Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres

2021 ◽  
pp. 12-29
Author(s):  
Vladimir Vyshnyepolskiy ◽  
E. Zavarihina ◽  
D. Peh
Keyword(s):  
Cylindrical Surface ◽  
Second Order ◽  
Mutual Position ◽  
Geometric Figures ◽  
Straight Line ◽  
Zero Radius ◽  
Ellipsoid Of Revolution ◽  
Order Degenerate ◽  
The Common ◽  
Equidistant Points

The article deals with the geometric locations of points equidistant from two spheres. In all variants of the mutual position of the spheres, the geometric places of the points are two surfaces. When the centers of the spheres coincide with the locus of points equidistant from the spheres, there will be spheres equal to the half-sum and half-difference of the diameters of the original spheres. In three variants of the relative position of the initial spheres, one of the two surfaces of the geometric places of the points is a two-sheet hyperboloid of revolution. It is obtained when: 1) the spheres intersect, 2) the spheres touch, 3) the outer surfaces of the spheres are removed from each other. In the case of equal spheres, a two-sheeted hyperboloid of revolution degenerates into a two-sheeted plane, more precisely, it is a second-order degenerate surface with a second infinitely distant branch. The spheres intersect - the second locus of the points will be the ellipsoid of revolution. Spheres touch - the second locus of points - an ellipsoid of revolution, degenerated into a straight line, more precisely into a zero-quadric of the second order - a cylindrical surface with zero radius. The outer surfaces of the spheres are distant from each other - the second locus of points will be a two-sheet hyperboloid of revolution. The small sphere is located inside the large one - two coaxial confocal ellipsoids of revolution. In all variants of the mutual position of spheres of the same diameters, the common geometrical place of equidistant points is a plane (degenerate surface of the second order) passing through the middle of the segment perpendicular to it, connecting the centers of the original spheres. The second locus of points equidistant from two spheres of the same diameter can be either an ellipsoid of revolution (if the original spheres intersect), or a straight (cylindrical surface with zero radius) connecting the centers of the original spheres when the original spheres touch each other, or a two-sheet hyperboloid of revolution (if continue to increase the distance between the centers of the original spheres).

scholarly journals Analysis of Linear Surfaces of Building Structures

2020 ◽  
Vol 24 (3) ◽  
pp. 111-120
Author(s):  
S. N. Volkova ◽  
A. V. Shleenko ◽  
V. V. Morozova ◽  
E. E. Sivak
Keyword(s):  
Cylindrical Surface ◽  
Construction Material ◽  
Second Order ◽  
Polar Coordinate System ◽  
Arbitrary Parameter ◽  
Building Structures ◽  
Straight Line ◽  
Polar Coordinate ◽  
Third Dimension ◽  
Straight Lines

Purpose of reseach is to analyze the practice in the application of surfaces formed by the movement of a straight line. It is known that among the second-order surfaces cones, cylinders, hyperboloids of one sheet and hyperbolic paraboloids, as well as lines represented in the polar coordinate system in the form of intricate shapes that can be represented in space by the above-mentioned surfaces, adding a third dimension, have rectilinear generators. The strength resulting from covering each point of the listed surfaces with straight lines from different families does not make the structure heavier but strengthens it and makes it light compared to monoliths without reinforcements made of other materials, in which stability is not based on Shukhov calculation formulas. Methods Finding families of rectilinear generators for second-order surfaces calculation of which is based on the separation of equations that represent a second-order surface as a difference of squares in one part of the equation and as a product with an arbitrary parameter in the other part. Results. Analyzing second-order surfaces, we came to the conclusion that cones, cylinders are prone to this method of Shukhov calculations; equation of the form F (x,y)=0 in space defines a cylindrical surface whose generators are parallel to axis oz. Similarly, F (x, z)=0 defines a cylindrical surface with generators parallel to axis oy and F (y;z)=0 is a cylindrical surface with generators parallel to axis ox. A hyperboloid of one sheet, hyperbolic paraboloid, i.e. 10 surfaces out of 14, make up more than 70%. Conclusion. As a result of applying these formulas for calculating reinforced building structures, city buildings will acquire a new appearance, which will create a comfortable environment for residents, as well as lead to saving construction material resources.

A Simple Practical Criterion to Guarantee Second Order Smoothness of Blend Surfaces

1989 ◽  
Author(s):  
J. Pegna ◽  
F.-E. Wolter
Keyword(s):  
Second Order ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Common Boundary ◽  
First Order ◽  
Computer Aided ◽  
The Common ◽  
Blend Surfaces ◽  
Continuous Curves ◽  
Order Smoothness

Abstract Computer Aided Geometric Design of surfaces sometimes presents problems that were not envisioned by mathematicians in differential geometry. This paper presents mathematical results that pertain to the design of second order smooth blending surfaces. Second order smoothness normally requires that normal curvatures agree along all tangent directions at all points of the common boundary of two patches, called the linkage curve. The Linkage Curve Theorem proved here shows that, for the blend to be second order smooth when it is already first order smooth, it is sufficient that normal curvatures agree in one direction other than the tangent to a first order continuous linkage curve. This result is significant for it substantiates earlier works in computer aided geometric design. It also offers simple practical means of generating second order blends for it reduces the dimensionality of the problem to that of curve fairing, and is well adapted to a formulation of the blend surface using sweeps. From a theoretical viewpoint, it is remarkable that one can generate second order smooth blends with the assumption that the linkage curve is only first order smooth. This property may be helpful to the designer since linkage curves can be constructed from low order piecewise continuous curves.

scholarly journals Learned Video Compression via Joint Spatial-Temporal Correlation Exploration

2020 ◽  
Vol 34 (07) ◽  
pp. 11580-11587
Author(s):  
Haojie Liu ◽  
Han Shen ◽  
Lichao Huang ◽  
Ming Lu ◽  
Tong Chen ◽  
...  
Keyword(s):  
Video Coding ◽  
Video Compression ◽  
High Efficiency ◽  
Temporal Correlation ◽  
Second Order ◽  
Compression Method ◽  
High Efficiency Video Coding ◽  
First Order ◽  
Coding Efficiency ◽  
The Common

Traditional video compression technologies have been developed over decades in pursuit of higher coding efficiency. Efficient temporal information representation plays a key role in video coding. Thus, in this paper, we propose to exploit the temporal correlation using both first-order optical flow and second-order flow prediction. We suggest an one-stage learning approach to encapsulate flow as quantized features from consecutive frames which is then entropy coded with adaptive contexts conditioned on joint spatial-temporal priors to exploit second-order correlations. Joint priors are embedded in autoregressive spatial neighbors, co-located hyper elements and temporal neighbors using ConvLSTM recurrently. We evaluate our approach for the low-delay scenario with High-Efficiency Video Coding (H.265/HEVC), H.264/AVC and another learned video compression method, following the common test settings. Our work offers the state-of-the-art performance, with consistent gains across all popular test sequences.

A new kinematic method for mapping seismic reflectors

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1594-1602 ◽  
Author(s):  
Jingping Zhe ◽  
Stewart A. Greenhalgh
Keyword(s):  
Numerical Scheme ◽  
Graphical Method ◽  
Common Tangent ◽  
Curved Boundaries ◽  
Curved Boundary ◽  
Straight Line ◽  
Kinematic Method ◽  
The Common ◽  
Seismic Reflectors ◽  
Arbitrary Velocity

This paper presents a modern version of an old technique of common tangent reflection migration. Rather than using the graphical method of swinging arcs and looking for the envelope of touching tangents on widely separated geophones, we use a numerical scheme of searching along each isochron, constructed by a wave‐equation‐based modeling scheme for arbitrary velocity media, to find the common tangent points. The assumption is made that the receivers are close together so that the interface can be locally approximated by a straight line, although different pairs of receivers permit different tangents to be found, enabling a curved boundary to be migrated. The technique can be extended to several shot gathers to map multiple curved boundaries. Synthetic examples are used to illustrate the capabilities of the method and the effect of using an erroneous velocity distribution.

Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces

2018 ◽  
Vol 61 (4) ◽  
pp. 717-737 ◽  
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Second Order ◽  
Linear Operators ◽  
Bounded Linear ◽  
Order Degenerate ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Equations With Delay ◽  
Closed Linear Operators

AbstractWe give necessary and sufficient conditions of the Lp-well-posedness (resp. -wellposedness) for the second order degenerate differential equation with finite delayswith periodic boundary conditions (Mu)(0) = (Mu)(2π), (Mu)′ (0) = (Mu)′ (2π), where A, B, and M are closed linear operators on a complex Banach space X satisfying D(A) ∩ D(B) ⊂ D(M), F and G are bounded linear operators from into X.

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