Geometry
We show that the fact that three heights of triangle intersect at the one point
follows from the Jacoby identity for cross-product.
We show relation between Mercator projection and function Logarithm. (in Russian)
(This etude is the part of my article with Sasha Borovik:
A. Borovik, O.M. Khudaverdian. Merkator projection, logarithm
and...(in Russian) Matematiqeskoe Prosvewe- nie no. 14 (2010), 58–82.
)
We study shape operator for surface of codimension 1 in Euclidean space and show
why for even-dimensional surface Gaussian curvature is invariant of isometries.
The group SO(3) have a memory about projective space..
(This etude was initiated many many years ago by S.Borovik after remarkable
seminar)
We try to answer on the question: why an arbitrary transformation which transforms straight lines to straight lines are projective trasnformations. Our considerations
are elementary.
It is elementary geometrical fact that two polygons of equal area are equipartial.
It is deep result of XX-th centure mathematics that cube and tetrahedron of equal
volumes are not equipartial (Hilbert's problem). In other words one does not need
Calculus to find the volume of triangle and one does need Calculus to find the volume of
tetrahedron (22.04.09). (I corrected erratum and added
Remark on 28.12.13)
Explicit formula defining globally affine connection on manifold (without use of existence of
Riemannian metric) is presented.
(22.02.10)
We discuss here one of possible definitions of Schwarzian.
(26.06.12)
We present here nice calculations of Killings of sphere which possess
information of Runge-Lentz vectors for Coulomb problem .
(28.02.13)
We discuss here one integral which reveals the basic
properties of localisation formula.
(12.10.13)
It is very natural to formulate
Duistermaat-Heckman localisation formula in the language of
"supermathematics". We discuss here this formulation
and give an idea of proof of Duistermaat-Heckman formula
in this approach. This text is the continuation of the previous etude.
(28.10.13)
We consider here geometrical nature of some
surface integrals.
(06.11.13)
This is sketch of the lecture "Galois group" for students
(see also the file "Chebyshev approx...." in subsection "Algebra" of
Etudes)(26.11.13)
If $C$ is a circle then the set of points which are verticis
of circular cone over $C$ is a
line passing through the centre of this circle. How looks this locus
in the case if $C$ is just an arbitrary quadric on the plane?
In this etude I try to asnwer this question
which I heard from Gabor Megyesi (Autumn 14).
Comment (Spring 2017): In the file above doing hard calculations
three years ago I came to very beautiful answer:
Let $C$ be an ellipse (hyperbola)
with foci at the points $F_1,F_2$. Then take the plane orthogonal
to the curve $C$ which passes through the foci. Consider cofocal
hyperbola (ellipse)
$C^*$ in this plane, i.e. the hyperbola (ellispe)
which passes via foci $F_1,F_2$ and has foci
at the points $F_1^*,F_2^*$ which are intersection of $F_1F_2$
with intial curve $C$. Answer is very beautiful.
Just two days ago I found very short and elegant explanation of
this fact, using Dandelen spheres: If a point $K$ belongs to the locus
then consider triangle $KF_1^*F_2^*$ and the Dandelen circle
in this triangle which touches the edge $F_1^*F_2^*$ at the point
$F_1$, we see that difference of the sides $KF_1^*$
and $KF_2^*$ is exactly equal to the difference of
the segments $F_1^*F_1$ and $F_1F_2^*$. Thus $K$ belongs to this hyperbola
(we suppose that $C$ is ellipse.)
THis is really short and beautiful.
It becomes fashionable to quote the Ptolemy's Theorem
during ``presentation'' talk on cluster algebras.
This proof of Ptolemy's Theorem I heard from
Anna Felikson (Durham University). (21.05.15)
Stereographic projection has many beautiflu properties.
This is nice to see that this is a restriction of
more elementary map, invrsion. (29.01.17)
I consider here the problem which looks as a standard
not very easy question in Euclidean geometry, but it has
beatiful interpretation in terms of hyperbolic geometry
(22.06.17)
Orbits planets are ellipses. Can we explain it without caclulus?
(18.07.17)
We make a bridge between vector field and
Poisson bracket including these structures in the triple...
(10.12.17)
We consider a solution of an elementary
"pursuit" problem using conic sections
(14.04.18)
We consider a simple and beautiful inequality
on determinants
(12.06.18)
We discuss the formula for distance
in half-plane model of Lobachevsky geometry
(26.06.18)
We discuss how looks infinitesimal transformation
of contact space which generates Legendre transformation, and
apply it to Clairaut equation
(04.05.19)
We return again to localisation (see notes in
2013). Here I try to fomulate localisation principle in
a more `rational' way
(12.10.19)
We return to one question in stereometry which I was
thinking about over 30 years ago. Surprisingly it is related
with orthocentre.
(16.06.20)
We return to the definition of geometrical obejct.
We learned it long ago, but I understood it much later.
(24.09.20)