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A simplex in a projective space of dimension n is expressed by a matrix of order n + 1, where each row represents the homogeneous coordinates of a vertex of the simplex with respect to a reference frame. In the present study, a block Toeplitz matrix is used to express a simplex which forms a Mobius pair along with the reference simplex. A pair of mutually inscribed, circumscribed tetrahedrons is called a Mobius pair. The existence of such pairs of simplexes in higher-dimensional (odd) projective spaces is already established. In the present study an existence of an infinite chain of simplexes in a five-dimensional projective space is established where any two simplexes from the chain form a Mobius pair in some order of their vertices. This is studied with the help of powers of a block Toeplitz matrix. Also, attempt has been made to generalize this result to 2n + 1-dimensional projective space.

We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction., Final version to appear in Discrete & Computational Geometry

We extended the study of the linear fractional self maps (e.g., by Cowen–MacCluer and Bisi–Bracci on the unit balls) to a much more general class of domains, called generalized type I domains, which includes in particular the classical bounded symmetric domains of type I and the generalized balls. Since the linear fractional maps on the unit balls are simply the restrictions of the linear maps of the ambient projective space (in which the unit ball is embedded) on a Euclidean chart with inhomogeneous coordinates, and in this article we always worked with homogeneous coordinates, here the term linear map was used in this more general context. After establishing the fundamental result which essentially says that almost every linear self map of a generalized type I domain can be represented by a matrix satisfying the “expansion property” with respect to some indefinite Hermitian form, we gave a variety of results for the linear self maps on the generalized balls, such as the holomorphic extension across the boundary, the normal form and partial double transitivity on the boundary for automorphisms, the existence and the behavior of the fixed points, etc. Our results generalize a number of known statements for the unit balls, including, for example, a theorem of Bisi–Bracci saying that any linear fractional map of the unit ball with more than two boundary fixed points must have an interior fixed point.

We introduce in this paper an equivalence notion for submersions [Formula: see text], [Formula: see text] open in [Formula: see text], which makes it possible to identify a smooth planar curve with a unique class of submersions. This idea, which extends to the nonlinear setting the construction of a dual projective space, provides a systematic way to handle global implicit descriptions of smooth planar curves. We then apply this framework to model nonlinear electrical devices as classes of equivalent functions. In this setting, linearization naturally accommodates incremental resistances (and other analogous notions) in homogeneous terms. This approach, combined with a projectively-weighted version of the matrix-tree theorem, makes it possible to formulate and address in great generality several problems in nonlinear circuit theory. In particular, we tackle unique solvability problems in resistive circuits, and discuss a general expression for the characteristic polynomial of dynamic circuits at equilibria. Previously known results, which were derived in the literature under unnecessarily restrictive working assumptions, are simply obtained here by using dehomogenization. Our results are shown to apply also to circuits with memristors. We finally present a detailed, graph-theoretic study of certain stationary bifurcations in nonlinear circuits using the formalism here introduced.

Tento příspěvek představuje novou formulaci Plückerových souřadnic pomocí geometrické algebry a standardní lineární algebry s projektivní reprezentací. Plückerovy souřadnice se obvykle používají pro zobrazení čar v prostoru, které je dáno dvěma body. Čára však může být také dána průnikem dvou rovin ve vesmíru. Princip duality vede k jednoduché formulaci pro oba případy. Prezentovaný přístup využívá homogenní souřadnice s aplikací principu duality. Je vhodný pro aplikaci také na GPU. Plückerovy souřadnice se používají v mnoha aplikacích, např. v robotice, počítačově podporovaném návrhu a algoritmech počítačové grafiky atd. This contribution presents a new formulation of Plücker coordinates using geometric algebra and standard linear algebra with projective representation. The Plücker coordinates are usually used for a line representation in space, which is given by two points. However, the line can be also given as an intersection of two planes in space. The principle of duality leads to a simple formulation for both cases. The presented approach uses homogenous coordinates with the duality principle application. It is convenient for application on GPU as well. The Plücker coordinates are used in many applications, e.g. in robotics, computer aided design and computer graphics algorithms etc.

Nová formulace Pluckerových souřadnic s použitím geometrické algebry a standardní lineárni algebry s projektivní reprezentaci. A new formulation of Plücker coordinates using geometric algebra and standard linear algebra with projective representation.

We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi--Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Neto's classification of degree-two foliations on projective space. Corresponding to the ``exceptional'' component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles., 27 pages, 1 figure, 1 table

Projective shape consists of the information about a configuration of points that is invariant under projective transformations. It is an important tool in machine vision to pick out features that are invariant to the choice of camera view. The simplest example is the cross ratio for a set of four collinear points. Recent work involving ideas from multivariate robustness enables us to introduce here a natural preshape on projective shape space. This makes it possible to adapt the Procrustes analysis that forms the basis of much methodology in the simpler setting of similarity shape space. Copyright 2012, Oxford University Press.

Many geometrically oriented problems lead to intersection computation or to its dual problems. In many cases the problem is reduced to intersection computation of two planes in E3, e.g. intersection of two triangles. However in several cases triangles are given by vertices in the homogeneous coordinates. The usual approach is to convert coordinates to the Euclidean space and make intersection computation in the Euclidean space. This leads to extensive use of division operations and to decreased precision of computation. Another approach is an application of Plucker coordinates which are not commonly recognized in computer graphics or direct computing in the projective space. In this paper we present a relation between the extended cross product and the Plucker coordinates. The extended cross product is especially convenient for GPU application. Also a new formulation for the extended cross product using matrix notation in n-dimensional space is introduced. The presented approach leads to simple, robust and fast intersection computation of two planes on GPU. Also the advantage of the projective representation for geometrical problems solution is presented as it actually offers "doubled" mantissa length naturally and saves division operations.

Many problems, not only in computer vision and visualization, lead to a system of linear equations Ax = 0 or Ax = b and fast and robust solution is required. A vast majority of computational problems in computer vision, visualization and computer graphics are three dimensional in principle. This paper presents equivalence of the cross–product operation and solution of a system of linear equations Ax = 0 or Ax = b using projective space representation and homogeneous coordinates. This leads to a conclusion that division operation for a solution of a system of linear equations is not required, if projective representation and homogeneous coordinates are used. An efficient solution on CPU and GPU based architectures is presented with an application to barycentric coordinates computation as well.

Article, Formalized Mathematics. 20(1): 87-95 (2012)

The natural homogeneous coordinate system is the analog of the Cartesian coordinate system for projective geometry. Roughly speaking a projective geometry adds an axiom that parallel lines meet at a point at infinity. This removes the impediment to line-point duality that is found in traditional Euclidean geometry. The natural homogeneous coordinate system is surprisingly useful in a number of applications including computer graphics and statistical data visualization. In this article, we describe the axioms of projective geometry, introduce the formalism of natural homogeneous coordinates, and illustrate their use with four applications. WIREs Comp Stat 2010 2 678–685 DOI: 10.1002/wics.122 For further resources related to this article, please visit the WIREs website.

A method for investigating the equations of static models describing the scattering of a relativistic particle on a fixed center in projective spaces is proposed. The basis of the method is a construction of a set of affine coordinates of the problem at zero total energy. This method is exemplified by the three-row crossing-symmetry matrix. Each element of the set results in a solution with the same Riemann surface on which the behavior of the separatrix at infinity is considered.

Computer graphics is ostensibly based on projective geometry. The graphics pipeline—the sequence of functions applied to 3D geometric primitives to determine a 2D image—is described in the graphics literature as taking the primitives from Euclidean to projective space, and then back to Euclidean space. This is a weak foundation for computer graphics. An instructor is at a loss: one day entering the classroom and invoking the established and venerable theory of projective geometry while asserting that projective spaces are not separable, and then entering the classroom the following week to tell the students that the standard graphics pipeline performs clipping not in Euclidean, but in projective space—precisely the operation (deciding sidedness, which depends on separability) that was deemed nonsensical. But there is no need to present Blinn and Newell’s algorithm (Comput. Graph. 12, 245–251, 1978; Commun. ACM 17, 32–42, 1974)—the crucial clipping step in the graphics pipeline and, perhaps, the most original knowledge a student learns in a fourth-year computer graphics class—as a clever trick that just works. Jorge Stolfi described in 1991 oriented projective geometry. By declaring the two vectors $(x,y,z,w)^{{\ensuremath {\mathsf {T}}}}$and $(-x,-y,-z,-w)^{{\ensuremath {\mathsf {T}}}}$distinct, Blinn and Newell were already unknowingly working in oriented projective space. This paper presents the graphics pipeline on this stronger foundation.

This paper describes algorithms for fast and robust GPU computation of barycentric coordinates and intersection of two planes. The presented algorithms are based on matrix-vector operations which make the algorithms convenient for GPU or SSE based architectures. Also a new formula for finding the closest point of two planes intersection to the given point is given.

The singularities along the polar axis induced by the spherical coordinates can be circumvented with the use of the projective coordinates, replacing besides cumbersome trigonometric manipulations by elementary algebraic operations of vectorial nature. New formulas in projective coordinates for the gradient, curl, divergence, Jacobian, Laplacian and Hessian are also provided.

The Fourier and Radon hyperplane transforms are closely related, and one such relation was established by Brylinski [4] in the framework of holonomic D-modules. The integral kernel of the Radon hyperplane transform is associated with the hypersurface SCP P of pairs ðx; yÞ; where x is a point in the n-dimensional complex projective space P belonging to the hyperplane yAP : As it turns out, a useful variant is obtained by considering the integral transform associated with the open complement U of S in P P : In the first part of this paper, we generalize Brylinski’s result in order to encompass this variant of the Radon transform, and also to treat arbitrary quasi-coherent D-modules, as well as (twisted) abelian sheaves. Our proof is entirely geometrical, and consists in a reduction to the onedimensional case by the use of homogeneous coordinates. The second part of this paper applies the above result to the quantization of the Radon transform, in the sense of [7]. First we deal with line bundles. More precisely, let P 1⁄4 PðVÞ be the projective space of lines in the vector spaceV; denote by ð Þ 3 D R

The Fisher information and the Rao measure are obtained in closed form for a family of probability density functions parametrized by the manifold PSL(2, R) of projective transformations of t...

In part 1, we were talking about points, planes, and lines in 3D, more particularly in projective 3-space. The idea is to find algebraic expressions for the various geometric relationships between these objects. We were just on the verge of discovering what would be a good algebraic formulation for lines in projective 3D space. My goal here is to update my original paper to see how the results look using tensor diagram notation. I start by reviewing what we did last time, but will say some things a bit differently. You might pick up more insight from this different viewpoint.

A framework is presented for the visualization of high-dimensional systems, particularly multicomponent phase diagrams. It is based on geometric modeling with homogeneous coordinates of the transformations among geometric varieties in high-dimensional space. By reduction of the dimensionality of the system through such transforms, the resulting linear cuts and projections, in their aggregate, provide a mental picture of the system in its entirety. A general way to perform cuts and projections is presented, including how to calculate the transforms and how to define the projective space. Specifically, a procedure is presented for the development of a set of canonical coordinates for the projective space with reduced dimensions. Also, it is shown that the seemingly different transformed coordinates for reactive systems in the literature can be unified under our framework. Examples are provided to highlight the procedure and to demonstrate the visualization of phase diagrams for nonreactive and reactive systems.

Today's computer graphics is ostensibly based upon insights from projective geometry and computations on homogeneous coordinates. Paradoxically, however, projective spaces and homogeneous coordinates are incompatible with much of the algebra and a good deal of the geometry currently in actual use in computer graphics. To bridge this gulf between theory and practice, Grassmann spaces are proposed here as an alternative to projective spaces. We establish that unlike projective spaces, Grassmann spaces do support all the algebra and geometry needed for contemporary computer graphics. We then go on to explain how to exploit this algebra and geometry for a variety of applications, both old and new, including the graphics pipeline, shading algorithms, texture maps, and overcrown surfaces.

Virtual reality and haptic systems use geometric transformations with points represented in homogeneous coordinates extensively. In many cases interpolation and barycentric coordinates are used. However, developers do not fully use properties of projective representation to make algorithms stable, robust and faster. This paper describes geometric transformations and principle of duality which enables to solve some problems effectively.

A well-known method proposed by Quan to compute projective invariants of 3D points uses six points in three 2D images. The method is nonlinear and complicated. It usually produces three possible solutions. It is noted previously that the problem can be solved directly and linearly using six points in five images. This paper presents a method to compute projective invariants of 3D points from four uncalibrated images directly. For a set of six 3D points in general position, we choose four of them as the reference basis and represent the other two points under this basis. It is known that the cross ratios of the coefficients of these representations are projective invariant. After a series of linear transformations, a system of four bilinear equations in the three unknown projective invariants is derived. Systems of nonlinear multivariable equations are usually hard to solve. We show that this form of equations can be solved linearly and uniquely. This finding is remarkable. It means that the natural configuration of the projective reconstruction problem might be six points and four images. The solutions are given in explicit formulas.

Given multiple image data from a set of points in 3D, there are two fundamental questions that can be addressed: What is the structure of the set of points in 3D? What are the positions of the cameras relative to the points? In this paper we show that, for projective views and with structure and position defined projectively, these problems are dual because they can be solved using constraint equations where space points and camera positions occur in a reciprocal way. More specifically, by using canonical projective reference frames for all points in space and images, the imaging of point sets in space by multiple cameras can be captured by constraint relations involving three different kinds of parameters only, coordinates of: (1) space points, (2) camera positions (3) image points. The duality implies that the problem of computing camera positions fromp points in q views can be solved with the same algorithm as the problem of directly reconstructing q+4 points in p-4 views. This unifies different approaches to projective reconstruction: methods based on external calibration and direct methods exploiting constraints that exist between shape and image invariants.

In (Mace, 1996) we have shown how Cayley's Algebra can be used to obtain expressions of generation and intersection of lines and planes. This tool is of great interest in homogenous coordinates. This paper shows how the introduction of line and plane as kernels of linear mappings leads to tensorial formulation easier to manipulate than traditional Plucker-Grassmann coordinates. After extensors and duality for antisymmetric tensors are defined, we show that line and plane generation in projective space is always an orthogonality problem solved by an extensor. The use of duality gives us analogous formulas for intersections. The results are simple and have a wide area of applications.

21st Signal Processing and Communications Applications Conference (SIU) -- APR 24-26, 2013 -- CYPRUS, WOS: 000325005300116, The aim of this study is to quantify the projective distortion of candidate quadrilaterals found in a square-framed fiducial marker detection algorithm. Based on the quantified value, candidates can be eliminated in such a way that only the quadrilaterals that may be a projective transformation of a square remain. In the first part of the study, it is shown that under a projective transform, the line at infinity of a plane corresponds to a line that is not at infinity. Two methods to find the equation of the corresponding line are proposed. The first method uses the homography matrix that represents a particular projective transformation. The correspondant of the line at infinity can be found using this homography matrix. The second method is a direct algorithm that utilizes the parallelism of the opposing edges of the square frame. At the last section, the obtained line is used to quantify the projective distortion and an elimination is performed on this basis.

A new modified method based on the Gaussian elimination method for solution of linear system of equations in the projective space is formulated. It is based on application of projective extension of the Euclidean space and use of homogeneous coordinates. It leads to an elimination of division operation and higher precision due to division operation elimination. The approach is based on understanding that a solution of the linear system Ax = b is equivalent to the extended cross-product, i.e. x = a1 × ... × an. As it can be seen there no division is needed. Use of the projective representation enables to avoid division operation and use advantages of the matrix-vector architectures. Division operations have to be used only if the final result of computation has to be in the Euclidean representation. The proposed method was implemented in C♯ and C++ and experimentally verified. It is especially convenient for computations on GPUs based architectures.

The traditional four-position method of mechanism synthesis focuses on the poles corresponding to four displacements. From these poles arise equations for the center-points and circle-points of possible four-bar linkages. However, if one of the poles is infinite (the associated displacement is a pure translation), the established derivations for the pole curve equation break down. This problem is rectified by expressing the pole curve equation in the projective plane, because all points, including points at infinity, have finite homogeneous coordinates. In Part I of this paper, this form of the pole curve equation is applied to opposite-pole quadrilaterals formed by four finite poles. This projective derivation is an analytic expression of Alt’s graphical construction of the pole curve. The pole curve is the intersection of a pair of projective pencils of circles (one pencil for each side of a pair of opposite sides in the opposite-pole quadrilateral) which are defined by the homogeneous coordinates of the poles. The resulting equation for the pole curve is a function of the locations of the poles and of the orientation of one opposite side of the opposite-pole quadrilateral relative to the other.

The dual intersection test is described. It is a new method for deciding whether a line intersects a window. The concept of semi-homogeneous coordinates is introduced. It allows us to define line segments in projective space, and to derive a generalized Cohen-Sutherland end-point t test for such segments. When this is combined with the dual intersection test, we obtain new clipping algorithms for 2D and 3D projective space.

The tutorial gives a practical overview of projective geometry and its applications in geometry, GPU computations and games. It will show how typical geometrical and computational problems can be solved easily if reformulated using the projective geometry. Presented algorithms are easy to understand, implement and they are robust. Homogeneous coordinates and projective geometry are mostly connected with geometric transformations only. However the projective extension of the Euclidean system allows reformulation of geometrical problems which can be easily solved. In many cases quite complicated formulae are becoming simple from the geometrical and computational point of view. In addition they lead to simple parallelization and to matrix-vector operations which are convenient for matrix-vector hardware architecture like GPU. In this short tutorial we will introduce "practical theory" of the projective space and homogeneous coordinates. We will show that a solution of linear system of equations is equivalent to generalized cross product and how this influences basic geometrical algorithms. The projective formulation is also convenient for computation of barycentric coordinates, as it is actually one cross-product implemented as one clock instruction on GPU. Additional speed up can be expected, too. Moreover use of projective representation enables to postpone division operations in many geometrical problems, which increases robustness and stability of algorithms. There is no need to convert coordinates of points from the homogeneous coordinates to the Euclidean one as the projective formulation supports homogeneous coordinates natively. The presented approach can be applied in computational problems, games and visualization applications as well. The tutorial is targeted to algorithm developers in geometry and graphics, visualization and games. The course is also intended for educators and attendees interested in computational issues in general.

The projective invariants used in computer vision today are permutation-sensitive since their value depends on the order in which the features were considered in the computation. We derive, using tools from representation theory, the projective and permutation ( p 2 ) invariants of the four collinear and the five coplanar points configurations. The p 2 -invariants are insensitive to both projective transformations and changes in the labeling of the points. When used as model database indexing functions in object recognition systems, the p 2 -invariants yield a significant speedup. Permutation invariants for affine transformations are also discussed.

Heyden et al. introduced an iterative factorization method for projective reconstruction from image sequences. In their formulation, the projective structure and motion are computed by using an iterative factorization based on 4D subspace. In this paper, the problem is reformulated based on fact that the x, y, and z coordinates of each feature in projective space are known from their projection. The projective reconstruction, i.e., the relative depths w and the 3D motion, is obtained by a simple iterative factorization based on 1D subspace. This allows the use of very fast algorithms even when using a large number of features and large number of frames. The experiments with both simulate and real data show that the method presented in the paper is efficient and has good convergency.

In the previous chapter we laid the foundations for representing Euclidean concepts (transformations, angles, distances, orthogonality, cocircularity…) in a projective framework. In this chapter we will apply these concepts. We will give a loose and by no means complete collection of interesting constructions/ calculations/theorems in Euclidean geometry that can be nicely carried out in our projective framework.

Harmonic conjugate is one of the most important and basic theories in projective geometry. According to the theory of projective transformation, the vanishing point's coordinates can be obtained, after that, combined with the corollary of Laguerre theorem: the infinity points of which the two mutually perpendicular lines and their circular points harmonic conjugate, the image coordinates of the circular points can be obtained. In the experiment, using the positive tri-prism as the experimental object and according to the same primal and the cross-ratio invariance, the coordinates of infinity points can be obtained. Then, according to the corollary of Laguerre theorem, the corresponding coordinates of the circular points can be got. Established the equations which constraint on the intrinsic parameters, then the intrinsic parameters can be solved. So, this theory application can be verified in computer vision.

A basic problem in computer vision is to recover the projective structure of a set of 3D points from its 2D images. It is known that 3D projective invariants of six points can be computed from three uncalibrated view images. In the previous method, three homogeneous polynomial equations in four variables relating the geometry of the six 3D points and their 2D projections were derived first. Then an eighth degree polynomial equation in single variable was derived by means of the classical resultant technique. Numerical method was applied to obtain an equation of a third degree. So the form of the equation is implicit and hard to apply in real applications. This paper adopts a novel method to eliminate variables. A third degree polynomial equation in single variable is derived symbolically. The equation is presented in explicit form. It can be used in real applications directly.

A computational formalism is given to computer vision problems involving collinearity and concurrency of points and lines on a 2-D plane from the viewpoint of projective geometry. The image plane is regarded as a 2-D projective space, and points and lines are represented by unit vectors consiting of homogeneous coordinates, called N-vectors. Fundamental notions of projective geometry such as collineations, correlations, polarities, poles, polars, and conis are reformulated as “computational” processes in terms of N-vectors. They are also given 3-D interpretations by regarding 2-D images as perspective projection of 3-D scenes. This N-vector formalism is further extended to infer 3-D translational motions from 2-D motion images. Stereo is also viewed as a special type of translational motion. Three computer vision applications are briefly discussed—interpretation of a rectangle, interpretation of a road, and interpretation of planar surface motion.

We investigate how projective plane geometry can be formalized in a proof assistant such as Coq. Such a formalization increases the reliability of textbook proofs whose details and particular cases are often overlooked and left to the reader as exercises. Projective plane geometry is described through two different axiom systems which are formally proved equivalent. Usual properties such as decidability of equality of points (and lines) are then proved in a constructive way. The duality principle as well as formal models of projective plane geometry are then studied and implemented in Coq. Finally, we formally prove in Coq that Desargues' property is independent of the axioms of projective plane geometry.