92 results on '"Homogeneous coordinates"'
Search Results
2. A Brief Survey of Clipping and Intersection Algorithms with a List of References (including Triangle-Triangle Intersections)✩.
- Author
-
Skala, Vaclav
- Subjects
- *
ALGORITHMS , *TRIANGLES , *PROJECTIVE spaces - Abstract
This contribution presents a brief survey of clipping and intersection algorithms in E 2 and E 3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which support GPU and SSE use. This survey is intended to help researchers, students, and practitioners dealing with intersection and clipping algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. A Brief Survey of Clipping and Intersection Algorithms with a List of References (including Triangle-Triangle Intersections)✩.
- Author
-
Skala, Vaclav
- Subjects
ALGORITHMS ,TRIANGLES ,PROJECTIVE spaces - Abstract
This contribution presents a brief survey of clipping and intersection algorithms in E 2 and E 3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which support GPU and SSE use. This survey is intended to help researchers, students, and practitioners dealing with intersection and clipping algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. Optimized line and line segment clipping in E2 and Geometric Algebra.
- Author
-
Skala, Vaclav
- Subjects
- *
ALGEBRA , *HOMOGENEOUS spaces , *COMPUTER graphics , *PROJECTIVE spaces , *ALGORITHMS - Abstract
Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the E² or E³ space. The projection operation from the E³ to the E² space leads to the necessity to convert coordinates to the Euclidean space if the clipping operation is to be used. In this contribution, an optimized simple algorithm for line and line segment clipping in the E² space, which works directly with homogeneous representation and not requiring the conversion to the Euclidean space, is described. It is based on Geometric Algebra (GA) formulation for projective representation. The proposed algorithm is simple, efficient and easy to implement. The algorithm can be efficiently modified for the SSE4 instruction use or the GPU application, too. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
5. Scientific Computing and Computer Graphics with GPU: Application of Projective Geometry and Principle of Duality.
- Author
-
Skala, V., Karim, S. A. A., and Kadir, E. A.
- Subjects
- *
PROJECTIVE geometry , *SCIENTIFIC computing , *VECTOR algebra , *ALGEBRA , *LINEAR equations , *COMPUTER graphics - Abstract
Geometric problems are usually solved in the Euclidean space by using the standard vector algebra techniques. In this study, principles of the projective geometry and geometric algebra will be introduced via a novel method that significantly simplifies the solution of geometrical problems. Also, it supports the GPU parallel computation application. Besides that, an application of the principle of duality leads to a simple solution of the dual problems. We show that, the equivalence of the extended cross-product (outer product) and the solution of the system of linear equations. This gives a direct impact to scientific computation, solution of geometrical problems, robotics, computer graphics algorithms and virtual reality via fast computation through GPU parallel systems. Some numerical and graphical results are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2020
6. Associate Submersions and Qualitative Properties of Nonlinear Circuits with Implicit Characteristics.
- Author
-
Riaza, Ricardo
- Subjects
- *
PROJECTIVE spaces , *NONLINEAR equations , *NONLINEAR theories , *MATHEMATICAL equivalence , *ERROR rates - Abstract
We introduce in this paper an equivalence notion for submersions U → ℝ , U open in ℝ 2 , 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. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
7. Geometric Transformations and Duality for Virtual Reality and Haptic Systems
- Author
-
Skala, Vaclav, Junqueira Barbosa, Simone Diniz, editor, Chen, Phoebe, editor, Cuzzocrea, Alfredo, editor, Du, Xiaoyong, editor, Filipe, Joaquim, editor, Kara, Orhun, editor, Kotenko, Igor, editor, Sivalingam, Krishna M., editor, Ślęzak, Dominik, editor, Washio, Takashi, editor, Yang, Xiaokang, editor, and Stephanidis, Constantine, editor
- Published
- 2014
- Full Text
- View/download PDF
8. Use of Block Toeplitz Matrix in the Study of Möbius Pairs of Simplexes in Higher-Dimensional Projective Space
- Author
-
Saroj Kanta Misra and Golak Bihari Panda
- Subjects
Combinatorics ,Matrix (mathematics) ,Simplex ,Homogeneous coordinates ,Chain (algebraic topology) ,General Mathematics ,Block (permutation group theory) ,Mathematics::Metric Geometry ,Projective space ,Toeplitz matrix ,Vertex (geometry) ,Mathematics - Abstract
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.
- Published
- 2020
9. A New Approach to Line - Sphere and Line - Quadrics Intersection Detection and Computation.
- Author
-
Skala, Vaclav
- Subjects
- *
QUADRICS , *COMPUTATIONAL mathematics , *POLYHEDRA , *ALGORITHMS , *GEOMETRIC analysis - Abstract
Line intersection with convex and un-convex polygons or polyhedron algorithms are well known as line clipping algorithms and very often used in computer graphics. Rendering of geometrical problems often leads to ray tracing techniques, when an intersection of many lines with spheres or quadrics is a critical issue due to ray-tracing algorithm complexity. A new formulation of detection and computation of the intersection of line (ray) with a quadric surface is presented, which separates geometric properties of the line and quadrics that enables pre-computation. The presented approach is especially convenient for implementation with SSE instructions or on GPU. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
10. Barycentric coordinates computation in homogeneous coordinates
- Author
-
Skala, Vaclav
- Subjects
- *
ARTIFICIAL intelligence , *PATTERN recognition systems , *COMPUTER vision , *COMPUTER graphics , *IMAGE processing , *DIGITAL image processing , *LINEAR systems - Abstract
Abstract: Homogeneous coordinates are often used in computer graphics and computer vision applications especially for the representation of geometric transformations. The homogeneous coordinates enable us to represent translation, rotation, scaling and projection operations in a unique way and handle them properly. Today''s graphics hardware based on GPU offers a very high computational power using pixel and fragment shaders not only for the processing of graphical elements, but also for the general computation using GPU as well. It is well known that points, triangles and strips of triangles are mostly used in computer graphics processing. Generally, triangles and tetrahedra are mostly represented by vertices. Several tests like “point inside…” or “intersection of…” are very often used in applications. On the other hand, barycentric coordinates in E 2 or E 3 can be used to implement such tests, too. Nevertheless, in both cases division operations are used that potentially lead to the instability of algorithms. The main objective of this paper is to show that if the vertices of the given polygon and/or a point itself are given in homogeneous coordinates the barycentric coordinates can be computed directly without transferring them from the homogeneous [w≠1] to the Euclidean coordinates. Instead of solving a linear system of equations, the cross-product can be used and the division operation is not needed. This is quite convenient approach for GPU computation. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
11. Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations
- Author
-
Jose A. Perea
- Subjects
Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,Homogeneous coordinates ,Collineation ,Projective unitary group ,Complex projective space ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Real projective line ,Computational Theory and Mathematics ,Projective line ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Computer Science - Computational Geometry ,Discrete Mathematics and Combinatorics ,Projective space ,Mathematics - Algebraic Topology ,Geometry and Topology ,0101 mathematics ,Quaternionic projective space ,Mathematics - Abstract
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
- Published
- 2017
12. LENGTH, AREA AND VOLUME COMPUTATION IN HOMOGENEOUS COORDINATES.
- Author
-
SKALA, VACLAV
- Subjects
- *
COMPUTER graphics , *DATA visualization , *GRAPHICAL projection , *NON-Euclidean geometry , *COORDINATES , *TRIANGLES , *ALGORITHMS - Abstract
Many problems solved in computer graphics, computer vision, visualization etc. require fast and robust computation of an area of a triangle or volume of a tetrahedron. These very often used algorithms are well known and robust if vertices coordinates of triangles or tetrahedrons are given in Euclidean coordinates. The homogeneous coordinates are often used for the representation of geometric transformations. They enable us to represent translation, rotation, scaling and projection operations in a unique way and handle them properly. Today's graphics hardware based on GPU offers very high computational power using pixel shaders and fragment shaders not only for graphical elements processing, but also for general computation using GPU as well. This paper presents simple methods for the area of a triangle and the volume of a tetrahedron computation if vertices are given in homogeneous coordinates without the need to use the division operation for vertices coordinates transformation from the homogeneous coordinates to the Euclidean coordinates. Area or volume computation is transferred to the cross product computation that is fast, simple, and robust and can be supported in hardware or implemented on GPU that uses vector operations with homogeneous coordinates natively. The presented formula can be used directly for Euclidean representation just setting w equal to 1. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
13. Generalizations of linear fractional maps for classical symmetric domains and related fixed point theorems for generalized balls
- Author
-
Aeryeong Seo, Yun Gao, and Sui-Chung Ng
- Subjects
Unit sphere ,Pure mathematics ,Homogeneous coordinates ,Sesquilinear form ,Mathematics - Complex Variables ,Applied Mathematics ,010102 general mathematics ,32M10, 32M15 ,Fixed-point theorem ,Boundary (topology) ,Fixed point ,01 natural sciences ,Linear map ,0103 physical sciences ,FOS: Mathematics ,Projective space ,010307 mathematical physics ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2019
14. Associate submersions and qualitative properties of nonlinear circuits with implicit characteristics
- Author
-
Ricardo Riaza
- Subjects
Equilibrium point ,Pure mathematics ,Class (set theory) ,Homogeneous coordinates ,Applied Mathematics ,020208 electrical & electronic engineering ,02 engineering and technology ,Memristor ,Systems and Control (eess.SY) ,01 natural sciences ,Electrical Engineering and Systems Science - Systems and Control ,010305 fluids & plasmas ,law.invention ,law ,Modeling and Simulation ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,FOS: Electrical engineering, electronic engineering, information engineering ,Projective space ,94C05, 94C15, 37G10, 53A04 ,Engineering (miscellaneous) ,Equivalence (measure theory) ,Bifurcation ,Mathematics ,Characteristic polynomial - Abstract
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.
- Published
- 2019
15. A New Formulation of Plücker Coordinates Using Projective Representation
- Author
-
Vaclav Skala and Michal Smolik
- Subjects
dualita ,Computer science ,robotika ,rozšířený křížový produk ,GPU ,Plücker coordinates ,Geometric algebra ,Intersection ,Plückerovy souřadnice ,projektivní prostor ,homogeneous coordinates ,Representation (mathematics) ,Projective representation ,robotics ,Homogeneous coordinates ,geometrická algebra ,projective space ,geometric algebra ,Algebra ,extended cross-produc ,computer graphics ,počítač grafika ,Line (geometry) ,Linear algebra ,homogenní souřadnice ,duality - Abstract
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.
- Published
- 2018
16. Geometrická algebra a Plückerovy souřadnice: Geometrický, vnější a vnitřní součin
- Author
-
Skala, Václav
- Subjects
geometric product ,geometrická algebra ,Plückerovy souřadnice ,dualita ,projective space ,geometrický součin ,Plücker coordinates ,duality ,projektivní prostor ,geometric algebra ,homogeneous coordinates - Abstract
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.
- Published
- 2018
17. Quantum deformations of projective three-space
- Author
-
Brent Pym
- Subjects
High Energy Physics - Theory ,Pure mathematics ,General Mathematics ,FOS: Physical sciences ,14A22, 53D17, 37F75 ,Space (mathematics) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Quantization (physics) ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Projective space ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Homogeneous coordinates ,010102 general mathematics ,Noncommutative geometry ,Unimodular matrix ,High Energy Physics - Theory (hep-th) ,Mathematics - Symplectic Geometry ,Projective line ,Bimodule ,Symplectic Geometry (math.SG) ,010307 mathematical physics - Abstract
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
- Published
- 2015
18. A geometric approach to projective shape and the cross ratio
- Author
-
Kanti V. Mardia and John T. Kent
- Subjects
Statistics and Probability ,Homogeneous coordinates ,Applied Mathematics ,General Mathematics ,Cross-ratio ,Topology ,Agricultural and Biological Sciences (miscellaneous) ,Algebra ,Real projective line ,Duality (projective geometry) ,Projective space ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences ,Pencil (mathematics) ,Mathematics ,Twisted cubic ,Projective geometry - Abstract
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.
- Published
- 2012
19. Plücker Coordinates and Extended Cross Product for Robust and Fast Intersection Computation
- Author
-
Vaclav Skala
- Subjects
Discrete mathematics ,Homogeneous coordinates ,Euclidean space ,Computation ,020207 software engineering ,02 engineering and technology ,Cross product ,Plücker coordinates ,Algebra ,Geometric algebra ,Intersection ,0202 electrical engineering, electronic engineering, information engineering ,Projective space ,020201 artificial intelligence & image processing ,Mathematics - Abstract
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.
- Published
- 2016
20. 'Extended Cross-Product' and Solution of a Linear System of Equations
- Author
-
Vaclav Skala
- Subjects
Homogeneous coordinates ,Underdetermined system ,Computer science ,Independent equation ,Linear system ,Relaxation (iterative method) ,020207 software engineering ,02 engineering and technology ,System of linear equations ,Overdetermined system ,Algebra ,Nonlinear system ,Geometric algebra ,Simultaneous equations ,0202 electrical engineering, electronic engineering, information engineering ,Projective space ,020201 artificial intelligence & image processing ,Coefficient matrix ,Projective representation - Abstract
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.
- Published
- 2016
21. Operations of Points on Elliptic Curve in Projective Coordinates
- Author
-
Yasunari Shidama, Daichi Mizushima, Hiroyuki Okazaki, and Yuichi Futa
- Subjects
Homogeneous coordinates ,Applied Mathematics ,Mathematical analysis ,Rational normal curve ,Computational Mathematics ,Jacobian curve ,QA1-939 ,Projective space ,Algebraic curve ,Line coordinates ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Tripling-oriented Doche–Icart–Kohel curve ,Mathematics ,Twisted cubic - Abstract
Article, Formalized Mathematics. 20(1): 87-95 (2012)
- Published
- 2012
22. Natural homogeneous coordinates
- Author
-
Yasmin H. Said and Edward J. Wegman
- Subjects
Statistics and Probability ,Pure mathematics ,Analytic geometry ,Homogeneous coordinates ,Computer science ,Duality (projective geometry) ,Coordinate system ,Projective space ,Line at infinity ,Line coordinates ,Projective geometry - Abstract
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.
- Published
- 2010
23. Behavior of separatrix of one static model at infinity
- Author
-
V. I. Zhuravlev and V. A. Meshcheryakov
- Subjects
Circular points at infinity ,Physics ,Nuclear and High Energy Physics ,Radiation ,Homogeneous coordinates ,Mathematical analysis ,Line at infinity ,Riemann sphere ,Atomic and Molecular Physics, and Optics ,symbols.namesake ,Real projective line ,Projective line ,symbols ,Projective space ,Radiology, Nuclear Medicine and imaging ,Point at infinity - Abstract
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.
- Published
- 2010
24. Sense and sidedness in the graphics pipeline via a passage through a separable space
- Author
-
Sherif Ghali
- Subjects
Discrete mathematics ,Homogeneous coordinates ,Computer science ,Complex projective space ,Oriented projective geometry ,Computer Graphics and Computer-Aided Design ,Separable space ,Projective space ,Computer Vision and Pattern Recognition ,2D computer graphics ,Software ,Pencil (mathematics) ,ComputingMethodologies_COMPUTERGRAPHICS ,Projective geometry - Abstract
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.
- Published
- 2008
25. GPU fast and robust computation for barycentric coordinates and intersection of planes using projective representation
- Author
-
Vaclav Skala
- Subjects
Computer Science::Performance ,Computer Science::Graphics ,Homogeneous coordinates ,Intersection ,Orthogonal coordinates ,Computer science ,Log-polar coordinates ,Computer Science::Mathematical Software ,Projective space ,Line coordinates ,Barycentric coordinate system ,Plücker coordinates ,Algorithm - Abstract
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.
- Published
- 2014
26. Differential operators in projective coordinates
- Author
-
André Deprit and Jesús F. Palacián
- Subjects
Homogeneous coordinates ,General Mathematics ,Applied Mathematics ,Log-polar coordinates ,Mathematical analysis ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Action-angle coordinates ,Plücker coordinates ,Orthogonal coordinates ,Projective space ,Line coordinates ,Mathematics ,Bipolar coordinates - Abstract
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.
- Published
- 2004
27. Radon and Fourier transforms for D-modules
- Author
-
Andrea D'Agnolo and Michael Eastwood
- Subjects
Pure mathematics ,Mathematics(all) ,Homogeneous coordinates ,Radon transform ,General Mathematics ,Complex projective space ,Integral transform ,symbols.namesake ,Hypersurface ,Fourier transform ,Hyperplane ,Calculus ,symbols ,Projective space ,Mathematics - Abstract
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
- Published
- 2003
- Full Text
- View/download PDF
28. Fisher information and model selection for projective transformations of the line
- Author
-
Stephen J. Maybank
- Subjects
Homogeneous coordinates ,General Mathematics ,General Engineering ,General Physics and Astronomy ,Statistics::Other Statistics ,Rational normal curve ,Mathematics::Geometric Topology ,Combinatorics ,symbols.namesake ,Real projective line ,Homography ,symbols ,Projective space ,Line coordinates ,Fisher information ,Mathematics ,Twisted cubic - Abstract
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...
- Published
- 2003
29. Lines in space. 2. The line formulation
- Author
-
James F. Blinn
- Subjects
Pure mathematics ,Homogeneous coordinates ,Collineation ,Algebraic geometry of projective spaces ,Computer science ,Complex projective space ,One-dimensional space ,Dimension of an algebraic variety ,Computational geometry ,Computer Graphics and Computer-Aided Design ,Algebra ,Real projective line ,Blocking set ,Duality (projective geometry) ,Projective line ,Homography ,Projective space ,Tensor ,Projective plane ,Algebraic expression ,Quaternionic projective space ,Bézout's theorem ,Software ,Pencil (mathematics) ,Projective geometry - Abstract
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.
- Published
- 2003
30. Visualization of High-Dimensional Systems via Geometric Modeling with Homogeneous Coordinates
- Author
-
Christianto Wibowo and Ka Ming Ng
- Subjects
Homogeneous coordinates ,Computer science ,General Chemical Engineering ,Canonical coordinates ,Projective space ,General Chemistry ,Space (mathematics) ,Plücker coordinates ,Geometric modeling ,Algorithm ,Industrial and Manufacturing Engineering ,Curse of dimensionality ,Visualization - Abstract
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.
- Published
- 2002
31. On the algebraic and geometric foundations of computer graphics
- Author
-
Ron Goldman
- Subjects
Discrete mathematics ,Homogeneous coordinates ,Plücker coordinates ,Computer Graphics and Computer-Aided Design ,Algebra ,Computer graphics ,Computer Science::Graphics ,Duality (projective geometry) ,Projective space ,2D computer graphics ,Pencil (mathematics) ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics ,Projective geometry - Abstract
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.
- Published
- 2002
32. A new approach to line and line segment clipping in homogeneous coordinates
- Author
-
Skala, Vaclav
- Published
- 2005
- Full Text
- View/download PDF
33. Geometric Transformations and Duality for Virtual Reality and Haptic Systems
- Author
-
Vaclav Skala
- Subjects
Algebra ,Transformation matrix ,Homogeneous coordinates ,Computer science ,Duality (projective geometry) ,Projective space ,Plücker coordinates ,System of linear equations ,Transformation geometry ,Projective representation ,Haptic technology ,Interpolation - Abstract
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.
- Published
- 2014
34. A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images
- Author
-
Bin Zhang, Ying Wang, Xingwei Wang, and Yuanbin Wang
- Subjects
Pure mathematics ,Projective harmonic conjugate ,Homogeneous coordinates ,Article Subject ,lcsh:T ,lcsh:R ,Cross-ratio ,lcsh:Medicine ,General Medicine ,Models, Theoretical ,Complete quadrangle ,lcsh:Technology ,General Biochemistry, Genetics and Molecular Biology ,Imaging, Three-Dimensional ,Real projective line ,Duality (projective geometry) ,Projective space ,lcsh:Q ,lcsh:Science ,Pencil (mathematics) ,Research Article ,General Environmental Science ,Mathematics - Abstract
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.
- Published
- 2014
35. [Untitled]
- Author
-
Stefan Carlsson and Daphna Weinshall
- Subjects
Homogeneous coordinates ,Camera matrix ,business.industry ,Epipolar geometry ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Cardinal point ,Real projective line ,Artificial Intelligence ,Duality (projective geometry) ,Projective space ,Computer vision ,Computer Vision and Pattern Recognition ,Artificial intelligence ,business ,Software ,Projective representation ,Mathematics - Abstract
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.
- Published
- 1998
36. Tensorial calculus of line and plane in homogeneous coordinates
- Author
-
Pierre Macé
- Subjects
Pure mathematics ,Homogeneous coordinates ,Antisymmetric relation ,Computer science ,Duality (projective geometry) ,General Engineering ,Projective space ,Algebraic geometry ,Line coordinates ,Tensor calculus ,Projective geometry - Abstract
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.
- Published
- 1997
37. Quantification of projective distortion for fiducial markers
- Author
-
Burak Benligiray, Cuneyt Akinlar, Cihan Topal, Yakup Genc, Anadolu Üniversitesi, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü, Benligiray, Burak, Topal, Cihan, and Akınlar, Cüneyt
- Subjects
Homogeneous coordinates ,Homography ,Line at infinity ,Topology ,Real projective line ,Fiducial Markers ,Duality (projective geometry) ,Projective Transformation ,Projective space ,Line coordinates ,Algorithm ,Pencil (mathematics) ,Mathematics - Abstract
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.
- Published
- 2013
38. Modified Gaussian elimination without division operations
- Author
-
Vaclav Skala
- Subjects
Discrete mathematics ,Gaussova eliminace ,Homogeneous coordinates ,Euclidean space ,Linear system ,parciální diferenciální rovnice ,Division (mathematics) ,System of linear equations ,symbols.namesake ,numerické metody ,linear algebra ,Gaussian elimination ,numerical methods ,partial differential equations ,symbols ,Projective space ,lineární algebra ,Algorithm ,Projective representation ,Mathematics - Abstract
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.
- Published
- 2013
39. A Derivation of the Pole Curve Equations in the Projective Plane
- Author
-
Steven W. Peterson and Michael H. Brady
- Subjects
Homogeneous coordinates ,Quadrilateral ,Plane (geometry) ,Mechanical Engineering ,Mathematical analysis ,Geometry ,Rigid body ,Computer Graphics and Computer-Aided Design ,Displacement (vector) ,Computer Science Applications ,Mechanics of Materials ,Projection method ,Projective space ,Projective plane ,Locus (mathematics) ,Pencil (mathematics) ,Mathematics - Abstract
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.
- Published
- 1995
40. Line Clipping Using Semi-Homogeneous Coordinates
- Author
-
Hans Peter Nielsen
- Subjects
Real projective line ,Homogeneous coordinates ,Line clipping ,Line segment intersection ,One-dimensional space ,Line at infinity ,Projective space ,Line coordinates ,Computer Graphics and Computer-Aided Design ,Algorithm ,Mathematics - Abstract
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.
- Published
- 1995
41. Projective coordinates and compactification in elliptic, parabolic and hyperbolic 2-D geometry
- Author
-
Debapriya Biswas
- Subjects
Homogeneous coordinates ,Applied Mathematics ,Mathematical analysis ,Lie group ,Geometry ,symbols.namesake ,Computational Theory and Mathematics ,symbols ,Projective space ,Compactification (mathematics) ,Line coordinates ,Statistics, Probability and Uncertainty ,Quaternionic projective space ,Mathematical Physics ,Projective geometry ,Möbius transformation ,Mathematics - Published
- 2012
42. Projective Geometry and Duality for Graphics, Games and Visualization
- Author
-
Vaclav Skala
- Subjects
Homogeneous coordinates ,Computer science ,Plücker coordinates ,System of linear equations ,Algebra ,projective geometry ,Real projective line ,projektivní geometrie ,Duality (projective geometry) ,Computer graphics (images) ,computer graphics ,Euclidean geometry ,Projective space ,počítačová grafika ,Transformation geometry ,vizualizace ,visualization ,Projective representation ,Projective geometry - Abstract
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.
- Published
- 2012
43. Point configuration invariants under simultaneous projective and permutation transformations
- Author
-
Peter Meer and Reiner Lenz
- Subjects
Discrete mathematics ,Homogeneous coordinates ,Collineation ,Cross-ratio ,Real projective line ,Artificial Intelligence ,Signal Processing ,Projective space ,Computer Vision and Pattern Recognition ,Projective linear group ,Projective differential geometry ,Software ,Projective geometry ,Mathematics - Abstract
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.
- Published
- 1994
44. An iterative method based on 1D subspace for projective reconstruction
- Author
-
Z. Zeng, Y. Peng, C. Han, and S. Liu
- Subjects
Radiation ,Homogeneous coordinates ,Real projective line ,Hyperplane ,Projection (mathematics) ,Computer science ,Iterative method ,Homography ,Projective space ,General Materials Science ,Electrical and Electronic Engineering ,Algorithm ,Subspace topology - Abstract
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.
- Published
- 2011
45. Euclidean Structures from a Projective Perspective
- Author
-
Jürgen Richter-Gebert
- Subjects
Algebra ,Real projective line ,Homogeneous coordinates ,Collineation ,Duality (projective geometry) ,Complex projective space ,Projective space ,Geometry ,Pencil (mathematics) ,Mathematics ,Projective geometry - Abstract
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.
- Published
- 2010
46. Experimental teaching of harmonic conjugate and circular point in projective geometry
- Author
-
Jianping Li, Xiaohua Hu, and Yue Zhao
- Subjects
Circular points at infinity ,Projective harmonic conjugate ,Homogeneous coordinates ,Real projective line ,Duality (projective geometry) ,Mathematical analysis ,Projective space ,Line coordinates ,Computer Science::Databases ,Mathematics ,Projective geometry - Abstract
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.
- Published
- 2010
47. The Projective Invariants of Six 3D Points from Three 2D Uncalibrated Images
- Author
-
Bin Zhang, Fenghua Hou, and Yuanbin Wang
- Subjects
Algebra ,Discrete mathematics ,Homogeneous coordinates ,Duality (projective geometry) ,Homogeneous polynomial ,Projective space ,Degree of a polynomial ,Projective differential geometry ,Mathematics ,Twisted cubic ,Projective geometry - Abstract
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.
- Published
- 2010
48. Computational projective geometry
- Author
-
Kenichi Kanatani
- Subjects
Pure mathematics ,Homogeneous coordinates ,Collineation ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,General Engineering ,Fano plane ,Duality (projective geometry) ,General Earth and Planetary Sciences ,Projective space ,Projective plane ,Pencil (mathematics) ,General Environmental Science ,Mathematics ,Projective geometry - Abstract
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.
- Published
- 1991
49. Formalizing Projective Plane Geometry in Coq
- Author
-
Nicolas Magaud, Pascal Schreck, Julien Narboux, Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT), Université Louis Pasteur - Strasbourg I-Centre National de la Recherche Scientifique (CNRS), Thomas Sturm, and ANR: Galapagos,Galapagos
- Subjects
formalization ,projective plane geometry ,Geometry ,02 engineering and technology ,Fano plane ,[INFO.INFO-SE]Computer Science [cs]/Software Engineering [cs.SE] ,01 natural sciences ,models ,Real projective plane ,0202 electrical engineering, electronic engineering, information engineering ,Projective space ,Coq ,0101 mathematics ,homogeneous coordinates ,Pencil (mathematics) ,Non-Desarguesian plane ,Mathematics ,Projective geometry ,010102 general mathematics ,[INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO] ,[INFO.INFO-IA]Computer Science [cs]/Computer Aided Engineering ,16. Peace & justice ,[INFO.INFO-PF]Computer Science [cs]/Performance [cs.PF] ,Blocking set ,duality ,fano plane ,020201 artificial intelligence & image processing ,Projective plane - Abstract
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.
- Published
- 2008
50. Homogeneous Coordinates for Projective Geometry
- Author
-
Sherif Ghali
- Subjects
Physics ,Homogeneous coordinates ,Duality (projective geometry) ,Mathematical analysis ,Projective space ,Geometry ,Line coordinates ,Projective differential geometry ,Quaternionic projective space ,Rational normal curve ,Projective geometry - Published
- 2008
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.