Projective geometry

perfect_radio

I saw projective geometry being used on this site to solve many geo problems. Can someone tell me what it is and maybe some properties/theorems?

Or perhaps give a link?

_________________
"Germany has offered to send troops to the Lebanon border. I bet Israel's breathing a sigh of relief there. Nothing makes Jewish people feel safer and more secure than the German Army marching on their border."

fanaticsm · Poincare Conjecture Offline Joined: 26 Oct 2004 Posts: 155

Here is a brief outline of some properties and some links/notes for further info:

Some basic axioms...

Plane Properties:

1. If A and B are distinct points on a plane, there is at least one line containing both A and B.
2. If A and B are distinct points on a plane, there is not more than one line containing both A and B.
3. Any two lines in a plane have at least one point of the plane (which may be the point at infinity) in common.
4. There is at least one line on a plane.
5. Every line contains at least three points of the plane.
6. All the points of the plane do not belong to the same line.

Space Properties:

1. Three points not all coincident with the same line are coincident with a unique plane.
2. Three planes not all coincident with the same line are coincident with a unique point.
3. For a line and a plane not coincident with it, there's exactly one point that is coincident with both.
4. For a line and a point not coincident with it, there's exactly one plane that is coincident with both.
5. Two distinct planes are coincident with exactly one common line.
6. Two distinct points are coincident with exactly one common line.

There are also a number of resources on the web, and I have provided some here. I think that much more can be found by simply searching for 'Projective Geometry' using Google. Also, Kiran Kedlaya's geometry notes contain an entire section devoted to this subject as well. A copy of his notes can be found on the AMC website.

Links:

http://mathworld.wolfram.com/ProjectiveGeometry.html
http://www.cs.elte.hu/geometry/csikos/proj/proj.html
http://ai.stanford.edu/~birch/projective/
http://www.kuro5hin.org/story/2003/5/26/92949/1229
http://mathforum.org/library/topics/projective_g/
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/BEARDSLEY/beardsley.html

**Posted:** Mon May 02, 2005 4:49 pm

murgi

http://www.imomath.com/index.php?options=mbb|strsljenbfa&p=0
may also be an usefull link Smile

**Posted:** Sun Feb 22, 2009 9:12 am