The random tangential projection of a surface

1980 ◽  
Vol 12 (02) ◽  
pp. 425-446 ◽  
Author(s):  
R. E. Miles
Keyword(s):  
Cylindrical Surface ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Image Plane ◽  
Parallel Line ◽  
Tangential Projection ◽  
Light Beams ◽  
Boundary Curves ◽  
Isotropic Random

Suppose a smooth surface S is exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency with S, there is a corresponding point registration in an image plane H placed perpendicular to the beam. The registering rays form a cylindrical surface which registers in H as a smooth curve C (except possibly for a number of cusps). Properties of C when the beam is taken isotropic random in space are investigated, and stereological applications are noted. The phenomenon of screening of projecting rays, which occurs for example with opaque surfaces exposed to light beams, is considered. Limiting processes permit extensions of the formulae to the cases in which S contains curved edges and in which any boundary curves of S also register. Finally, various related types of random tangent to a surface are considered.

The random tangential projection of a surface

1980 ◽  
Vol 12 (2) ◽  
pp. 425-446 ◽  
Author(s):  
R. E. Miles
Keyword(s):  
Cylindrical Surface ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Image Plane ◽  
Parallel Line ◽  
Tangential Projection ◽  
Light Beams ◽  
Boundary Curves ◽  
Isotropic Random

Suppose a smooth surface S is exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency with S, there is a corresponding point registration in an image plane H placed perpendicular to the beam. The registering rays form a cylindrical surface which registers in H as a smooth curve C (except possibly for a number of cusps). Properties of C when the beam is taken isotropic random in space are investigated, and stereological applications are noted. The phenomenon of screening of projecting rays, which occurs for example with opaque surfaces exposed to light beams, is considered. Limiting processes permit extensions of the formulae to the cases in which S contains curved edges and in which any boundary curves of S also register. Finally, various related types of random tangent to a surface are considered.

ESTIMATING SURFACE NORMALS IN NOISY POINT CLOUD DATA

2004 ◽  
Vol 14 (04n05) ◽  
pp. 261-276 ◽  
Author(s):  
NILOY J. MITRA ◽  
AN NGUYEN ◽  
LEONIDAS GUIBAS
Keyword(s):  
Point Cloud ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Local Information ◽  
Least Square ◽  
Neighborhood Size ◽  
Point Cloud Data ◽  
Cloud Data ◽  
Normal Estimation ◽  
Least Square Fitting

In this paper we describe and analyze a method based on local least square fitting for estimating the normals at all sample points of a point cloud data (PCD) set, in the presence of noise. We study the effects of neighborhood size, curvature, sampling density, and noise on the normal estimation when the PCD is sampled from a smooth curve in ℝ2or a smooth surface in ℝ3, and noise is added. The analysis allows us to find the optimal neighborhood size using other local information from the PCD. Experimental results are also provided.

scholarly journals Complete intersections primitive structures on space curves

2015 ◽  
Vol 26 (12) ◽  
pp. 1550104
Author(s):  
Philippe Ellia
Keyword(s):  
Complete Intersection ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Complete Intersections ◽  
Multiple Structure ◽  
Space Curves ◽  
Structure Formula ◽  
Primitive Structure

A multiple structure [Formula: see text] on a smooth curve [Formula: see text] is said to be primitive if [Formula: see text] is locally contained in a smooth surface. We give some numerical conditions for a curve [Formula: see text] to be a primitive set theoretical complete intersection (i.e. to have a primitive structure which is a complete intersection).

scholarly journals CALCULATION OF A GLARE STOP FOR TWO-MIRROR EXTRA-FOCAL OBJECTIVE

2017 ◽  
Vol 8 (4) ◽  
pp. 327-333
Author(s):  
L. F. Zambrano ◽  
N. K. Artyukhina ◽  
L. V. Peroza
Keyword(s):  
Image Quality ◽  
Image Plane ◽  
Stray Light ◽  
Geometric Constructions ◽  
Calculation Algorithm ◽  
Second Stage ◽  
Mathematical Expressions ◽  
Intermediate Image ◽  
Two Stages ◽  
Light Beams

Recently, efforts to improve optical characteristics in canonical mirror systems, including aspherical surfaces and corrective aberration capabilities. At the same time, much attention is paid to the development of new optical schemes of two-mirror objectives. Development measures to protect the image plane from stray light and harmful flows with minimal vignetting and screening is one of the most perspective ways for improving the image quality objectives. The only method to eliminate or even reduce these non-constructive rays is to set glare stops. The aim of the work was an improving method for constructing a glare stop to protect the image plane and the creation of a calculation algorithm of glare stop for protecting the image plane based on two-mirror extra-focal objectives.The study was conducted in two stages. In the course of the first stage, the positions of screening and intermediate image plane were obtained, as well as the central screening coefficient. At the second stage, an arrangement for the position of glare stop is proposed using the algorithm calculation. Thus, mathematical expressions were achieved by geometric constructions. The relation of the screening coefficient with the distance between the surfaces of the mirrors and the height of the paraxial rays is established. А representation of vignetting diagram for two-mirror extra-focal objective with D/f´ = 1 : 1,3 and 2ω = 4° was realized. TheQestimation of vignetting of inclined light beams is k= 0,56.

scholarly journals Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface

2017 ◽  
Vol 28 (13) ◽  
pp. 1750099 ◽  
Author(s):  
Hirokazu Nasu
Keyword(s):  
Elliptic Curves ◽  
Hilbert Scheme ◽  
Smooth Surface ◽  
Smooth Curve ◽  
K3 Surface ◽  
Sufficient Condition ◽  
Infinitesimal Deformation ◽  
First Order ◽  
Irreducible Components

We study the deformations of a smooth curve [Formula: see text] on a smooth projective [Formula: see text]-fold [Formula: see text], assuming the presence of a smooth surface [Formula: see text] satisfying [Formula: see text]. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of [Formula: see text] in [Formula: see text] to be primarily obstructed. In particular, when [Formula: see text] is Fano and [Formula: see text] is [Formula: see text], we give a sufficient condition for [Formula: see text] to be (un)obstructed in [Formula: see text], in terms of [Formula: see text]-curves and elliptic curves on [Formula: see text]. Applying this result, we prove that the Hilbert scheme [Formula: see text] of smooth connected curves on a smooth quartic [Formula: see text]-fold [Formula: see text] contains infinitely many generically non-reduced irreducible components, which are variations of Mumford’s example for [Formula: see text].

scholarly journals Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures

2016 ◽  
Vol 27 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Ziv Ran
Keyword(s):  
Poisson Structure ◽  
Hilbert Scheme ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Smooth Surfaces ◽  
Poisson Structures ◽  
Hilbert Schemes ◽  
Hilbert Scheme Of Points

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.

Shape equations for two-dimensional manifolds with nonempty boundary based on a variational method

2020 ◽  
Vol 17 (06) ◽  
pp. 2050082
Author(s):  
Paul Bracken
Keyword(s):  
Free Energy ◽  
Smooth Surface ◽  
Differential Forms ◽  
Energy Functional ◽  
Total Free Energy ◽  
Moving Frame ◽  
Two Dimensional ◽  
Curved Boundary ◽  
Free Energy Functional ◽  
Boundary Curves

A smooth surface is considered which has a curved boundary. A system of exterior differential forms is introduced which describes the surface and boundary curves completely in the moving frame approach. A total free energy functional is defined based on these forms for which an equilibrium equation and boundary conditions of the surface are derived by calculating the variation of the total free energy. These results can be applied to a surface with several freely exposed edges.

The Polishing Process of Cylindrical Rollers by Using a Double-Side Lapping Machine

2013 ◽  
Vol 589-590 ◽  
pp. 447-450 ◽  
Author(s):  
Wen Hua Zhou ◽  
Wei Feng Yao ◽  
Ming Feng ◽  
Bing Hai Lv ◽  
Qian Fa Deng
Keyword(s):  
Surface Quality ◽  
Cylindrical Surface ◽  
Smooth Surface ◽  
Great Influence ◽  
Working Life ◽  
Geometric Accuracy ◽  
Polishing Process ◽  
Rolling Bearings ◽  
Lapping Machine

The surface quality of cylindrical rollers has a great influence on the working life of rolling bearings. In the paper, double-side lapping machine and flannelette pads are applied to polish the cylindrical surface of rollers. After 20 min polishing, rollers with smooth surface are obtained and the roundness of the rollers have a improvement of about 0.3 μm, which means that the polishing process based on double-side lapping method can effectively improve the surface quality and geometric accuracy of cylindrical rollers.

Non-Abrasive Cryogenic Polishing Device on Cylindrical Surface Based on Recurdyn Dynamic Simulation

2011 ◽  
Vol 418-420 ◽  
pp. 861-864
Author(s):  
Tian Zhu Liang ◽  
Shou Ping Qu
Keyword(s):  
Machine Tool ◽  
Dynamic Simulation ◽  
Rotational Speed ◽  
Cylindrical Surface ◽  
Smooth Surface ◽  
3D Model ◽  
Precision Machine ◽  
Machine Tool Spindle ◽  
Aero Engine ◽  
Ultra Precision

With the increasing rotational speed of the aero-engine shaft and ultra-precision machine tool spindles, there has been the machine tool spindle with ten thousand Rpm. It demands that the spindles should have high rotational accuracy and ultra-smooth surface. In this paper, it takes use of the new ice tool polishing technology to polish metal cylindrical surfaces. The 3D model has been designed. And also the model has been analyzed by Recurdyn software. It verified that the experiment is feasible and effective.

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