Geometry

PROJECTIVE GEOMETRY

Contents Introduction to projective geometry Projective invariants, infinity, collinearity, cross ratio Axioms for Projective Geometry Principle of duality Pappus Theorem Desargues Theorem Projective conic sections Analysis of statements related to projective geometry

Introduction to projective geometry 01

Introduction In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice versa.

Projective invariants, infinity, collinearity, cross ratio

Plane projective geometry treats properties of geometric figures that are invariant under certain transformations, the "projectivitiest'. The classical theorems mainly deal with incidences of points and lines. Metrical concepts, such as distance and area, have less natural sites in projective geometry. One well-known exception is the cross-ratio, relating distances between collinear points. On the other hand, area invariants have been discovered in connection with robot control and vehicle guidance. There the problem was to find objects suited as signposts or marking symbols. For reasons of error robustness and existing hardware, øreø, measurements were preferable. Figure 1 shows one object that features area-invariants, and a possible image of it.

Next list of properties of the cross-ratio gives, among other things, the reason for the validity of the relations between the 24 possible orderings of the table of the preceding section. The first two properties are discussed in the file Projective line. The rest is proved by easy calculations. The next

1. Given three pairwise different real numbers {1 , 2 , 3 } the “cross ratio function” = () = (12 , 3) defines an invertible “homographic relation”, whose graph is a rectangular hyperbola. 2. Given three pairs of real numbers {(1 , 1 ), (2 , 2 ), (3 , 3 )} in general position, the equation (12 , 3) = (12 , 3) solved for defines uniquely a homographic re- lation = (), such that { (1 ) = 1 , (2 ) = 2 , (3 ) = 3 .} 3. Every permutation, which is product of two “transpositions” of the letters {, , , } leaves the cross ratio invariant i.e. (, ) = (, ) = (, ) = (, ). Hence from the 4! in total permutations of the 4 letters, the various resulting cross ratios obtain only 6 different values (seen in the table of section 1). 4. (, ) = (, )−1 and (, ) + (, ) = 1. From these follow the last equalities in the rows of the table of section 1.

Boolean algebra is a branch of algebra that deals with binary numbers and binary variables. The principle of duality is a kind of pervasive property of algebraic structure in which two principles or concepts are interchangeable only if all outcomes held true in one formulation are also held true in another. This concept is also referred to as "dual formulation". The principle of duality is a kind of pervasive property of algebraic structures in maths in which two principles or concepts are interchangeable only if all outcomes held true in one formulation are also held true in another. This concept is also referred to as "dual formulation." We interchange unions (U) into intersections (∩) or intersections (∩) into the union (U). We also sometimes interchange the universal set with the null set (\phi) or the null set with the universal set to obtain the dual statement. If we interchange the symbols and obtain this statement itself, it will be called the self-dual statement. Physicists also find the principle of duality important. Let me help you recall the dual properties of light as a wave and particle.

Pappus Alexandrinus, Greek mathematician, approximately 3rd or 4th century AD.) 1.If a plane area is rotated about an axis in its plane, but which does not cross the area, the volume swept out equals the area times the distance moved by the centroid. 2.If a plane curve is rotated about an axis in its plane, but which does not cross the curve, the area swept out equals the length times the distance moved by the centroid. These theorems enable us to work out the volume of a solid of revolution if we know the position of the centroid of a plane area, or vice versa; or to work out the area of a surface of revolution if we know the position of the centroid of a plane curve or vice versa. It is not necessary that the plane or the curve be rotated through a full 360o . We prove the theorems first. We then follow with some examples.

In projective geometry, Desargues's theorem, named after Girard Desargues, states: Two triangles are in perspective axially if and only if they are in perspective centrally. Denote the three vertices of one triangle by a, b and c, and those of the other by A, B and C. Axial perspectivity means that lines ab and AB meet in a point, lines ac and AC meet in a second point, and lines bc and BC meet in a third point, and that these three points all lie on a common line called the axis of perspectivity. Central perspectivity means that the three lines Aa, Bb and Cc are concurrent, at a point called the center of perspectivity.

Conic sections can be regarded as plane sections of a right circular cone (see the figure). By regarding a plane perpendicular to the cone’s axis as the reality plane (RP), a “cutting” plane as the picture plane (PP), and the cone’s apex as the projective “eye,” each conic section can be seen to correspond to a projective image of a circle (see the figure). Depending on the orientation of the cutting plane, the image of the circle will be a circle, an ellipse, a parabola, or a hyperbola.

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