Geometric modeling in architecture and technical of conjugate surfaces of the second order

Point projection on an implicit surface is essential for the geometric modeling and graphics applications of it. This paper presents a method for computing the principle curvatures and principle directions of an implicit surface. Using the principle curvatures and principle directions, we construct a torus patch to approximate the implicit surface locally. The torus patch is second order osculating to the implicit surface. By taking advantage of the approximation torus patch, this paper develops a second order geometric iterative algorithm for point projection on the implicit surface. Experiments illustrate the efficiency and less dependency on initial values of our algorithm.

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Second Order Curves on Computer Screen

Geometry & Graphics ◽

10.12737/article_5b55a829cee6c0.74112002 ◽

2018 ◽

Vol 6 (2) ◽

pp. 100-112 ◽

Cited By ~ 2

Author(s):

Виктор Короткий ◽

Viktor Korotkiy ◽

Е. Усманова ◽

E. Usmanova

Keyword(s):

Computer Graphics ◽

Computer Program ◽

Geometric Modeling ◽

Main Tool ◽

Second Order ◽

Modern Computer ◽

Order Curve ◽

Junction Points ◽

Tools And Techniques ◽

Straight Lines

Modern computer graphics is based on methods of computational geometry. The curves and surfaces’ description is based on apparatus of spline functions, which became the main tool for geometric modeling. Methods of projective geometry are almost not applying. One of the reasons for this is impossibility to exactly construct a second-order curve passing through given points and tangent to given straight lines. To eliminate this defect a computer program for second order curves construction has been developed. The program performs the construction of second-order curve’s metric (center, vertices, asymptotes, foci) for following combinations: • The second-order curve is given by five points; • The second-order curve is given by five tangent lines; • The second-order curve is given by a point and two tangent lines with points of contact indicated on them; • The parabola is given by four tangent lines; • The parabola is given by four points. In this paper are presented algorithms for construction a metric for each combination. After construction the metric the computer program written in AutoLISP language and using geometrically exact projective algorithms which don’t require algebraic computations draws a second-order curve. For example, to construct vertices and foci of two parabolas passing through four given points, it is only necessary to draw an arbitrary circle and several straight lines. To construct a conic metric passing through five given points, it is necessary to perform only three geometrically exact operations: to construct an involution of conjugate diameters, to find the main axes and asymptotes; to note the vertices of desired second-order curve. Has been considered the architectural appearance of a new airport in Simferopol. It has been demonstrated that a terminal facade’s wavelike form can be obtained with a curve line consisting of conic sections’ areas with common tangent lines at junction points. The developed computer program allows draw second-order curves. The program application will promote the development of computer graphics’ tools and techniques.

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A Unified Constructive Algorithm For Second- Order Curves’ Foci Creation

Geometry & Graphics ◽

10.12737/article_5b559dc3551f95.26045830 ◽

2018 ◽

Vol 6 (2) ◽

pp. 47-54 ◽

Cited By ~ 5

Author(s):

Д. Волошинов ◽

Denis Voloshinov

Keyword(s):

Geometric Modeling ◽

Second Order ◽

Geometric Constructions ◽

Practical Applications ◽

Geometric Problems ◽

Geometric Concepts ◽

Unified Algorithm ◽

To Come ◽

Scientific Value ◽

Practical Feasibility

While using conventional tools for solving geometric problems, it is difficult to obtain and analyze results where imaginary geometric images appear. Despite the recognition of legitimacy and scientific value of imaginary solutions presenting in geometric constructions, the question on such solutions’ appropriateness and practical feasibility remains no completely clear up till now. That’s why, for most practitioners imaginary solutions are presented as something unattainable or unimportant. However, the introduction of imaginary geometric images into the practice of geometric modeling makes it possible to obtain solutions in an exhaustiveness, to develop unified algorithms for solving problems that were usually presented as either not solvable or reduced to solutions in partial settings. The use of computer technologies and the paradigm of constructive geometric modeling allow eliminate this problem’s acuteness, and direct efforts both at geometric theory’s improvement and introduction of scientific achievements in this area at the field of practical applications. Automation means for geometric experiment make it possible to find new regularities in seemingly well-known mathematical facts, to come to more general understanding of geometric concepts and images. This paper is devoted to analysis of some geometric schemes and to discussion of arising from it questions related to the theory of second-order curves creation by the methods of constructive synthesis. In the paper it has been demonstrated that the currently used definitions of second-order curves’ center and diameters contradict the principle of conics indistinguishability in projective geometry. The ways for eliminating of these contradictions have been proposed, and a unified algorithm for the second-order curves’ foci creation has been developed based on these ways.

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On Spatial Euler–Savary Equations for Envelopes

Journal of Mechanical Design ◽

10.1115/1.2735339 ◽

2006 ◽

Vol 129 (8) ◽

pp. 865-875 ◽

Cited By ~ 14

Author(s):

David B. Dooner ◽

Michael W. Griffis

Keyword(s):

Direct Contact ◽

Order Approximation ◽

Spatial Relations ◽

Second Order ◽

Gear Pair ◽

Second Order Approximation ◽

Planar Curves ◽

Gear Teeth ◽

Kinematic Geometry ◽

Conjugate Surfaces

Presented are three equations that are believed to be original and new to the kinematics community. These three equations are extensions of the planar Euler–Savary relations (for envelopes) to spatial relations. All three spatial forms parallel the existing well established planar Euler–Savary equations. The genesis of this work is rooted in a system of cylindroidal coordinates specifically developed to parameterize the kinematic geometry of generalized spatial gearing and consequently a brief discussion of such coordinates is provided. Hyperboloids of osculation are introduced by considering an instantaneously equivalent gear pair. These analog equations establish a relation between the kinematic geometry of hyperboloids of osculation in mesh (viz., second-order approximation to the axode motion) to the relative curvature of conjugate surfaces in direct contact (gear teeth). Planar Euler–Savary equations are presented first along with a discussion on the terms in each equation. This presentation provides the basis for the proposed spatial Euler–Savary analog equations. A lot of effort has been directed to establishing generalized spatial Euler–Savary equations resulting in many different expressions depending on the interpretation of the planar Euler–Savary equation. This work deals with the interpretation where contacting surfaces are taken as the spatial analog to the contacting planar curves.

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APPLICATION OF CIRCLES FOR CONJUGATION OF FLAT CONTOURS OF THE FIRST ORDER OF SMOOTHNESS

APPLIED GEOMETRY AND ENGINEERING GRAPHICS ◽

10.32347/0131-579x.2021.100.162-171 ◽

2021 ◽

pp. 162-171

Author(s):

Galyna Koval ◽

Margarita Lazarchuk ◽

Liudmila Ovsienko

Keyword(s):

Coordinate System ◽

Geometric Modeling ◽

Second Order ◽

First Order ◽

Order Curve ◽

First Case ◽

Affine System ◽

Coordinate Lines ◽

Point Of Intersection ◽

Projective Coordinate

In geometric modeling of contours, especially for conjugation of sections of flat contours of the first order of smoothness, arcs of circles can be applied. The article proposes ways to determine the equations of a circle for two ways of its problem: the problem of a circle with a point and two tangents, none of which contains a given point, and the problem of a circle with three tangents. The equations of the circles were determined in both cases using a projective coordinate system. In the first case, when a circle is given by a point and two tangents, neither of which contains this point, the center of the conjugation circle is defined as the point of intersection of two locus of points - the bisector of the angle between the tangents and the parabola, the focus of which is a given point. given tangents. In the general case, there are 2 conjugation circles for which canonical equations are defined. Parametric equations of conjugate circles, the parameters of which are equal to 0 and ∞ on tangents and equal to one at a given point, with the help of affine and projective coordinates of points of contact are determined first in the projective coordinate system, and then translated into affine system. For the second case, when specifying a circle using three tangent lines, the equation of the second-order curve tangent to these lines is first determined in the projective coordinate system. The tangent lines are taken as the coordinate lines of the projective coordinate system. The unit point of the projective coordinate system is selected in the metacenter of the thus obtained base triangle. The equation of the tangent to the base lines of the second order contains two unknown variables, positive or negative values which determine the location of four possible tangents of the second order. After writing the vector-parametric equation of the tangent curve of the second order in the affine coordinate system, the equation is written to determine the parameters of cyclic points. In order for the equation of the tangent curve of the second order obtained in the projective plane to be an equation of a circle, it must satisfy the coordinates of the cyclic points of the plane, which allows to write the second equation to determine the parameters of cyclic points. By solving a system of two equations, we obtain the required equations of circles tangent to three given lines.

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Disappearance Voltage Determinations Using a Convergent Beam

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100064979 ◽

1971 ◽

Vol 29 ◽

pp. 184-185

Author(s):

W. L. Bell

Keyword(s):

Second Order ◽

Objective Method ◽

Uniform Thickness ◽

Rocking Curve ◽

Crystal Perfection ◽

Kikuchi Line ◽

Central Maximum ◽

Diffraction Patterns ◽

Convergent Beam ◽

Beam Technique

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.

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