Difference between revisions of "Projective geometry (simplest cases)"
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5. There is a central projection that maps a circle to a circle, and a chosen interior point of the first circle to the center of the second circle. This central projection maps the polar of the chosen point to the line at infinity. | 5. There is a central projection that maps a circle to a circle, and a chosen interior point of the first circle to the center of the second circle. This central projection maps the polar of the chosen point to the line at infinity. | ||
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6. The relationships of segments belonging to lines parallel to the exceptional line are the same for images and preimages. | 6. The relationships of segments belonging to lines parallel to the exceptional line are the same for images and preimages. | ||
Revision as of 04:02, 2 November 2024
Projective geometry contains a number of intuitively obvious statements that can be effectively used to solve some Olympiad mathematical problems.
Useful simplified information
Let two planes and
and a point
not lying in them be defined in space. To each point
of plane
we assign the point
of plane
at which the line
intersects this plane. We want to find a one-to-one mapping of plane
onto plane
using such a projection.
We are faced with the following problem. Let us construct a plane containing a point and parallel to the plane
Let us denote the line along which it intersects the plane
as
No point of the line
has an image in the plane
Such new points are called points at infinity.
To solve it, we turn the ordinary Euclidean plane into a projective plane. We consider that the set of all points at infinity of each plane forms a line. This line is called the line at infinity. The plane supplemented by such line is called the projective plane, and the line for which the central projection is not defined is called (in Russian tradition) the exceptional line of the transformation. We define the central projection as follows.
Let us define two projective planes and
and a point
For each point of plane
we assign either:
- the point of plane
at which line
intersects
- or a point at infinity if line does not intersect plane
We define the inverse transformation similarly.
A mapping of a plane onto a plane is called a projective transformation if it is a composition of central projections and affine transformations.
Properties of a projective transformation
1. A projective transformation is a one-to-one mapping of a set of points of a projective plane, and is also a one-to-one mapping of a set of lines.
2. The inverse of a projective transformation is projective transformation. The composition of projective transformations is a projective transformation.
3. Let two quadruples of points and
be given. In each quadruple no three points lie on the same line: Then there exists a unique projective transformation that maps
to
to
to
to
4. There is a central projection that maps any quadrilateral to a square. A square can be obtained as a central projection of any quadrilateral.
5. There is a central projection that maps a circle to a circle, and a chosen interior point of the first circle to the center of the second circle. This central projection maps the polar of the chosen point to the line at infinity.
6. The relationships of segments belonging to lines parallel to the exceptional line are the same for images and preimages.
Projection of a circle into a circle
Let a circle with diameter
and a point
on this diameter
be given.
Find the prospector of the central projection that maps the circle into the circle
and the point
into point
- the center of
Solution
Let be the center of transformation (perspector) which is located on the perpendicular through the point
to the plane containing
Let
be the diameter of
and plane
is perpendicular to
Spheres with diameter and with diameter
contain a point
, so they intersect along a circle
Therefore the circle is a stereographic projection of the circle
from the point
That is, if the point lies on
, there is a point
on the circle
along which the line
intersects
It means that is projected into
under central projection from the point
is antiparallel
in
is the symmedian.
Corollary
Let
The inverse of a point
with respect to a reference circle
is
The line throught in plane of circle
perpendicular to
is polar of point
The central projection of this line to the plane of circle from point
is the line at infinity.
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