scholarly journals Epihypocurves and epihypocyclic surfaces with arbitrary base curve

2021 ◽  
Vol 17 (4) ◽  
pp. 404-413
Author(s):  
Vyacheslav N. Ivanov
Keyword(s):  
Space Curve ◽  
Form Finding ◽  
Rolling Circle ◽  
Curves And Surfaces ◽  
Normal Plane ◽  
Straight Line ◽  
Arbitrary Base ◽  
Base Curve ◽  
Base Circle ◽  
Cyclic Surface

If a circle rolls around another motionless circle then a point bind with the rolling circle forms a curve. It is called epicycloid, if a circle is rolling outside the motionless circle; it is called hypocycloid if the circle is rolling inside the motionless circle. The point bind to the rolling circle forms a space curve if the rolling circle has the constant incline to the plane of the motionless circle. The cycloid curve is formed when the circle is rolling along a straight line. The geometry of the curves formed by the point bind to the circle rolling along some base curve is investigated at this study. The geometry of the surfaces formed when the circle there is rolling along some curve and rotates around the tangent to the curve is considered as well. Since when the circle rotates in the normal plane of the base curve, a point rigidly connected to the rotating circle arises the circle, then an epihypocycloidal cyclic surface is formed. The vector equations of the epihypocycloid curve and epihypocycloid cycle surfaces with any base curve are established. The figures of the epihypocycloids with base curves of ellipse and sinus are got on the base of the equations obtained. These figures demonstrate the opportunities of form finding of the surfaces arised by the cycle rolling along different base curves. Unlike epihypocycloidal curves and surfaces with a base circle, the shape of epihypocycloidal curves and surfaces with a base curve other than a circle depends on the initial rolling point of the circle on the base curve.

Mathematical Model for Face-Hobbed Straight Bevel Gears

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Yi-Pei Shih
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Tooth Contact Analysis ◽  
Rolling Circle ◽  
High Productivity ◽  
Bevel Gears ◽  
Straight Line ◽  
Base Circle ◽  
Straight Line Mechanism ◽  
Straight Lines

Face hobbing, a continuous indexing and double-flank cutting process, has become the leading method for manufacturing spiral bevel gears and hypoid gears because of its ability to support high productivity and precision. The method is unsuitable for cutting straight bevel gears, however, because it generates extended epicycloidal flanks. Instead, this paper proposes a method for fabricating straight bevel gears using a virtual hypocycloidal straight-line mechanism in which setting the radius of the rolling circle to equal half the radius of the base circle yields straight lines. This property can then be exploited to cut straight flanks on bevel gears. The mathematical model of a straight bevel gear is developed based on a universal face-hobbing bevel gear generator comprising three parts: a cutter head, an imaginary generating gear, and the motion of the imaginary generating gear relative to the work gear. The proposed model is validated numerically using the generation of face-hobbed straight bevel gears without cutter tilt. The contact conditions of the designed gear pairs are confirmed using the ease-off topographic method and tooth contact analysis (TCA), whose results can then be used as a foundation for further flank modification.

scholarly journals The Ruled Surfaces According to Bishop Frame in Minkowski 3-Space

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Nural Yüksel
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
Spacelike Curve ◽  
Bishop Frame ◽  
Straight Line ◽  
Base Curve ◽  
Distribution Parameters

We investigate the ruled surfaces generated by a straight line in Bishop frame moving along a spacelike curve in Minkowski 3-space. We obtain the distribution parameters, mean curvatures. We give some results and theorems related to be developable and minimal of them. Furthermore, we show that, if the base curve of the ruled surface is also an asymtotic curve and striction line, then the ruled surface is developable.

The projective geometry of ambiguous surfaces

1990 ◽  
Vol 332 (1623) ◽  
pp. 1-47 ◽  
Keyword(s):  
Projective Geometry ◽  
Space Curve ◽  
Scene Reconstruction ◽  
Original Surface ◽  
Twisted Cubic ◽  
Straight Line ◽  
Cubic Polynomial ◽  
Ambiguous Case ◽  
Polynomial Constraint

The projective geometry underlying the ambiguous case of scene reconstruction from image correspondences is developed. The am biguous case arises when reconstruction yields two or more essentially different surfaces in space, each capable of giving rise to the image correspondences. Such surfaces naturally occur in complementary pairs. Ambiguous surfaces are examples of rectangular hyperboloids. Complementary ambiguous surfaces intersect in a space curve of degree four, which splits into two components, namely a twisted cubic (space curve of degree three), and a straight line. For each ambiguous surface compatible with a given set of image correspondences, a complementary surface compatible with the same image correspondences can always be found such that both the original surface and the twisted cubic contained in the intersection of the two surfaces are invariant under the same rotation through 180°. In consequence, each ambiguous surface is subject to a cubic polynomial constraint. This constraint is the basis of a new proof of the known result that there are, in general, exactly ten scene reconstructions compatible with five given image correspondences. Ambiguity also arises in reconstruction based on image velocities rather than on image correspondences. The two types of ambiguity have m any sim ilarities because image velocities are obtained from image correspondences as a limit, when the distances between corresponding points become small. It is shown that the amount of similarity is restricted, in that when passing from image correspondences to image velocities, some of the detailed geometry of the ambiguous case is lost.

Feature-Based Identification and Reconstruction of Regular Curves and Surfaces

2012 ◽  
Vol 184-185 ◽  
pp. 206-209
Author(s):  
Tie Li Ye ◽  
He Li ◽  
Qing Liang Zeng
Keyword(s):  
Line Segment ◽  
Cylindrical Surface ◽  
Straight Line Segment ◽  
Geometric Feature ◽  
Surface Point ◽  
Regular Curve ◽  
Curves And Surfaces ◽  
Straight Line ◽  
Feature Based ◽  
Point Set

Based on the preprocessed measuring data, this paper proposes a method for identification and reconstruction of regular curves and surfaces including straight line(segment), circle(arc), four-sided plane and right cylindrical surface. Considering the geometric feature of each kind of regular curve or surface, the paper studies the corresponding algorithm for identifying the curve or surface point set, reconstructs the curve or surface and gives the parameter equation.

scholarly journals Algorithms for Geometrical Models in Borromini's San Carlino alle Quattro Fontane

2016 ◽  
pp. 642-665 ◽  
Author(s):  
Corrado Falcolini
Keyword(s):  
Mathematical Models ◽  
Point Cloud ◽  
Measured Data ◽  
Point Cloud Data ◽  
Simple Point ◽  
Parametric Curves ◽  
Cloud Data ◽  
Curves And Surfaces ◽  
Base Curve ◽  
Different Levels

Construction of mathematical models of the vault of Borromini's San Carlino alle Quattro Fontane based on parametric curves and surfaces, including the shape of the vault and rules for its tessellation with crosses and octagonal coffers. Several models of different complexity are optimized and tested measuring their distance from the point cloud of a very accurate 3D survey and the analysis of such measured data is proposed to validate hypothesis of construction procedures by checking symmetries of coffers shape, scale and position in different levels and sectors. Some original algorithms are discussed to produce regular tessellations on a surface with a generic base curve and to construct regular parametric curves section out of simple point cloud data.

Formation of Cyclic Surfaces in Kinetic Geometry

2017 ◽  
Vol 5 (4) ◽  
pp. 24-36 ◽  
Author(s):  
Николай Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Machine Tool ◽  
General Position ◽  
Space Curve ◽  
Rotational Movement ◽  
Space Navigation ◽  
Straight Line ◽  
Cyclic Surfaces ◽  
Made In

This paper is an evolution of the "Kinematic Compliance of Rotating Spaces" paper, previously published in the "Geometry and Graphics" journal №1, 2013. A great many of mechanisms are making rotational movement, wherein rotating parts of one mechanism are "invading" into a rotation zone belonging to parts of another rotating mechanism. The challenge is to prevent the collision of rotating parts belonging to two or more details with each other. This problem is particularly sensitive for machine engineering. In space navigation, where, in principle, there are no objects that are at rest, the problem of satellites collision with astronomical bodies rotating around their axes is also the urgent one. Therefore, the theory of kinematic matching for rotating spaces R31 and R32 when they are moving independently from each other is urgent too. Each of two considered spaces may have a uniform or non-uniform movement in a given direction, a curved movement or a rotational movement around the axis specified for each space. In this paper has been considered the formation of cyclic surfaces obtained by rotation of one space relative to another one and different orientations of the generating line relative to the axes. Has been considered one of the options for rotating spaces, when their axes are parallel. In such a case the generating line is located in the following positions: it is straight and parallel to the axis; it is straight and intersects the axis; the rectilinear generator is in a plane that is parallel to the plane of the axes; the generating line is a straight line of general position; the generating line is a space curve. Has been demonstrated application of the rotating spaces theory in mining, chemical and machine tool industries, made in the form of inventions, confirmed by copyright certificates of the USSR.

scholarly journals Capturing information on curves and surfaces from their projected images

2020 ◽  
pp. 2050004
Author(s):  
Masaru Hasegawa ◽  
Yutaro Kabata ◽  
Kentaro Saji
Keyword(s):  
Differential Geometry ◽  
Singularity Theory ◽  
Complete Information ◽  
Space Curve ◽  
Geometric Information ◽  
Orthogonal Projections ◽  
Curves And Surfaces

Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal projections in a number of directions. We discuss relations between (1) a space curve and the projected curves from several distinct directions, and (2) a surface and the apparent contours of projections from several distinct directions, in terms of differential geometry and singularity theory. In particular, formulae for recovering certain information on the original curves or surfaces from their projected images are given.

The form of the tangent-developable at points of zero torsion on space curves

1980 ◽  
Vol 88 (3) ◽  
pp. 403-407 ◽  
Author(s):  
J. P. Cleave
Keyword(s):  
Power Series ◽  
Space Curve ◽  
Arc Length ◽  
Space Curves ◽  
Normal Plane ◽  
Tangent Developable ◽  
Cuspidal Edge ◽  
Curvature And Torsion ◽  
Image Position ◽  
Classical Derivation

A tangent-developable is a surface generated by the tangent lines of a space curve. The intersection of a tangent-developable with the normal plane at a point P of the curve generally has a cusp at that point. Thus the tangent-developable of a space curve has a cuspidal edge along the curve. The classical derivation of this phenomenon takes the trihedron (t, n, b) at P as coordinate axes to which the curve is referred. Then the intersection of the part of the tangent-developable generated by tangent lines at points close to P with the normal plane at P (i.e. the plane through P containing n and b) is given parametrically by power serieswhere K, T are the curvature and torsion, respectively, of the curve at P and s is arc-length measured from P ((2) p. 68). It is tacitly understood in this analysis that curvature and torsion are both defined and non-zero.

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