is the pdf file of the lecture C10 of second part of the course. In this lecture we consider projective lines on projective planes. Recall that a point of $RP^2$= line in $R^3 passing through origin. Projective line in $RP^2$=set of lines which belong to the same plane, i.e. projective line is represented by plane in $R^3$ passing through origin. If $A=[x_A:y_A:z_A]$, $B=[x_B:y_B:z_B]$ are two point on $RP^2$ the the line $AB$ passing thorug these points is represented by the plane $mr_A+nr_B$ spanned by the vectors $r_A=(x_A,y_A,z_A)$ and $r_B=(x_B,y_B,z_B)$.
Then we consider
the conception of collinearity
for points: three points $A,B,C$ in the
projective plane
are collinear if they belong to the same
projective line. We formulate the
criterion that three
points are collinear in terms of
homogeneous coordinates
and in terms of affine coordinates.
(This lecture
began and finished at 3-rd May)
Here it is the pdf file of the lecture C11 of second part of the course. In this lecture we calculate cross-ratio of four distinct points $A,B,C,D$ which are collinear, i.e. which belong to the same projective plane. Considering an arbitrary affine coordinate of these points (the same coordinate for all points !) we calculate the cross-ratio of these points. The answer does not depend on the choice of how we have chosen an affine coordinate, since cross-ratio is the invariant of projective transformations which include the affine transformations.
How to calculate the cross-ratio if one of the points is at infinity. Sure we may use homogeneous coordinates, or to do it considering the point moving to infinity (see also the exercise 3 from tthe Homework 8). This part of the lecture and the rest of the lecture is not compulsory
At the end of the lecture
we consider the projective
transformation of projective plane $RP^2$.
Recalling projective transformations of
projective line $RP^1$ (see the lecture C7)
we define projective transformation
by linear transformation in $R^3$:
in the following way: to linear transformation
$x\to x'=ax+by+cz$,
$y\to y'=dx+ey+fz$
$z\to z'=gx+hy+iz$
we assign transformation
in homogeneous coordinates by
$[x:y:z]\to [x':y':z']=
[ax+by+cz:dx+ey+fz:gx+hy+iz]$
(we suppose that corresponding linear
transformation is non-degenerate).
Respectively in affine coordinate
(u=x/z, v=y/z) we come to
linear fractional transformations:
$u'=x'/z'=(au+bv+c)/(gu+hv+i)$.
$v'=y'/z'=(du+ev+f)/(gu+hv+i)$.
These transformations
include affine transformation of affine plane:
$u'=au+bv+e, v'=cu+dv+f$.
Here is
the pdf file of the last
lecture, the lecture C12 of the
second part of the course.
In this lecture we return to classification of
conic sections on
the base of the projective geometry.
First of all we note again that ellipse and circle
are affine equivalent (see also the lecture C5)
Then we consider the following example: