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.
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 , so 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 , then is projected into under central projection from the point