Projective Geometry: Exploring Projective Geometry in Computer Vision

By Fouad Sabry

What is Projective Geometry

Projective geometry is a branch of mathematics that focuses on the study of geometric qualities that remain unchanged regardless of the transformations that are being applied to them. This indicates that, in contrast to simple Euclidean geometry, projective geometry is characterized by a distinct environment, a space that is the subject of the project, and a limited collection of fundamental geometric notions. For a given dimension, the fundamental intuitions are that projective space has a greater number of points than Euclidean space does, and that geometric transformations are allowed that change the extra points into Euclidean points, and vice versa.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Projective geometry

Chapter 2: Projective plane

Chapter 3: Projective space

Chapter 4: Affine geometry

Chapter 5: Desargues's theorem

Chapter 6: Duality (projective geometry)

Chapter 7: Complete quadrangle

Chapter 8: Homography

Chapter 9: Desargues configuration

Chapter 10: Conic section

(II) Answering the public top questions about projective geometry.

(III) Real world examples for the usage of projective geometry in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Projective Geometry.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Apr 30, 2024

Book preview

Chapter 1: Projective geometry

Projective geometry is the branch of mathematics concerned with the study of geometric qualities which are invariant under projective transformations. Projective geometry, then, uses a different environment from that of traditional Euclidean geometry and employs a smaller subset of the fundamental notions of geometry. There are more points in projective space than in Euclidean space of the same dimension, and geometric transformations are allowed that convert the extra points (termed points at infinity) to Euclidean points, and vice versa.

This new concept of transformation, more radical in its consequences than can be stated by a transformation matrix and translations, but maintains properties meaningful for projective geometry (the affine transformations). The first problem faced by mathematicians in uncharted territory is determining what sort of geometry is appropriate. As can be observed in perspective drawing, angles are not invariant with regard to projective transformations, hence they cannot be referred to in the same way in projective geometry as they are in Euclidean geometry. The notion of perspective was an inspiration for projective geometry. When translated into the language of projective geometry, the idea that two parallel lines meet at infinity takes on a new meaning. Again, this idea is grounded in common sense; for example, in a perspective drawing, train lines converge towards the horizon. For a primer on projective geometry in two dimensions, check out the projective plane.

Although the concepts were available earlier, projective geometry did not really take off until the nineteenth century. Among these areas is the theory of complex projective space, where complex numbers are utilized as coordinates (homogeneous coordinates). Projective geometry was the impetus for the development of several important branches of more abstract mathematics, such as invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme, which led to the study of the classical groups. As synthetic geometry, the field attracted numerous experts in its own right. Finite geometry is another area that sprang out of axiomatic research into projective geometry.

Many subfields of study have developed from the original field of projective geometry; for instance, projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).

Basic non-metrical geometry, projective geometry is not reliant on the measurement of distance. Starting with point and line arrangements in two dimensions. Desargues and others, investigating the foundations of perspective art, were the first to discover that there is, in fact, some geometric appeal in this barren context. The theorems that can be applied to projective geometry are found to be shorter and more straightforward. Some theorems regarding circles can be seen as particular examples of these general theorems, and in (complex) projective geometry, the various conic sections are equal to one another.

Projective geometry emerged as its own branch of mathematics thanks to the efforts of mathematicians like Jean-Victor Poncelet and Lazare Carnot in the early 19th century. Like affine and Euclidean geometry, projective geometry can be derived from Felix Klein's Erlangen program; projective geometry is distinguished by invariants under transformations of the projective group.

As a result of intensive study of the vast body of theorems in the field, the foundations of projective geometry were established. There are two fundamental invariants under projective transformations: the incidence structure and the cross-ratio. If we add a line (hyperplane) at infinity to the affine plane (or affine space) and then handle it as though it were ordinary, we have a model for projective geometry. Axiomatic research, on the other hand, has uncovered non-Desarguesian planes as proof that the axioms of incidence can be modeled (in just two dimensions) by structures inaccessible to reasoning via homogeneous coordinate systems.

Both projective geometry and ordered geometry can serve as the basis for affine and Euclidean geometry, making them fundamentally simple. thereby providing a unique premise upon which geometry can be built.

Pappus of Alexandria, who lived in the third century, is credited with discovering the first projective geometric properties. Desargues created a new method of drawing perspective that accounts for the circumstance when the vanishing point is infinity away. He extended the scope of geometry to include the general situation of Euclidean geometry, in which parallel lines are indeed parallel. Blaise Pascal, at 16 years old, was inspired to develop Pascal's theorem by Desargues's work on conic sections. The following growth of projective geometry owes a great deal to Gaspard Monge's contributions around the turn of the nineteenth century. Desargues's work was forgotten until 1845, when Michel Chasles discovered a copy in a drawer. In the meantime, the seminal work on projective geometry by Jean-Victor Poncelet appeared in 1822. Using the concrete pole and polar relation with regard to a circle, Poncelet established a connection between metric and projective qualities by investigating the invariance of objects under central projection. Models, such as the Klein model of hyperbolic space, were eventually proved to exist for the newly discovered non-Euclidean geometries.

In 1855 A.

F.

Möbius wrote an article about permutations, now called Möbius transformations, of complex-plane generalized circles.

The projectivities of the complex projective line are represented here by these transformations.

The study of spatial lines, Julius Plücker used homogeneous coordinates in his description, next we looked at the collection of lines on the Klein quadric, one of projective geometry's earliest contributions to a developing discipline termed algebraic geometry, a branch of geometry inspired by projective techniques.

Lobachevski's and Bolyai's hypotheses about hyperbolic geometry were proven correct in large part due to the models for the hyperbolic plane provided by projective geometry. case in point, the Poincaré disc model where generalised circles perpendicular to the unit circle correspond to hyperbolic lines (geodesics), and the translations of this model are described by Möbius transformations that map the unit disc to itself.

A Cayley-Klein metric is used to calculate the distance between two points, based on cross-ratio, hence it is known to be translation-invariant, central invariant of projection.

In the theory of metric spaces, the translations are classified as a variety of isometries, partial linear transforms, the projective linear group, and as linear transformations of that group, As an example, SU(1), 1).

Poncelet, Jakob Steiner, and others did not set out to broaden analytic geometry with their work. Synthetic methods were meant to be implemented, with projective space as we know it now added axiomatically. Therefore, it might be challenging to reformulate early work in projective geometry so that it is rigorous by today's standards. Axiomatic methods can provide models that defy linear algebra description, even in the simple instance of the projective plane.

Clebsch, Riemann, Max Noether, and others pushed the limits of existing methods in geometry by studying universal algebraic curves; this was followed by the development of invariant theory. The Italian school of algebraic geometry (Enriques, Segre, Severi) branched out from the traditional subject at the century's conclusion, into a realm that required more advanced techniques.

Although there is a large body of written material on the subject, projective geometry fell out of favor in the latter half of the 19th century. In particular, Schubert did some work in enumerative geometry that is now seen as foreshadowing the theory of Chern classes, which are understood to describe the algebraic topology of Grassmannians.

Paul Dirac's development of quantum mechanics relied heavily on projective geometry. Heisenberg was shaken and discouraged by the idea that quantum observations might fail to commute, while Dirac, who had already studied projective planes over noncommutative rings, was probably less affected by this new development. In his latter work, Dirac relied heavily on projective geometry drawings to grasp the intuitive meaning of his equations before committing his findings to paper in a purely algebraic approach.

When compared to Euclidean geometry and affine geometry, projective geometry is more flexible. It is a geometry in which the facts stand on their own without the need for a metric framework. The incidence structure and the relationship of projective harmonic conjugates are unaffected by the projective transformations. The groundwork is a one-dimensional projective range. One of the most fundamental ideas of perspective is the meeting of parallel lines at infinity, and projective geometry formalizes this idea. In principle, a projective geometry can be understood as an extension of Euclidean geometry in which the direction of each line is incorporated into the line as an additional point, and a horizon of directions corresponding to coplanar lines is viewed as a line. So, if two lines are traveling in the same direction, they will eventually cross at the horizon.

Points at infinity denote idealized directions, whereas lines at infinity denote idealized horizons. Consequently, these lines are flat at infinity in the plane. However, infinity is a metric term, therefore in a strictly projective geometry, all points, lines, and planes are regarded equally regardless of their distance from the origin.

Due to the fact that projective geometry has a simpler basis than Euclidean geometry, general results in Euclidean geometry can be derived in a more comprehensible fashion within the framework of projective geometry, where distinct but analogous theorems of Euclidean geometry can be dealt with collectively. For instance, homogeneous coordinates can be used to place an arbitrary projective plane at infinity, eliminating the requirement to distinguish between parallel and nonparallel lines.

The Theorem of Desargues and the Theorem of Pappus are two further properties of vital relevance. It is possible to prove Desargues' Theorem using a specific construction in projective spaces of size 3 or higher. In contrast, a separate postulate is required for dimension 2.

Desargues' Theorem is utilized, when taken together with the other axioms, The rudimentary steps of arithmetic can be defined, geometrically.

Axioms of the field are met by the resulting operations, with the exception that commutativity of multiplication requires Pappus's hexagon theorem.

This means, Each line's nodes are directly proportional to their respective variables, F, accompanied by an extra component, ∞, such that r ⋅ ∞ = ∞, −∞ = ∞,