A Generalised Solution to the Point to Target Problem Using the Minimum Curvature Method

2021 ◽  
Author(s):  
Steven J. Sawaryn
Keyword(s):  
General Expression ◽  
Quadratic Forms ◽  
Geometric Interpretation ◽  
General Point ◽  
Target Problem ◽  
Polynomial Type ◽  
Circular Arcs ◽  
Straight Line ◽  
Solution Methods ◽  
Minimum Curvature

Abstract An explicit solution to the general 3D point to target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multi-valued and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact, or polynomial type solution methods to be employed. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point to target problem can be represented as a 10th order self-intersecting geometric surface, characterised by the trajectory's start and end points, the radii of the two arcs and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided and together these indicate the most convenient solution method for each case. In the presence of a tangent section the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point to target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.

A Generalized Solution to the Point-to-Target Problem Using the Minimum Curvature Method

2021 ◽  
pp. 1-15
Author(s):  
Steven J. Sawaryn
Keyword(s):  
General Expression ◽  
Quadratic Forms ◽  
Generalized Solution ◽  
Geometric Interpretation ◽  
General Point ◽  
Target Problem ◽  
Polynomial Type ◽  
Circular Arcs ◽  
Solution Methods ◽  
Minimum Curvature

Summary An explicit solution to the general 3D point-to-target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multivalued, and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact or polynomial-type solution methods to be used. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point-to-target problem can be represented as a 10th-orderself-intersecting geometric surface, characterized by the trajectory's start and end points, the radii of the two arcs, and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided, and together these indicate the most convenient solution method for each case. In the presence of a tangent section, the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point-to-target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.

Grouping of Straight Line Segments and Circular Arcs for Scene Analysis

1995 ◽  
pp. 163-185
Author(s):  
Jean-Luc Arseneault ◽  
Robert Bergevin ◽  
Denis Laurendeau
Keyword(s):  
Scene Analysis ◽  
Line Segments ◽  
Circular Arcs ◽  
Straight Line

scholarly journals On the theory of the elliptic transcendents

1837 ◽  
Vol 3 ◽  
pp. 60-60
Keyword(s):  
Elliptic Function ◽  
Elliptic Integral ◽  
Elliptic Functions ◽  
Algebraic Expression ◽  
Algebraic Equations ◽  
Similar Function ◽  
Circular Arcs ◽  
Straight Line ◽  
Regular Theory ◽  
General Method

Fagnani discovered that the two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and proved that any arc of a lemniscate, like that of a circle, may be multiplified any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. What he had accomplished with respect to the arcs of the lemniscates, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. Landen showed that the arcs of the hyperbola may be reduced, by a proper transformation, to those of an ellipse. Lagrange furnished us with a general method for changing an elliptic function into another having a different modulus; a process which greatly facilitates the numerical calculation of this class of integrals. Legendre distributed the elliptic functions into distinct classes, and reduced them to a regular theory, developing many of their properties which were before unknown, and introducing many important additions and improvements in the theory. Mr. Abel of Christiana happity conceived the idea of expressing the amplitude of an elliptic function in terms of the function itself, which led to the discovery of many new and useful properties. Mr. Jacobi proved, by a different method, that an elliptic function may be transformed in innumerable ways into another similar function, to which it bears constantly the same proportion. But his demonstrations require long and complicated calculations; and the train of deductions he pursues does not lead naturally to the truths which are proved, nor does it present in a connected view all the conclusions which the theory embraces. The author of the present paper gives a comprehensive view of the theory in its full extent, and deduces all the connected truths from the same principle. He finds that the sines or cosines of the amplitudes, used in the transformations, are analogous to the sines or cosines of two circular arcs, one of which is a multiple of the other; so that the former quantities are changed into the latter when the modulus is supposed to vanish in the algebraic expression. Hence he is enabled to transfer to the elliptic transcendents the same methods of investigation that succeed in the circle: a procedure which renders the demonstrations considerably shorter, and which removes most of the difficulties, in consequence of the close analogy that subsists between the two cases.

scholarly journals Subjective Probability and Geometry: Three Metric Theorems Concerning Random Quantities

2018 ◽  
Vol 10 (3) ◽  
pp. 7
Author(s):  
Pierpaolo Angelini ◽  
Angela De Sanctis
Keyword(s):  
Random Quantity ◽  
Real Space ◽  
Geometric Interpretation ◽  
Extensive Study ◽  
Dimensional Structure ◽  
Interpretation Of Probability ◽  
Straight Line ◽  
Probability Concept ◽  
Deep Meaning ◽  
Unit Vectors

Affine properties are more general than metric ones because they are independent of the choice of a coordinate system. Nevertheless, a metric, that is to say, a scalar product which takes each pair of vectors and returns a real number, is meaningful when $n$ vectors, which are all unit vectors and orthogonal to each other, constitute a basis for the $n$-dimensional vector space $\mathcal{A}$. In such a space $n$ events $E_i$, $i = 1, \ldots, n$, whose Cartesian coordinates turn out to be $x^i$, are represented in a linear form. A metric is also meaningful when we transfer on a straight line the $n$-dimensional structure of $\mathcal{A}$ into which the constituents of the partition determined by $E_1, \ldots, E_n$ are visualized. The dot product of two vectors of the $n$-dimensional real space $\mathbb{R}^n$ is invariant: of these two vectors the former represents the possible values for a given random quantity, while the latter represents the corresponding probabilities which are assigned to them in a subjective fashion.We deduce these original results, which are the foundation of our next and extensive study concerning the formulation of a geometric, well-organized and original theory of random quantities, from pioneering works which deal with a specific geometric interpretation of probability concept, unlike the most part of the current ones which are pleased to keep the real and deep meaning of probability notion a secret because they consider a success to give a uniquely determined answer to a problem even when it is indeterminate.Therefore, we believe that it is inevitable that our references limit themselves to these pioneering works.

Stable Computation of the 2D Medial Axis Transform

1998 ◽  
Vol 08 (05n06) ◽  
pp. 577-598 ◽  
Author(s):  
Guy Evans ◽  
Alan Middleditch ◽  
Nick Miles
Keyword(s):  
Medial Axis ◽  
Medial Axis Transform ◽  
Element Mesh ◽  
Alternative Representation ◽  
Circular Arcs ◽  
Straight Line ◽  
Key Points ◽  
Stable Computation ◽  
Mathematical Definitions ◽  
Biological Shape

The medial axis transform of a 2D region was introduced by Blum in the 1960's as an aid to the description of biological shape. It is an alternative representation of a region which is often more amenable to analysis. This property has led to its use in diverse fields including pattern recognition and automatic finite element mesh generation. There are two widely agreed mathematical definitions for the medial axis transform which are closely related. It is shown that these definitions are not in general equivalent, despite being so far many types of region. In this paper, precise mathematical definitions of the medial axis transform and its key points (atoms) are given, and an O(n2) algorithm for its computation via those atoms presented. This algorithm is described in terms of simple polygons whose sole boundary consists of circular arcs and straight line segments, then extended to polygons with holes. It is shown how more complex edges could be accommodated. In comparison with existing algorithms it is simple to implement and stable in the presence of geometric degeneracy.

scholarly journals II. Second memoir on plane stigmatics

1867 ◽  
Vol 15 ◽  
pp. 192-203
Keyword(s):  
General Expression ◽  
Fundamental Lemma ◽  
Double Points ◽  
Straight Line ◽  
Straight Lines ◽  
Stigmatic Point ◽  
Coordinate Geometry

Let there be two groups of points upon a plane, termed, for distinction, indices and stigmata respectively, bearing such relations to each other that any one index determines the position of n stigmata, and any one stigma determines the position of m indices. The theory of these relations between indices and stigmata constitutes plane stigmatics . Each related pair of index X and stigma Y constitutes a stigmatic point , henceforth written “the s. point ( xy )." The straight lines joining any index with each of its corresponding stigmata are termed ordinates . If, when the index moves upon a straight line, the ordinate remains parallel to some other straight line, the relation between index and stigma is that expressed by the relation between abscissa and ordinate in the coordinate geometry of Descartes. When only one index corresponds to one stigma and conversely, and both indices and stigmata lie always on one and the same straight line, or the indices upon one and the stigmata upon another, the relations between indices and stigmata are those between homologous points in the homographic geometry of Chasles. The general expression of the stigmatic relation is obtained by a generalization of Chasles’s fundamental lemma in his theory of characteristics ( Comptes Rendus , June 27, 1864, vol. lviii. p. 1175), clinants being substituted for scalars. It results that in certain forms of the law of coordination , which “ coordinates ” the stigmata with the indices, there may be solitary indices which have no corresponding stigmata, and solitary stigmata which have no corresponding indices, and also double points in which the index coincides with its stigma (76). The particular case in which one index corresponds to one stigma and conversely, and no solitary index or stigma occurs, is termed a stigmatic line (henceforth written “s. line”), because the Cartesian case is that of a Cartesian straight line in ordinary coordinate geometry, but in the general s. line the figures described by index and stigma may be any directly similar plane figures (77). The investigation of this particular case occupies almost the whole of the Introductory Memoir . When one index corresponds to one stigma and conversely, but there is one solitary index and one solitary stigma, we have s. homography , provided the solitary index is distinct from the solitary stigma (79), and s. involution when the solitary index coincides with the solitary stigma (78), so called because they generalize the relations treated of under these names by Chasles.

scholarly journals TOPOLOGY-PRESERVING WATERMARKING OF VECTOR GRAPHICS

2014 ◽  
Vol 24 (01) ◽  
pp. 61-86 ◽  
Author(s):  
STEFAN HUBER ◽  
MARTIN HELD ◽  
PETER MEERWALD ◽  
ROLAND KWITT
Keyword(s):  
Input Data ◽  
Two Dimensional ◽  
Statistical Features ◽  
Embedding Capacity ◽  
Vector Graphics ◽  
Line Segments ◽  
Circular Arcs ◽  
Straight Line ◽  
Watermark Embedding ◽  
Topological Correctness

Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques.

scholarly journals On the generation of spiral-like paths within planar shapes

2017 ◽  
Vol 5 (3) ◽  
pp. 348-357 ◽  
Author(s):  
Martin Held ◽  
Stefan de Lorenzo
Keyword(s):  
Simple Algorithm ◽  
Basic Algorithm ◽  
Line Segments ◽  
Spiral Path ◽  
Curvature Variation ◽  
Circular Arcs ◽  
Straight Line ◽  
Planar Shapes ◽  
Double Spiral ◽  
Step Over

Abstract We simplify and extend prior work by Held and Spielberger [CAD 2009, CAD&A 2014] to obtain spiral-like paths inside of planar shapes bounded by straight-line segments and circular arcs: We use a linearization to derive a simple algorithm that computes a continuous spiral-like path which (1) consists of straight-line segments, (2) has no self-intersections, (3) respects a user-specified maximum step-over distance, and (4) starts in the interior and ends at the boundary of the shape. Then we extend this basic algorithm to double-spiral paths that start and end at the boundary, and show how these double spirals can be used to cover complicated planar shapes by composite spiral paths. We also discuss how to improve the smoothness and reduce the curvature variation of our paths, and how to boost them to higher levels of continuity. Highlights The algorithm computes a spiral path within planar shapes with and without islands. It respects a user-specified maximum step-over distance. Double spirals and composite spiral paths can be computed. Heuristics for smoothing the spirals are discussed. The algorithm is simple and easy to implement, and suitable for various applications.

A Three-Dimensional Morphometrical Study of the Distal Human Femur

1992 ◽  
Vol 206 (3) ◽  
pp. 147-157 ◽  
Author(s):  
M Zoghi ◽  
M S Hefzy ◽  
K C Fu ◽  
W T Jackson
Keyword(s):  
Distal Femur ◽  
Sagittal Plane ◽  
Femoral Condyle ◽  
Three Dimensional ◽  
Distal Portion ◽  
Average Radius ◽  
Anterior Portion ◽  
Circular Arcs ◽  
Straight Line ◽  
Circular Base

The objective of this paper is to present a method to describe the three-dimensional variations of the geometry of the three portions forming the distal part of the human femur: the medial and lateral femoral condyles and the intercondylar fossa. The contours of equally spaced sagittal slices were digitized on the distal femur to determine its surface topography. Data collection was performed using a digitizer system which utilizes low-frequency, magnetic field technology to determine the position and orientation of a magnetic field sensor in relation to a specified reference frame. The generalized reduced gradient optimization method was used to reconstruct the profile of each slice utilizing two primitives: straight-line segments and circular arcs. The profile of each slice within the medial femoral condyle was reconstructed using two circular arcs: posterior and distal. The profile of each slice within the lateral femoral condyle was reconstructed using three circular arcs: posterior, distal and anterior. Finally, the profile of each slice within the intercondylar fossa was reconstructed using two circular arcs: proximal-posterior and anterior, and a distal-posterior straight-line segment tangent to the proximal-posterior circular arc. Combining the data describing the profiles of the different slices forming the distal femur, the posterior portions of each of the medial and lateral femoral condyles were modelled using parts of spheres having an average radius of 20 mm. The anterior portion of the lateral condyle was approximated to a right cylinder having its circular base parallel to the sagittal plane with an average radius of 26 mm. The anterior portion of the intercondylar fossa was modelled using an oblique cylinder having its circular base parallel to the sagittal plane with an average radius of 22 mm. Furthermore, it is suggested that the distal portion of the lateral femoral condyle could be modelled using parts of two oblique cones while the distal portion of the medial femoral condyle could be modelled using a part of a single oblique cone, all cones having their circular bases parallel to the sagittal plane. It is also suggested that the posterior portion of the intercondylar fossa could be modelled using two oblique cones: a proximal cone having its base parallel to the sagittal plane and a distal cone having its base parallel to the frontal plane.

Modelling of circular wave guides

1992 ◽  
Vol 70 (10-11) ◽  
pp. 1092-1098 ◽  
Author(s):  
A. Delage ◽  
K. A. McGreer ◽  
E. Rainville
Keyword(s):  
Integrated Circuit ◽  
Beam Propagation Method ◽  
Two Dimensions ◽  
Effective Index ◽  
Radius Of Curvature ◽  
Index Method ◽  
Photonic Integrated Circuit ◽  
Circular Arcs ◽  
Straight Line ◽  
Wave Guides

In many circumstances the design of interconnects in a photonic integrated circuit can be simplified by using low loss curved wave guides in the shapes of circular arcs. Radiative losses associated with the curvature have been computed as a function of the radius of curvature. The technique takes advantage of the effective index method to reduce the problem from two dimensions to one dimension (1D) and uses a change of coordinate that transforms an arc of circle into a straight line. This transformation results in a monotonous increase of the refractive index as function of r (the distance from the centre of the circle) for original constant index regions. The new system is solved by discretizing this varying effective index onto many small layers of constant index over a window large enough to contain the region where the field is not negligible. A multilayer algorithm in 1D is then used to find complex propagation constants in which the imaginary part is related to the fundamental energy loss owing to the curvature. The solution also gives the shape of the field necessary to match the mode profiles at the junction between the straight and curved part of the wave guide. The basic change of variable has been extended to the finite difference solution of the scalar wave equation and to the beam propagation method.

Sign in / Sign up

Export Citation Format

Share Document