Extended Euclidean Plane
Geometry is founded by a set of basic elements and basic relations between them (which are not defined, but are intuitively clear) and a system of axioms (statements which are not proved, but are considered to be true and are intuitively obvious) from which other figures are defined and all corresponding consequences (theorems) are deduced.
We can describe projective geometry as the study of the geometric properties which are invariant under a central projection. The central projection is the closest to the image that develops in our eye and later it will be defined. Note that the axioms A_{1} i A_{2} can be obtained from each other by the replacement of the words line and point. This fact conditions the principle of duality on the projective plane, i.e. if a statement is true in projective geometry it will remain true if we interchange the words line and point as well as the words join and intersect (lie on and pass through). Therefore, if we prove a theorem in the projective geometry, we also prove its dual statement. Notions whose definitions are connected in such a way are referred to as dual notions. For example: range of points (p)  a set of all points in the plane incident with a line p, pencil of lines (P)  a set of all lines in the plane incident with a point P. We extend the Euclidean plane in the following way: The elements of the plane at infinity (line and points that lie on it) are real as are all the other elements of the plane. The described Euclidean plane is called the extended Euclidean plane or real projective plane. In the projective plane every two lines intersect (parallel lines intersect at a point at infinity). Projective geometry does not use measurements (the distance of points and measure of angles). Measure is a characteristic feature of the Euclidean geometry, and is indispensable in the field of engineering. Therefore we shall use the extended Euclidean plane (and later space) in our course for the deduction of certain general conclusions about the plane (spatial) figures, while we shall use the Euclidean plane (space) with the Euclidean metric for the constructions (on the paper or by computer). ^{*} For the incidence relation we often use following expressions: "lie on" (point lies on the line), "pass through" (line passes through the point), "join" (line joins two points), "intersect" (two lines intersect) etc. Points incident with the same line are called collinear and the lines incident with the same point are called concurrent. Created by Sonja Gorjanc  3DGeomTeh  Developing project of the University of Zagreb.
