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. | ||
==Projection of a circle into a circle== | ==Projection of a circle into a circle== | ||
| − | Let a circle <math>\omega</math> with diameter <math>PQ | + | [[File:Circle to circle.png|350px|right]] |
| + | Let a circle <math>\omega</math> with diameter <math>PQ</math> and a point <math>A</math> on this diameter <math>(2AP < PQ)</math> be given. | ||
| − | Find the prospector of the central projection that maps the circle <math>\omega</math> into the circle <math>\omega'</math> and the point <math>A</math> into point <math>O</math> - the center of <math>\omega'.</math> | + | Find the prospector of the central projection that maps the circle <math>\omega</math> into the circle <math>\omega'</math> and the point <math>A</math> into point <math>O'</math> - the center of <math>\omega'.</math> |
| + | |||
| + | <i><b>Solution</b></i> | ||
| + | |||
| + | Let <math>S</math> be the center of transformation (perspector) which is located on the perpendicular through the point <math>P</math> to the plane containing <math>\omega.</math> Let <math>P = \omega \cap \omega', PQ'</math> be the diameter of <math>\omega', PO' = O'Q'</math> and plane <math>\omega'</math> is perpendicular to <math>SQ.</math> | ||
| + | |||
| + | Spheres with diameter <math>PQ</math> and with diameter <math>PQ'</math> contain a point <math>Q'</math>, so they intersect along a circle <math>\omega'</math>, so the circle <math>\omega</math> is a stereographic projection of the circle <math>\omega'</math> from the point <math>S,</math> that is, if the point <math>M</math> lies on <math>\omega</math>, there is a point <math>M'</math> on the circle <math>\omega'</math> along which the line <math>SM</math> intersects <math>\omega'</math>, then <math>\omega</math> is projected into <math>\omega'</math> under central projection from the point <math>S.</math> | ||
Revision as of 15:10, 31 October 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.
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 
