On Pedal Quadrics in Non-Euclidean Hyperspace

1925 ◽  
Vol 22 (5) ◽  
pp. 751-758
Author(s):  
J. P. Gabbatt
Keyword(s):  
Orthogonal Projection ◽  
Euclidean Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Elementary Geometry ◽  
Orthogonal Projections ◽  
Point Pair ◽  
Straight Line ◽  
Three Dimensional Space

1. The following are well-known theorems of elementary geometry: Given any euclidean plane triangle, A0 A1 A2, and any pair of points, X, Y, isogonally conjugate q. A0 A1 A2; then the orthogonal projections of X, Y on the sides of A0 A1 A2 lie on a circle, the pedal circle of the point-pair. If either of the points X,- Y describe a (straight) line, m, then the other describes a conic circumscribing A0 A1 A2, and the pedal circle remains orthogonal to a fixed circle, J; thus the pedal circles in question are members of an ∞2 linear system of circles of which the circle J and the line at infinity constitute the Jacobian. In particular, if the line m meet Aj Ak at Lt (i, j, k = 0, 1, 2), then the circles on Ai Li as diameter, which are the pedal circles of the point-pairs Ai, Li, are coaxial; the remaining circles of the coaxial system being the director circles of the conics, inscribed in the triangle A0 A1 A2, which touch the line m. If Mi denote the orthogonal projection on m of Ai, and Ni the orthogonal projection on Aj Ak of Mi, then the three lines Mi Ni meet at a point (Neuberg's theorem), viz. the centre of the circle J. Analogues for three-dimensional space of most of these theorems are also known ‖.

A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

1925 ◽  
Vol 22 (5) ◽  
pp. 694-699 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Quadratic Relation ◽  
Four Dimensions ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Set Up ◽  
Image Position ◽  
Three Dimensional Space

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relationwhich immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

scholarly journals A quadratic transformation of the three-dimensional space of all conics through two points of a projective plane

2009 ◽  
Vol 42 (3) ◽  
Author(s):  
Giorgio Donati
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Quadratic Transformation ◽  
Pencil Of Conics ◽  
Dimensional Projective Space ◽  
Three Dimensional Space

AbstractUsing the Steiner’s method of projective generation of conics and its dual we define two projective mappings of a double contact pencil of conics into itself and we prove that one is the inverse of the other. We show that these projective mappings are induced by quadratic transformations of the three-dimensional projective space of all conics through two distinct points of a projective plane.

scholarly journals The Third Dimention in a Landscape

Secreta Artis ◽  
2021 ◽  
pp. 74-82
Author(s):  
Daria Vladimirovna Fomicheva
Keyword(s):  
Land Surface ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Landscape Painting ◽  
The Third ◽  
Soviet Art ◽  
Other Hand ◽  
Third Dimension ◽  
Three Dimensional Space

The present study examines the principles of conveying the third dimension in landscape painting. The author analyzes the recommendations provided in J. Littlejohns’ manual entitled “The Composition of a Landscape” [London, 1931]. J. Littlejohns describes four methods of showing depth in a landscape painting, each illustrated with pictorial composition schemes: 1) portrayal of long roads, which allows one to unveil the plasticity of the land surface; 2) creation of a “route” for the viewer by means of a well-thought-out arrangement of natural landforms; 3) introduction of vertically and horizontally flowing streams of water on different picture planes; 4) depiction of cloud shadows on a distinctly hilly landscape. The author of the article compares the schemes contained in the manual of J. Littlejohns with the works of G. G. Nissky, which enables readers to comprehend and reflect on the compositions of the masterpieces created by a prominent figure in Soviet art; on the other hand, Nissky’s landscape paintings open for a deeper understanding of the meaning and effectiveness of the methods proposed by J. Littlejohns. The outlined composition techniques are certainly relevant for contemporary artists (painters, graphic artists, animators, designers, etc.) as they make it possible to achieve the plastic expressiveness of a three-dimensional space in a twodimensional image.

scholarly journals ANY FIVE POINTS LIE IN A 3-SPACE

2020 ◽  
Author(s):  
Zh. Nikoghosyan ◽  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Entire Space ◽  
Unique Line ◽  
Straight Line ◽  
Set Of Points ◽  
Three Dimensional Space

In axiomatic formulations, every two points lie in a (straight) line, every three points lie in a plane and every four points lie in a three-dimensional space (3-space). In this paper we show that every five points lie in a 3-space as well, implying that every set of points lie in a 3-space. In other words, the 3-space occupies the entire space. The proof is based on the following four axioms: 1) every two distinct points define a unique line, 2) every three distinct points, not lying on the line, define a unique plane, 3) if 𝐴 and 𝐵 are two distinct points in a 3-space, then the line defined by the points 𝐴, 𝐵, entirely lies in this 3-space, 4) if 𝐹1, 𝐹2, 𝐹3 are three distinct points in a 3-space, not lying in a line, then the plane defined by the points 𝐹1, 𝐹2, 𝐹3, lies entirely in this 3-space.

scholarly journals Constructive Geometric Models with Imaginary Objects

2020 ◽  
pp. short27-1-short27-9
Author(s):  
Denis Voloshinov ◽  
Alexandra Solovjeva
Keyword(s):  
Projective Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Order ◽  
Second Order ◽  
The Other ◽  
Significant Obstacle ◽  
The Relationship ◽  
Planar Drawing ◽  
Three Dimensional Space

The article is devoted to the consideration of a number of theoretical questions of projective geometry related to specifying and displaying imaginary objects, especially, conics. The lack of development of appropriate constructive schemes is a significant obstacle to the study of quadratic images in three-dimensional space and spaces of higher order. The relationship between the two circles, established by the inversion operation with respect to the other two circles, in particular, one of which is imaginary, allows obtain a simple and effective method for indirect setting of imaginary circles in a planar drawing. The application of the collinear transformation to circles with an imaginary radius also makes it possible to obtain unified algorithms for specifying and controlling imaginary conics along with usual real second-order curves. As a result, it allows eliminate exceptional situations that arise while solving problems with quadratic images in spaces of second and higher order.

scholarly journals Minimum action principle and shape dynamics

2017 ◽  
Vol 14 (130) ◽  
pp. 20170031 ◽  
Author(s):  
Patrice Koehl
Keyword(s):  
Elastic Energy ◽  
Dimensional Space ◽  
Shape Recognition ◽  
Three Dimensional ◽  
The Other ◽  
Success Rates ◽  
Evolutionary Patterns ◽  
The Cost ◽  
Better Than ◽  
Three Dimensional Space

In this paper, we propose a new method for computing a distance between two shapes embedded in three-dimensional space. Instead of comparing directly the geometric properties of the two shapes, we measure the cost of deforming one of the two shapes into the other. The deformation is computed as the geodesic between the two shapes in the space of shapes. The geodesic is found as a minimizer of the Onsager–Machlup action, based on an elastic energy for shapes that we define. Its length is set to be the integral of the action along that path; it defines an intrinsic quasi-metric on the space of shapes. We illustrate applications of our method to geometric morphometrics using three datasets representing bones and teeth of primates. Experiments on these datasets show that the variational quasi-metric we have introduced performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.

Geometry in three dimensions over GF (3)

1954 ◽  
Vol 222 (1149) ◽  
pp. 262-286 ◽  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Galois Field ◽  
The Other ◽  
Three Dimensions ◽  
Common Vertex ◽  
Line Geometry ◽  
Fundamental Properties ◽  
The Common ◽  
Three Dimensional Space

The object of this paper is to give some account of the geometry of the three-dimensional space S wherein the co-ordinates belong to a Galois field K of 3 marks. A description of the fundamental properties of quadrics is sufficiently long for one paper, and so an account of the line geometry is deferred. The early paragraphs (§§ 1 to 4) are necessarily concerned with geometry on a line or in a plane. A line consists of 4 points; these are self-projective under all 4! permutations. A plane consists of 13 points and has the same number, 234, of triangles, quadrangles, quadri-laterals and non-singular conics. A diagram is helpful, especially when we consider sections by planes in S . The space S has 40 points. Non-singular quadrics are of two kinds: either ruled, when we call them hyperboloids, or non-ruled, when we call them ellipsoids. A hyperboloid H consists of 16 points and has a pair of reguli; the 24 points of S not on H are the vertices of 6 tetra-hedra that form two allied desmic triads. The ellipsoid F is introduced in § 12; it consists of 10 points, the other 30 points of S being separated into two batches of 15 between which there is a symmetrical (3, 3) correspondence. Either batch can be arranged as a set of 6 pentagons, each of the 15 points being the common vertex of 2 of these. The pentagons of either set have all their edges tangents of F and, with their polar pentahedra, have significant properties and interrelations. By no means their least important attribute is that they afford, with F , so apposite a domain of operation for the simple group of order 360. In §§ 23 to 26 are described the operations of the group in this setting. Thereafter the 36 separations of the 10 points of F into complementary pentads are discussed, no 4 of either pentad being coplanar. During the work constructions for an ellipsoid are encountered; one is in § 16, another in § 30.

The Inversive Distance Between Two Circles

1967 ◽  
Vol 19 ◽  
pp. 1149-1152
Author(s):  
O. Bottema
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Plane ◽  
Dimensional Space ◽  
Single Point ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Inversive Geometry ◽  
The One ◽  
Three Dimensional Space

H. S. M. Coxeter (3) has recently studied the correspondence between two geometries the isomorphism of which was well known, but to which he was able to add some remarkable consequences. The two geometries are the inversive geometry of a plane E (the Euclidean plane completed with a single point at infinity or, what is the same thing, the plane of complex numbers to which ∞ is added) on the one hand, and the hyperbolic geometry of three-dimensional space S.

scholarly journals Nontrivial Isometric Embeddings for Flat Spaces

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 477
Author(s):  
Sergey Paston ◽  
Taisiia Zaitseva
Keyword(s):  
Minkowski Space ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Ambient Space ◽  
The Other ◽  
Field Limit ◽  
Isometric Embeddings ◽  
Weak Gravitational Field ◽  
Three Dimensional Space

Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be required to have the same symmetry as the metric. On the other hand, it is possible to require the embedding to be unfolded so that the surface in the ambient space would occupy the subspace of the maximum possible dimension. In the weak gravitational field limit, such a requirement together with a large enough dimension of the ambient space makes embedding gravity equivalent to general relativity, while at lower dimensions it guarantees the linearizability of the equations of motion. We discuss symmetric embeddings for the metrics of flat Euclidean three-dimensional space and Minkowski space. We propose the method of sequential surface deformations for the construction of unfolded embeddings. We use it to construct such embeddings of flat Euclidean three-dimensional space and Minkowski space, which can be used to analyze the equations of motion of embedding gravity.

scholarly journals Improved Total Least Square Algorithm

2015 ◽  
Vol 9 (1) ◽  
pp. 394-399 ◽  
Author(s):  
Deng Yonghe
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Least Square ◽  
Error Equation ◽  
Virtual Observation ◽  
Straight Line ◽  
Condition Equation ◽  
Total Least Square ◽  
Three Dimensional Space ◽  
Improved Algorithm

Aim to blemish of total least square algorithm based on error equation of virtual observation,this paper proposed a sort of improved algorithm which doesn’t neglect condition equation of virtual observation,and considers both error equation and condition equation of virtual observation.So,the improved algorithm is better.Finally,this paper has fitted a straight line in three-dimensional space based on the improved algorithm.The result showed that the improved algorithm is viable and valid.

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