The language of axiomatic projective geometry is a language consisting of two so

Question

The language of axiomatic projective geometry is a language consisting of two sorts: point and line, and one binary predicate I (p, L) we called "incidence" between points and lines. There are various ways to call I(p, L) : point p is incident with line L, L is incident with p, p lies on L, p is on L , or L passes through p. One model of projective geometry is the unit sphere (think about our Earth). A point means a point at the sphere. A line means a circle which is intersection of the sphere with a plane passing through the origin (for example, the equator, or a circle through the north pole and south pole, or any rotation of them). Let us list a few of the axioms in projective geometry: Axiom 1. Any two distinct points are incident with exactly one line. Axiom 3. There exist at least four points, no three of which are collinear. 2 In a geometry with two undefined primitive terms, the dual of an axiom or theorem is the statement with the two terms interchanged. For example, the dual of "A line contains at least two points," is "A point contains at least two lines." An axiom system in which the dual of any axiom or theorem is also an axiom or theorem is said to satisfy the principle of duality. Plane projective geometry is an example of a geometry that satisfies the principal of duality. What is the dual of Axiom 1? What is the dual of Axiom 3?

Explanation / Answer

dual of Axiom 1 is there will be one and one line will be passed through any two fixed point i.e. from given two point we can not connect them from more than one line and it will give the shortest distance between those point. dual of axiom 3 is for a projective geometry there is need of a closed surface which should be a quadrilateral. for any quadrilateral there is need of at least 4 point and any 3 points should not be collinear.

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