Showing papers on "Point (geometry) published in 1968"

Multisurface method of pattern separation

Abstract: Let two sets of patterns be represented by two finite point sets in an n -dimensional Euclidean space E^{n} . If the convex hulls of the two sets do not intersect, the sets can be strictly separated by a plane. Such a plane can be constructed by the Motzkin-Schoenberg error-correction procedure or by linear programming. More often than not, however, the convex hulls of the two point sets do intersect, in which case strict separation by a plane is not possible any more. One may then resort to separation by more than one plane. In this paper, we show how two sets can be strictly separated by one or more planes or surfaces (nonlinear manifolds) via linear programming. A computer program that implements the present method has been written and successfully tested on a number of real problems.

Letter to the Editor—-A Closed Form Solution of Certain Programming Problems

Abstract: A formula is given for the coordinates of the point that maximizes a given function F(x1, …, xn) over the closure of a bounded domain S in n-dimensional Euclidean space. The principal assumption made in deriving the formula is that F attains a global maximum at exactly one point of S. In certain cases the formula may be used to discuss the maximization problem as a function of the parameters involved. Some simple examples are given.

Space filling curves and mathematical programming

Abstract: The problem of finding {if314-1} in n dimensional Euclidean space such that {if314-2}, i = 1, 2, ···, N , is considered. The only assumption on the f i is that a solution exists in the quantized unit hypercube. Implicitly exhaustive solution procedures, which obtain solutions by implicitly considering every point in the quantized space without making computations at each point, are studied. The implicitly exhaustive feature is made possible by adapting “space filling curves≓ to discrete spaces of general dimensionality. Several space filling curves are surveyed, and Peano's continuous mapping from the unit interval onto the unit square is used as a basis for defining a mapping from the unit quantized interval onto the unit quantized hypercube, and inversely. Ternary arithmetic is the basis for the required functional relationships in the discrete mapping. The discrete mapping has attributes of quasi-continuity, and specific numerical bounds are derived in this respect. It is shown that these bounds are of optimal order dependence on the relevant variables. It is shown how to use knowledge of first and second order properties of the f i to obtain solutions on a digital computer using space filling curves as a basis for the concept of implicitly exhaustive search. The only global properties assumed are bounds on first, or possibly second, order variations. Concluding remarks bear on the ultimate practicality of the method, and present a limited amount of experimental data.

Multiple imaging by means of point holograms.

Abstract: Multiple imaging devices are capable of producing simultaneously a number of images of one object. Several methods have been suggested to realize such an optical system. They make use of the imaging properties of a fly's-eye lens or of an array of pinholes, or they utilize birefringence. More recently, multiple image generation has been achieved by forming the convolution of the object with an array of δ functions in an ordinary spatial filtering arrangement. The complex filter is realized by the Fourier transform hologram of the cor­ responding array of point sources. The second Fourier trans­ form lens, however, has to transmit more than twice the full information of the multiplied image array. Since the most im­ portant application of multiple imaging is semiconductor device fabrication as has been discussed earlier, the resolving power has to be very high, so that up to now no lenses are available that are capable of imaging the multiple array of images of a usual etching mask. I t is for this reason that to date, images of etching masks are transferred to the semiconductor wafer by contact printing. In this communication, we propose to use an arrangement dif­ ferent from that in Ref. 4, such that the multiplied images do not have to be transmitted by any additional optical component. Par t of this communication has already been mentioned in a paper presented at a holography meeting organized by the Battelle Institute at Frankfurt /M, Germany, on 2 December 1967. The principle of operation is illustrated by Fig. 1. The hologram of the array of point sources is produced in the usual way by recording the interference pattern caused by coherent superposition of the spherical waves emanating from the ref­ erence point and the signal points the spatial positions of which correspond to the desired positions of the images to be formed. Multiple imaging is then performed in the reconstruction process by replacing the reference point source by the real image to be multiplied. This can be achieved in the well-known way by means of a lens that images the object through the hologram into the surrounding of the reference point. In this case, an additional real image of the object is formed in the surrounding of each signal point. The behavior of the hologram can be easily understood by thinking of a point hologram as of a generalized Fresnel zone plate with its well-known lenslike properties. Indeed, this hologram of the distribution of point sources may be considered as some special kind of a fly's-eye lens, in which the lenses are replaced by Fresnel zone plates. But there is a rather important difference. Contrary to lenses, the single point holograms may overlap completely without disturbing each other. For this reason, the aperture of each point hologram determining the diffraction limited size of the image point can be made equal to the aperture of the hologram of the whole point distribution provided that suitable spherical waves are used in the recording process. There is no relationship between the diameter of this aperture and the distances of the images to be formed. Limitations to the resolution of this multiple imaging device are mainly introduced by aberrations. In the special case of small plane objects, which are of dominant interest in semicon-

Dynamtc geometry of a twist drill

Abstract: SUMMARY The effect of the feed velocity on the cutting geometry of the main cutting edges of a twist drill is considered. Equations are derived for calculating the dynamic cutting geometry (i.e. the geometry measured relative to the cutting velocity) of the chisel edge given the static geometry, or the geometry produced by point grinding. It is shown that the dynamic geometry of the chisel edge is markedly different to the static geometry of this region.

Some practical aspects of curve fitting

Abstract: An attempt is made to extend the number of possible methods for solving curve fitting problems beyond the few very common methods, and to point out several pitfalls that commonly arise.

Metrically homogeneous sets

Abstract: A subset of a metric space is called metrically homogeneous if the set of distances from a chosen point of the subset to all the other points of the subset is independent of the chosen point. The main result of the paper is a complete characterization of the compact, metrically homogeneous subsets of the Euclidean plane.

Minimum Time Control by Time Interval Optimization

Abstract: This paper presents a time interval optimization method for the determination of the minimum time for transfer of a system from an initial state to a desired state. The method is applicable to systems whose characteristic matrix has real or complex eigenvalues. The characteristic matrix and the control matrix may havo constant or time varying parameters. The time interval optimization (TIO method) provides a means of determining a sequence of trajectories. The successively determined trajectories are the result of corrections so that the respective sequence of end points of the trajectories will approach the desired point in state space. Each of the trajectories is time optimal from the initial point to whatever point is reached. When the correction routine brings the end point of a trajectory sufficiently near the desired point in state space the optimal control has been determined.

Inspection and correction apparatus

Abstract: A MACHINE FOR DETECTING AND CORRECTING DIMENSIONAL IRREGULARITIES OF CURVED SURFACES SUCH AS PISTON RINGS, GEARS, BEARINGS AND THE LIKE. THE WORKPIECE BEING INSPECTED IS ROTATED WITHIN A SET OF JAWS AND A SENSOR IS POSITIONED ADJACENT THE WORKPIECE WHICH IS CAPABLE OF GENERATING AN OUTPUT SIGNAL IN RESPONSE TO AN OUT OF ROUND CONDITION. THIS SIGNAL RESULTS IN THE ACTUATION OF ONE OF THE JAW MEMBERS WHICH IS CAPABLE OF COMPRESSING THE WORKPIECE TO THEREBY ELIMINATE THE HIGH POINT.

Driving point function synthesis using uniform transmission lines

Abstract: It is shown that the two classes of driving point functions realized by Wyadrum and by O'Shea, using uniform transmission lines, are identical and further that these driving point functions may always be cascade synthesized. An alternative set of necessary and sufficient conditions for this class of driving point functions is given.

Circuit synthesis utilizing digital variable-precision-integrating and summing elements

Abstract: Integrated circuit digital building blocks for computer syntheses of electrical transfer and driving point functions