Projective geometry (simplest cases)
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.
Contents
[hide]- 1 Projection of a circle into a circle
- 2 Butterfly theorem
- 3 Sharygin’s Butterfly theorem
- 4 Semi-inscribed circle
- 5 Fixed point
- 6 Sphere and two points
- 7 Projecting non-convex quadrilateral into rectangle
- 8 Projecting convex quadrilateral into square
- 9 Medians crosspoint
- 10 Six segments
- 11 Sines of the angles of a quadrilateral
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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Butterfly theorem
Let be the midpoint of a chord
of a circle
through which two other chords
and
are drawn;
and
intersect chord
at
and
correspondingly.
Prove that is the midpoint of
Proof
Let point be the center of
We make the central projection that maps the circle into the circle
and the point
into the center of
Let's designate the images points with the same letters as the preimages points.
Chords and
maps into diameters, so
maps into rectangle and in this plane
is the midpoint of
The exceptional line of the transformation is perpendicular to so parallel to
The relationships of segments belonging to lines parallel to the exceptional line are the same for images and preimages. We're done! .
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Sharygin’s Butterfly theorem
Let a circle and a chord
be given. Points
and
lyes on
such that
Chords
and
are drawn through points
and
respectively such that quadrilateral
is convex.
Lines and
intersect the chord
at points
and
Prove that
Proof
Let us perform a projective transformation that maps the midpoint of the chord to the center of the circle
. The image
will become the diameter, the equality
will be preserved.
Let and
be the points symmetrical to the points
and
with respect to line
the bisector
Denote (Sharygin’s idea.)
is cyclic
is cyclic
points
and
are collinear.
Similarly points and
are collinear.
We use the symmetry lines and
with respect
and get in series
symmetry
and
with respect
symmetry
and CB with respect
symmetry
and
with respect
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Semi-inscribed circle
Let triangle and circle
centered at point
and touches sides
and
at points
and
be given.
Point is located on chord
so that
Prove that points and
(the midpoint
are collinear.
Proof
Denote point on line
such that
Therefore line is the polar of
Let us perform a projective transformation that maps point to the center of
Image is the point at infinity, so images
and
are parallel.
Image is diameter, so image
is midpoint of image
and image
is midpoint of image
so image
is parallel to the line at infinity and the ratio
is the same as ratio of images.
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Fixed point
Let triangle and circle
centered at point
and touches sides
and
at points
and
be given.
The points and
on the side
are such that
The cross points of segments and
with
form a convex quadrilateral
Point lies at
and satisfies the condition
Prove that
Proof
Let us perform a projective transformation that maps point to the center of
Image is the point at infinity, so images
and
are parallel. The plane of images is shown, notation is the same as for preimages.
Image is diameter
image
is parallel to the line at infinity, so in image plane
Denote
is rectangle, so
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Sphere and two points
Let a sphere and points
and
be given in space. The line
does not has the common points with the sphere. The sphere is inscribed in tetrahedron
Prove that the sum of the angles of the spatial quadrilateral (i.e. the sum
does not depend on the choice of points
and
Proof
Denote points of tangency
and faces of
(see diagram),
It is known that
Similarly,
The sum not depend on the choice of points
and
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Projecting non-convex quadrilateral into rectangle
Let a non-convex quadrilateral be given. Find a projective transformation of points
into the vertices of rectangle.
Solution
WLOG, point is inside the
Let and
be the rays,
be any point on segment
Planes
and
are perpendicular, planes
and
are parallel, so image
is line at infinity and
is rectangle.
Let's paint the parts of the planes and
that maps into each other with the same color.
maps into
(yellow).
Green infinite triangle between and
maps into
where plane
is parallel to plane
Blue infinite quadrilateral between and
with side
maps into quadrilateral
Therefore inner part of quadrilateral maps into external part of rectangle
For example
maps into
where
is the intersection of planes
and
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Projecting convex quadrilateral into square
Let be a convex quadrilateral with no parallel sides.
Find the projective transformation of into the square
if the angle between the planes
and
is given. This angle is not equal to
or
Solution
Denote
Let be the point satisfying the conditions
The locus of such points is the intersection circle of spheres with diameters and
Let be the perspector and the image plane be parallel to plane
We use the plane contains
so image
Then image is the line at infinity, point
is point at infinity, so images
(line
) and
(line
) are parallel to
Similarly point is the point at infinity, so images
is the rectangle.
Point is the point at infinity, so
Point
is the point at infinity, so
is the square.
Let be such point that
The angle between and plane
is the angle we can choose. It is equal to the angle between planes
and
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Medians crosspoint
Let a convex quadrilateral and line
in common position be given (points
and
not belong
sides and diagonals are not parallel to
Denote
Denote
and
midpoints of
and
respectively.
Prove that lines
and
are collinear.
Proof
Six segments
Let a convex quadrilateral and line
in common position be given (points
and
not belong
sides and diagonals are not parallel to
Denote
Prove that
Proof
By applying the law of sines, we get:
(see Sines of the angles of a quadrilateral)
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Sines of the angles of a quadrilateral
Let a convex quadrilateral be given. Prove that
Proof
By applying the law of sines, we get:
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