arXiv subjects

Vaclav Skala

Publications and source records attributed to Vaclav Skala.

At least 19 recordsLinked to original sources

arXiv2023

Line-Torus Intersection for Ray Tracing: Alternative Formulations

Intersection algorithms are very important in computation of geometrical problems. Algorithms for a line intersection with linear or quadratic surfaces are quite efficient. However, algorithms for a line intersection with other surfaces are more complex and time consuming. In this case the object is usually closed into a simple bounding volume to speed up the cases when the given line cannot intersect the given object. In this paper new formulations of the line-torus intersection problem are given and new specification of the bounding volume for a torus is given as well. The presented approach is based on an idea of a line intersection with an envelope of rotating sphere that forms a torus. Due to this approach new bounding volume can be formulated which is more effective as it enables to detect cases when the line passes the "hole" of a torus, too.

cs.GR↗

arXiv2023

Robust Surface Reconstruction from Orthogonal Slices

The surface reconstruction problem from sets of planar parallel slices representing cross sections through 3D objects is presented. The final result of surface reconstruction is always based on the correct estimation of the structure of the original object. This paper is a case study of the problem of the structure determination. We present a new approach, which is based on considering mutually orthogonal sets of slices. A new method for surface reconstruction from orthogonal slices is described and the benefit of orthogonal slices is discussed too.

cs.GR↗

arXiv2023

A Comparison of Fundamental Methods for Iso-surface Extraction

In this paper four fundamental methods for an iso-surface extraction are compared, based on cell decomposition to tetrahedra. The methods are compared both on mathematically generated data sets as well as on real data sets. The comparison using mathematical data is made from different points of view such as area approximation, volume approximation. On the other hand, the Hausdorff distance and root mean square are used to compare methods on real data sets. The presented comparison can be helpful when deciding among tested methods which one to choose, as well as when we need to compare a newly developed method with other existing approaches.

cs.GR↗

arXiv2022

Poseidon: Non-server WEB Forms Off-line Processing System

The proposed Poseidon system is based on email services of filled forms instead of WEB server based services. This approach is convenient especially for small applications or small-medium companies. It is based on PDF forms that are available on a WEB page. PDF forms can be downloaded, off-line filled in, printed and finally sent by email for final processing. Data are actually stored in the local outbox waiting for a connection to a mail server. This follows an idea of the standard "paper" letter sending. Filled in data are actually sent when a user is on-line, therefore a user is "free" of being on-line when filling the forms. When the PDF form is processed on the recipient side, answer is sent back via email as well. Typical application is e.g. in conference management systems, systems for submission to journals etc. The great advantage of the PDF forms use is that they can be easily made or modified by a non-specialized administrative person easily.

cs.NI↗

arXiv2022

S-patch: Modification of the Hermite parametric patch

A new modification of the Hermite cubic rectangular patch is proposed: the S-Patch, which is based on the requirement that diagonal curves must be of degree 3 instead of degree 6 as it is in the case of the Hermite patch. Theoretical derivation of conditions is presented and some experimental results as well. The S-Patch is convenient for applications, where different tessellation of the u-v domain is needed, boundary and diagonal curves of different degrees are not acceptable.

cs.GR↗

arXiv2022

BS-Patch: Constrained Bezier Parametric Patch

Bezier parametric patches are used in engineering practice quite often, especially in CAD/CAM systems oriented to mechanical design. In many cases quadrilateral meshes are used for tessellation of parameters domain. We propose a new modification of the Bezier cubic rectangular patch, the BS-patch, which is based on the requirement that diagonal curves must be of degree 3 instead of degree 6 as it is in the case of the Bezier patch. Theoretical derivation of conditions is presented and some experimental results as well. The BS-Patch is convenient for applications where for different tessellation of the u-v domain different degrees of diagonal curves are not acceptable.

cs.GR↗

arXiv2022

HS-Patch: A New Hermite Smart Bicubic Patch Modification

Bicubic four-sided patches are widely used in computer graphics, CAD/CAM systems etc. Their flexibility is high and enables to compress a surface description before final rendering. However, computer graphics hardware supports only triangular meshes. Therefore, four-sided bicubic patches are approximated by a triangular mesh. The border curves of a bicubic patch are of degree 3, while diagonal and anti-diagonal curves are of degree 6. Therefore the resulting shape and texturing depend on the actual mapping, i.e. how the tessellation of a bicubic patch is made. The proposed new modification of the Hermite bicubic patch, the HS-patch, is a result of additional restriction put on the Hermite bicubic patch formulation - the diagonal and anti-diagonal curves are of degree 3. This requirement leads to a new Hermite based bicubic four-sided patch with 12 control points and another 4 control points, i.e. twist vectors, are computed from those 12 control points.

cs.GR↗

arXiv2022

Summation Problem Revisited -- More Robust Computation

Numerical data processing is a key task across different fields of computer technology use. However, even simple summation of values is not precise due to the floating point representation use. This paper presents a practical algorithm for summation of values convenient for medium and large data sets. The proposed algorithm is simple, easy to implement. Its computational complexity is O(N) in the contrary of the Exact Sign Summation Algorithm (ESSA) approach with O(N^2) run-time complexity. The proposed algorithm is especially convenient for cases when exponent data differ significantly and many small values are summed with higher values

cs.DS↗

arXiv2022

Solution of Linear Systems of Equations Ax=b and Ax=0 using Unifying Approach with Geometric Algebra: Outer Product Application and Angular Conditionality

A solution of linear systems of equations Ax=b and Ax=0 is a vital part of many computational packages. This paper presents a novel formulation based on the projective extension of the Euclidean space using the outer product (extended cross-product). This approach enables to solve the both cases, i.e. Ax=b and Ax=0. The proposed approach leads actually to an analytical solution of linear systems in the form on which the other vector operation can be applied before using the numerical evaluation. This contribution also proposes a new approach to the conditionality estimation of matrices applicable to non-squared matrices. It splits the conditionality to structural conditionality showing matrix property if nearly unlimited precision is used, numerical issue which depends on numerical representation with respect to the right-hand side influence, if given.

math.GM↗

arXiv2022

Polar, Spherical and Orthogonal Space Subdivisions for an Algorithm Acceleration: O(1) Point-in-Polygon/Polyhedron Test

Acceleration of algorithms is becoming a crucial problem, if larger data sets are to be processed. Evaluation of algorithms is mostly done by using computational geometry approach and evaluation of computational complexity. However in todays engineering problems this approach does not respect that number of processed items is always limited and a significant role plays also speed of read/write operations. One general method how to speed up an algorithm is application of space subdivision technique and usually the orthogonal space subdivision is used. In this paper non-orthogonal subdivisions are described. The proposed approach can significantly improve memory consumption and run-time complexity. The proposed modified space subdivision techniques are demonstrated on two simple problems Point-in-Convex Polygon and Point-in-Convex Polyhedron tests.

cs.CG↗

arXiv2022

Projective Geometry, Duality and Plucker Coordinates for Geometric Computations with Determinants on GPUs

Many algorithms used are based on geometrical computation. There are several criteria in selecting appropriate algorithm from already known. Recently, the fastest algorithms have been preferred. Nowadays, algorithms with a high stability are preferred. Also technology and computer architecture, like GPU etc., plays a significant role for large data processing. However, some algorithms are ill-conditioned due to numerical representation used; result of the floating point representation. In this paper, relations between projective representation, duality and Plucker coordinates will be explored with demonstration on simple geometric examples. The presented approach is convenient especially for application on GPUs or vector-vector computational architectures

cs.GR↗

arXiv2022

A New Robust Algorithm for Computation of a Triangle Circumscribed Sphere in E3 and a Hypersphere Simplex

There are many applications in which a bounding sphere containing the given triangle E3 is needed, e.g. fast collision detection, ray-triangle intersecting in raytracing etc. This is a typical geometrical problem in E3 and it has also applications in computational problems in general. In this paper a new fast and robust algorithm of circumscribed sphere computation in the -dimensional space is presented and specification for the E3 space is given, too. The presented method is convenient for use on GPU or with SSE or Intel AVX instructions on a standard CPU

cs.GR↗

arXiv2022

Fast O_{expected}(N) Algorithm for Finding Exact Maximum Distance in E^2 Instead of O(N^2) or O(N lg N)

This paper describes novel and fast, simple and robust algorithm with O(N) expected complexity which enables to decrease run-time needed to find an exact maximum distance of two points in E2. The proposed algorithm has been evaluated experimentally on larger different datasets. The proposed algorithm gives a significant speed-up to applications, when medium and large data sets are processed. It is over 10 000 times faster than the standard algorithm for 10^6 points randomly distributed points in E2. Experiments proved the advantages of the proposed algorithm over the standard algorithm and convex hull diameters approaches.

cs.CG↗

arXiv2022

New Geometric Continuity Solution of Parametric Surfaces

This paper presents a new approach to computation of geometric continuity for parametric bi-cubic patches, based on a simple mathematical reformulation which leads to simple additional conditions to be applied in the patching computation. The paper presents an Hermite formulation of a bicubic parametric patch, but reformulations can be made also for Bézier and B-Spline patches as well. The presented approach is convenient for the cases when valencies of corners are different from the value 4, in general.

cs.GR↗

arXiv2022

A New Approach to Line-Sphere and Line-Quadrics Intersection Detection and Computation

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

cs.GR↗

arXiv2022

Point-in-Convex Polygon and Point-in-Convex Polyhedron Algorithms with O(1) Complexity using Space Subdivision

There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees, octrees, kd-trees, bounding volume hierarchies, etc. However in some applications a non-orthogonal space subdivision can offer new ways for actual speed up. In the case of convex polygon in E3 a simple Point-in-Polygon test is of the O(N) complexity and the optimal algorithm is of O(lg N) computational complexity. In the E3 case, the complexity is O(N) even for the convex polyhedron as no ordering is defined. New Point-in-Convex Polygon and Point-in-Convex Polyhedron algorithms are presented based on space subdivision in the preprocessing stage resulting to O(1) run-time complexity. The presented approach is simple to implement. Due to the principle of duality, dual problems, e.g. line-convex polygon, line clipping, can be solved in a similarly.

cs.GR↗

arXiv2022

A New Formulation for Total Least Square Error Method in d-dimensional Space with Mapping to a Parametric Line

There are many practical applications based on the Least Square Error (LSE) or Total Least Square Error (TLSE) methods. Usually the standard least square error is used due to its simplicity, but it is not an optimal solution, as it does not optimize distance, but square of a distance. The TLSE method, respecting the orthogonality of a distance measurement, is computed in d-dimensional space, i.e. for points given in E2 a line p in E2, resp. for points given in E3 a plane in rho in E3, fitting the TLSE criteria are found. However, some tasks in physical sciences lead to a slightly different problem. In this paper, a new TSLE method is introduced for solving a problem when data are given in E3 and a line p in E3 is to be found fitting the TLSE criterion. The presented approach is applicable for a general -dimensional case, i.e. when points are given in E^d a line E^d is to be found. This formulation is different from the TLSE formulation.

math.GM↗

arXiv2022

Clipping and Intersection Algorithms: Short Survey and References

This contribution presents a brief survey of clipping and intersection algorithms in E2 and E3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which supports GPU and SSE use.

cs.GR↗