QG-Tf1: QG-Projective Square Transformation


A projective transformation in a plane is a transformation used in projective geometry.
According to Ref-20, pages 22 and 24:
Definition:
A projective mapping of a projective plane Π onto a projective plane Π’ is a one-one mapping of Π onto Π’ such that the images of three collinear points are collinear.
A projective mapping of Π onto itself is called a projective transformation of Π.
Theorem I.
Given any four points A, B, C, D of a projective plane Π, no three of which are collinear, and four points A’, B’, C’, D’ of a projective plane Π’, no three of which are collinear,
there exists one and only one projective mapping α of Π onto Π’ that takes A, B, C, D into A’, B’, C’, D’, respectively.
Square Transformation:
Using this projective transformation it is possible to transform 4 points of a square to any other set of 4 points. Since a square consists of 4 points where the notion of adjacent points is relevant it can be seen as a quadrigon. That’s why the image of the transformation also will be a quadrigon.
Since projective transformations are invertible, Quadrigons can be uniquely transformed into a Square and a Square can be uniquely transformed back into the Reference Quadrigon.
A projective transformation can be used to transform points of the whole plane. This makes it possible to transform the points of a circle circumscribing a square with the same projective transformation that converts the square into a Reference Quadrigon.
More popular:
A projective transformation describes what happens to the perceived positions of observed objects when the point of view of the observer changes. Projective transformations do not preserve sizes or angles but do preserve incidence (the intersection point of 2 lines is after transformation the intersection point of both transformed lines, the connecting line of 2 points is after transformation the connecting line of both transformed points).


 

Plaats reactie