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Encyclopedia > Fundamental theorem of projective geometry

In mathematics, the fundamental theorem of projective geometry states that if Pn is a projective space and F and F′ are frames of Pn, then there exists a unique projective transformation sending F to F′.


In case n = 1 this comes down to saying that given two ordered triples of distinct points, there is a projective transformation of the projective line taking the first triple to the second. This is a basic result on Möbius transformations, saying that the group they form is "triply" transitive.


  Results from FactBites:
 
Projective geometry - Wikipedia, the free encyclopedia (2706 words)
Projective geometry is a non-metrical form of geometry that emerged in the early 19th century.
Projective geometry is a non-Euclidean geometry that formalizes one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way.
In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line".
Collineation - Wikipedia, the free encyclopedia (313 words)
A collineation, roughly, is a map from one projective space to the other, preserving the geometric structure.
The fundamental theorem of projective geometry states the converse :
The fundamental theorem explains the different definition for projective lines.
  More results at FactBites »


 

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