Showing papers on "Tangent published in 1983"

Local Control of Bias and Tension in Beta-splines

Abstract: The Beta-spline introduced recently by Barsky is a generalization of the uniform cubic B-spline: parametric discontinuities are introduced in such a way as to preserve continuity of the unit tangent and curvature vectors at joints (geometric continuity) while providing bias and tension parameters, independent of the position of control vertices, by which the shape of a curve or surface can be manipulated. Using a restricted form of quintic Hermite interpolation, it is possible to allow distinct bias and tension parameters at each joint without destroying geometric continuity. This provides a new means of obtaining local control of bias and tension in piecewise polynomial curves and surfaces.

Applications of Vector Fields to Image Processing

Abstract: We use rotational and curvature properties of vector fields to identify critical features of an image. Using vector analysis and dif-ferential geometry, we establish the properties needed, and then use these properties in three ways. First, our results make it theoretically possible to identify extremal edges of an intensity function f(x, y) of two variables by considering the gradient vector field V = ?f. There is also enough information in ?f to find regions of high curvature (i.e., high curvature of the level paths of f). For color images, we use the vector field V = (I, Q). In application, the image is partitioned into a grid of squares. On the boundary of each square, V/|V| is sampled, and these unit vectors are used as the tangents of a curve ?. The rotation number (or topological degree) ?(?) and the average curvature f|??| are computed for each square. Analysis of these numbers yields infor-mation on edges and curvature. Experimental results from both simu-lated and real data are described.

Tangent cones and quasi-interiorly tangent cones to multifunctions

Abstract: R. T. Rockaiellar has proved a number of rules of subdifferential calculus for nonlocally lipschitzian real-valued functions by investigating the Clarke tangent cones to the epigraphs of such functions. Following these lines we study in this paper the tangent cones to the sum and the composition of two multifunctions. This will be made possible thanks to the notion of quasi-interiorly tangent cone which has been introduced by the author for vector-valued functions in [29] and whose properties in the context of multifunctions are studied. The results are strong enough to cover the cases of real-valued or vector-valued functions. Introduction. Rockafellar has introduced in [23] the very important notion of directionally lipschitzian behaviour for extended real-valued functions, and with the aid of this notion he has proved in [24] a number of rules of subgradient calculus of nonconvex functions. If / is a function from a topological vector space E into R U {-oo, +00} with/(x) G R and if /(epi /; x, fix)) denotes the interiorly tangent cone to epi/={(x,^)G£XR:/(x) 0 such that X n epi /+ ]0, e[VC epi /, then the proof of Theorem 3 of [23, p. 268] shows that / is directionally lipschitzian at x if and only if /(epi /; x, fix)) i= 0. However, if g is a mapping from E into an ordered topological vector space H, then the interior of the cone of positive elements of H must be nonempty whenever /(epi g; x, g(x)) is nonempty. This very unsatisfactory state of affairs has led us to introduce in [29 and 30] the quasi-interiorly tangent cone ß(epi g; x, g(x)). With the help of this cone we have established in [29] rules of subdifferential calculus for nonconvex vector-valued functions. The aim of the present paper is to study the properties of Clarke tangent cones and quasi-interiorly tangent cones to the graphs of multifunctions following the way opened by Rockafellar. In §1 we recall Rockafellar's definition of Clarke tangent cone and we give an interpretation in terms of generalized sequences which proves that Rockafellar's definition is the same as the one we have given in [27]. Connection with strictly compactly lipschitzian vector-valued mappings is also made. Received by the editors March 19, 1982 and, in revised form, May 10, 1982. 1980 Mathematics Subject Classification. Primary 90C30, 90C48; Secondary 58C06.

Apparatus for tuning PID controllers in process control systems

Abstract: By applying to a linear, time-invariant process or system a pulse of predetermined duration instead of a step function and measuring the area enclosed by the pulse and the response curve of the process, respectively, instead of measuring the slope of a tangent line to the point of inflection of the response curve, the time constant or constants of a closed-loop (self-balancing) or open-loop (non-self-balancing) process may be accurately determined and from such time constant or constants the process may be optimally tuned according to one of the accepted tuning methods

The Theory of Straight Homogeneous Generalized Cylinders

Abstract: : In recent years, Binford's generalized cylinders have become an important tool for imagine understanding. However, research has been hampered by a lack of analytical results for these shapes. In this paper, a definition is presented for Straight Homogeneous Generalized Cylinders, those generalized cylinders, with a straight axis and with cross-sections which have constant shape but vary in size. This class of shapes, while still quite large, has properties which make considerable analysis possible. The results begin with deriving formulae for points and surface normals for these shapes. Theorems are presented concerning the conditions under which multiple descriptions can exist for a single solid shape. Then projections and contour generators are analyzed for some subclasses of shapes. The strongest results are obtained for solids of revolution (which the authors name Right Circular SHGCs), for which a closed-form method for analyzing image contours is presented. It is seen that a picture of the contours of a solid of revolution is ambiguous, with one degree of freedom related to the angle between the line of sight and the solid's axis. (Author)

Transient stability analysis of electric power systems by the method of tangent hypersurfaces

Abstract: A direct method is developed for investigating the transient stability of power systems, based on the approximation of the stability boundary by tangent hypersurfaces at certain unstable equilibrium points. This is a development of the tangent hyperplane method of Yee and Spalding. It is compared with the hyperplane method by application to three representative systems and is found to give improved results and to overcome some limitations. Other approaches to estimation of the stability region are also studied, which provide some indication of the shape of the separatrix and the transient behaviour of the state trajectories.

Vertical magnetic recording medium

Abstract: PURPOSE:To obtain a large and satisfactory output of reproduction by setting the coercive force of a soft magnetic layer at a level less than the prescribed value in the intra-face direction of said magnetic layer and also keeping the ratio of the maximum sloping value to the minimum sloping value at a level less than the prescribed value for a tangent line at an intersecting point between the increasing magnetic curve of a magnetization curve in the intra-face direction and a magnetic field axis. CONSTITUTION:A vertical magnetic recording medium has a double layer structure containing a soft magnetic layer of an Ni-Fe alloy thin film, etc. and a vertical magnetic anisotropic layer of a Co-Cr alloy thin film, etc. laminated on a nonmagnetic substrate. For such a recording medium, the coercive force of a soft magnetic layer is set at =2.5 times as much as the minimum sloping value for a tangent line at an intersecting point between a magnetic field axis and the increasing magnetic curve of a magnetization curve measured in each direction of said intra-face direction. In this case, MD, TD, Hce and Hch show the transfer direction of the substrate, the direction orthogonal to the transfer direction of the substrate, the coercive force in the direction of a magnetization facilitating axis and the coercive force in the direction of a magnetization unfacilitating axis respectively. Thus the anisotropy of a reproduction output is eliminated with a high reproduction output level for a vertical magnetic recording medium just by controlling the sloping of a tangent line. Such recording medium is suited to a disk recording medium such as a floppy disk, etc.

Method for determining consistent water relative permeability values from dynamic displacement data

Abstract: Corrected water relative permeability values for a core sample are determined from dynamic displacement measurements by the steps of plotting a ratio of water rate output (q w ) to calculated water relative permeability values (k rw ) vs. average water saturation (S w ), determining if a straight line segment exists in areas of the plot corresponding to high average oil saturations, and if it does, extrapolating the straight line segment to the end of the plotted data, otherwise plotting a tangent to the plotted ratio from a beginning point of the plot, producing a line in parallel with the extrapolated straight line segment or the tangent line but shifted so as to pass through the irreducible water saturation point S iw , and correcting the calculated water relative permeability values k rw by multiplying the calculated values k rw by a ratio of a value on the q w /k rw plot to a value on the constructed straight line. The applied corrections eliminate distortions in the calculated water relative permeability values k rw due to an end effect at the core sample which causes distortions in the dynamic displacement measurements.

Forming method of curved surface

Abstract: PURPOSE: To obtain a curved surface even with a three-dimensional reference curve by producing the surface of a three-dimensional curved surface body from plural intermediate section curved surfaces. CONSTITUTION: At the outset, a three-dimensional curve 31a is specified and fed, and at the same time the data on sections 11 and 12 as well as section curves 11a and 12a are fed. Then a dividing point Si is obtained to divide the length of the curve 31a into m:n, and at the same time a tangent line TLN of the curve 31a is obtained at the point Si. An intermediate section 41 is formed so that it is vertical to the line TLN. This process to form the section 41 is repeated while changing the dividing ratio m:n successively from 0:1 through 1:0. As a result, many intermediate section curves 41a are obtained. These curves are connected to each other to obtain a smooth curved surface. COPYRIGHT: (C)1985,JPO&Japio

Closed (C2- and C3-continuous) Bézier and B-spline curves with given tangent polygons

Abstract: This paper considers planar segmented Bezier curves of degree 3 (4) which have all the edges of a given convex polygon as tangents and the segments of which are C^2-(C^3-) continuous in their common joins. Also, the nodes of the B-spline curves which have the same shapes as these Bezier curves are constructed.

Numerical control processing system

Abstract: PURPOSE:To decide automatically a starting point and a direction of processing at the time of face processing by means of numerical control, by deciding a moving limit and the outermost tangent through a processing shape, a tool diameter and a reference processing diretion of a work. CONSTITUTION:In case of processing of the surface of a fixed shape through numerical control, a processing shape 100 of a work 10, a tool 12 diameter and reference processing direction X-Y are inputted. A moving limit shape 102 of the tool 12 is obtained by adding a half of the tool 12 diameter to the processing shape 100. Then, a tangent in a Y axis direction of the minimum value in a X axis direction of the work 10 obtained by making a reference direction into a Y axis direction is made into the outermost tangent 104. Then, one side point among points of intersection P1 and P2 between the moving limit shape 102 and the outermost tangent 104 of the tool is made into a starting point of processing. After the tool 12 has been moved along the outermost tangent 104 processing is kept on by shifting it by a fixed pitch in the X axis direction.

A theorem for a class of motions in a spherical geometry

Abstract: Considering the motion, at classical level, of a point particle constrained to move under the influence of a conservative centre of force, on a N-sphere S(N), embedded in a Euclidean (N+1)-dimensional space, the author has shown that, if the motion on S(N) is obtained from the motion on a tangent N-plane Pi (N) by means of a central projection, then the equations of motion of S(N) can be obtained from those on the Euclidean Pi (N) by a local reparameterisation of time. Some consequences of the theorem are also discussed.

Search for saddle points of energy hypersurfaces using a multi-dimensional space of “guiding” coordinates

Abstract: For the determination of saddle points (SPs) of adiabatic potential surfaces a novel method denning a “pseudo reaction path” (PRP) is presented. The PRP consists of two components, the one is being the function of some selected “guiding” coordinates and the other is depending on the remaining ones. The tangent components of the PRP are parallel and antiparallel to the normals of the tangential planes of the equipotential surfaces defined by the two groups of coordinates. PRPs starting from points in an appropriately chosen domain of the configurational space arrive at the SP.

Taper machining method

Abstract: PURPOSE:To perform the taper machining with high accuracy by a method wherein the wire electrode of an electrical discharge machine is tilted and the wire and a work piece are controlled so that the tilting angle of the wire electrode is rendered to continuously change when the wire electrode travels along a calculated passage comprising in smoothly connecting between the straight lines perpendicularly intersecting each other in the lower surface of the work piece with a plurality of circular arcs. CONSTITUTION:A straight line B2B2', which is parallel to the tangent touching a circular arc a at a point B in the upper surface of the work piece and passes through a point B2 in the lower guide corresponding to the point B, and a straight line C2C2', which is parallel to the tangent touching the circular arc a at a point C and passes through a point C2 in the lower guide corresponding to the point C, are obtained. An intersection J of the bisectors of the angles made by the straight lines B2B2' and B2C2 and by the straight lines C2C2' and B2C2 is calculated. Perpendiculars m1, m2 and m3, which are dropped from the intersection J, the point B2 and the point C2 to the straight lines B2C2, B2B2' and C2C2' respectively, are calculated in order to obtain the intersections P1 and P2 of the perpendiculars m1 and m2 and the perpendiculars m1 and m3. A desired curve e is obtained by drawing a circular arc e1 with center at the point P1 and a circular arc e2 with center at the point P2. After that, the lengths L1 and L2 of the circular arcs e1 and e2 are calculated. The relative movement of wire electrode to the work piece is controlled so that its locus in the upper surface of the work piece is the circular arc BI while its passage in the lower guide is the circular arc B2J.

Linear approximations of nonlinear systems

Abstract: A method for designing an automatic flight controller for short and vertical take off aircraft is presently being developed at NASA Ames Research Center. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system called the modified tangent model, was recently introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this an approximation (about a state space point x0) of an exact transformation for the nonlinear system can be found. Here we show that a canonical expansion in Lie brackets about the point x0 yields the same modified tangent model.

The reversal point in the perspective road picture

Abstract: Using perspective pictures of roads - drawn by the tangent method - the reversal point curve on road curves is defined. Some of its properties are discussed. The position of the reversal point curve is indicated and it is shown that the reversal points on the road edge and shoulder lines do not lie at the same distance below the horizon. An optical illusion could result, which is further influenced by the existence of walls in cuttings. The steep wall on the inside of curves tends to enforce the impression of adverse crossfall. The roundness or pointiness of the reversal curve is discussed as a clue to the curvature of the curve approached by the driver. The form of the reversal curve is not dependent on the total deflection of the road curve, but on the radius of the curve and the road width. Finally it is shown that the road picture is no different for a road on a sloping plane than it is for a road on a horizontal plane. Only the observation point changes. (Author/TRRL)

Probe for coordinate measuring machine

Abstract: The spherical shaped probe for a coordinate measuring machine is modified by incorporating a pair of rods mounted in recesses formed in the center of the sphere at a side and bottom to define a straight edge whose measurement is equal to the sphere's radius measured at the tangent to the vertical and horizontal diameters. Both rods being mounted at right angles relative to each other.

Magnetic field line merging: basic concepts

Abstract: The concept of magnetic field line merging is defined and is illustrated by reference to various possible structures of magnetic topology in the magnetosphere. A key element is the electric field tangent to the magnetic separatrix surfaces; its role and its relation to the MHD approximation can be viewed both in a global and a local context.

Design of mirrors with segmented conical surfaces tangent to a discontinuous aspheric base

Abstract: A unique mirror surface profile has been developed to produce annular, radial, or circular images of uniform intensity from incident radiation of non-uniform illumination. The design principles are ideally suited for industrial high power laser applications as well as for concentrator/integrator mirrors in the Abbe type condenser systems. The reflective surface is a series of segmented conical flats tangent to a discontinuous base aspheric. Each conical segment of specified width redirects the input light to a common image. The net contribution from all of the segments is a highly uniform intensity integrated image. The design procedure is represented in three stages, (1) optimizing the curvature and aspheric coefficients of the base curve; (2) calculating the point of tangency and width of each conical segment; (3) shifting the coordinate system to obtain the required annular or radial image dimensions. To facilitate the design, a semi-automatic computer program was written since commercially available optical design software does not handle discontinuous surfaces. Three examples of mirror designs will be presented.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Adjustable radial curve

Abstract: An adjustable radial curve is provided having an end tangent bar to which there is attached an assemblage of stacked elements. The stacked elements are fastened together by pin assemblies so that they move independently of each other. The stacked elements are attached to the tangent bar and are adjustable relative thereto. A ruling edge is attached to the stacked elements for creating a curve upon manipulation of the stacked elements. A barscale is attached to the tangent bar and to the stacked elements so that the user of the instrument may set or determine the exact radius of a curve created by the instrument. The barscale may be removable by the provision of a locking device which would function to lock the ruling edge to the ruling arm assembly. A base having at least one guideway is provided for the radial curve whereby radii of particular dimensions may be constructed. A transparent cover is provided for the base.

Multistage rolling mill and its rolling method

Abstract: PURPOSE:To prevent a sheet to be rolled from generating camber by adjusting an angle of pass line to an optional angle, and to obtain a well finished product, by providing an adjusting device of pass line either to the outlet side or to the inlet one of an offset multistage rolling mill. CONSTITUTION:An angle formed between a tangent line touching at a contact point P and a horizontal line passing through the point P is previously calculated basing on the offset quantity of a small diametral work roll 2' from a large diametral work roll 2. Next, a piston rod of a cylinder 10 of a pass-line adjusting device 12 is stretched out to push a guide roll 8. And then, an angle between the horizontal line and a sheet to be rolled, i.e. pass line angle theta is made to approximately coincide with the angle between the tangent line and the horizontal line. When a sheet 6 is rolled in this way, rolling forces acting on both surfaces of the sheet 6 are equalized by the upper and lower work rolls 2, 2' made to have different diameters respectively and made to offset, therefore the sheet 6 is rolled without generating camber and lubrications at the upper and lower surface are equalized.