Establishing enveloping features for engineering surfaces
22 Sep 1993-Vol. 2101, pp 966-977
TL;DR: In this paper, a non-linear optimization technique is used to solve simple numerical examples to compare the values obtained by the least squares and minimum deviation methods, and the results show that the least square method of establishing the ideal feature does not yield a minimum value.
Abstract: Form error is specified on the basis of an ideal geometric feature established from the actual measurements such that the maximum deviation from the ideal feature is the least possible value. The least squares method of establishing the ideal feature does not yield a minimum value. A few attempts have been made to establish the feature with a minimum value of error. In case of roundness evaluation modern instruments specify the error based on minimum deviation and ring-and plug-gauge circles. However the instrument manufacturers do not reveal the algorithms used. This paper deals with the methods of establishing the enveloping features for different geometric features namely straight line circle plane and cylinder. A non-linear optimization technique is followed to solve simple numerical examples. The values are compared with the values obtained by the least squares and minimum deviation methods. NOMENCLATURE ECF Enveloping crest feature EVF Enveloping valley feature e. Deviation of a point from assessment feature f Function to be minimized K Number of points in a section L Number of sections LSF Least squares feature 1 m Slope values (l'' for transformed values) o o 0 0 MDF Minimum deviation feature N Total number of points r. O. Polar coordinates of a point R. Radius of a circle/cylinder/ x. y. z. Cartesian coordinates of a point(x 1 1 1 1 1 1 for transformed values) x y z Estimated coordinates
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01 Apr 1998-Precision Engineering-journal of The International Societies for Precision Engineering and Nanotechnology
TL;DR: In this paper, a new algorithm is proposed based on a sound mathematical background to obtain the functional boundaries that are best represented by enveloping features, which can be used to control parameters of the production process and evaluate the workpiece based on functional requirements.
Abstract: Workpieces make contact on their functional boundaries, not on the best fit surfaces. Better quality products can be made by controlling parameters of the production process and evaluating the workpiece based on functional requirements. A new algorithm is proposed in this paper, based on a sound mathematical background, to obtain the functional boundaries that are best represented by enveloping features.
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Book•
01 Jan 1983
TL;DR: Theoretical Equivalence of Mayer, Lagrange, and Bolza Problems of Optimal Control, and the Necessary Conditions and Sufficient Conditions Convexity and Lower Semicontinuity.
Abstract: 1 Problems of Optimization-A General View.- 1.1 Classical Lagrange Problems of the Calculus of Variations.- 1.2 Classical Lagrange Problems with Constraints on the Derivatives.- 1.3 Classical Bolza Problems of the Calculus of Variations.- 1.4 Classical Problems Depending on Derivatives of Higher Order.- 1.5 Examples of Classical Problems of the Calculus of Variations.- 1.6 Remarks.- 1.7 The Mayer Problems of Optimal Control.- 1.8 Lagrange and Bolza Problems of Optimal Control.- 1.9 Theoretical Equivalence of Mayer, Lagrange, and Bolza Problems of Optimal Control. Problems of the Calculus of Variations as Problems of Optimal Control.- 1.10 Examples of Problems of Optimal Control.- 1.11 Exercises.- 1.12 The Mayer Problems in Terms of Orientor Fields.- 1.13 The Lagrange Problems of Control as Problems of the Calculus of Variations with Constraints on the Derivatives.- 1.14 Generalized Solutions.- Bibliographical Notes.- 2 The Classical Problems of the Calculus of Variations: Necessary Conditions and Sufficient Conditions Convexity and Lower Semicontinuity.- 2.1 Minima and Maxima for Lagrange Problems of the Calculus of Variations.- 2.2 Statement of Necessary Conditions.- 2.3 Necessary Conditions in Terms of Gateau Derivatives.- 2.4 Proofs of the Necessary Conditions and of Their Invariant Character.- 2.5 Jacobi's Necessary Condition.- 2.6 Smoothness Properties of Optimal Solutions.- 2.7 Proof of the Euler and DuBois-Reymond Conditions in the Unbounded Case.- 2.8 Proof of the Transversality Relations.- 2.9 The String Property and a Form of Jacobi's Necessary Condition.- 2.10 An Elementary Proof of Weierstrass's Necessary Condition.- 2.11 Classical Fields and Weierstrass's Sufficient Conditions.- 2.12 More Sufficient Conditions.- 2.13 Value Function and Further Sufficient Conditions.- 2.14 Uniform Convergence and Other Modes of Convergence.- 2.15 Semicontinuity of Functionals.- 2.16 Remarks on Convex Sets and Convex Real Valued Functions.- 2.17 A Lemma Concerning Convex Integrands.- 2.18 Convexity and Lower Semicontinuity: A Necessary and Sufficient Condition.- 2.19 Convexity as a Necessary Condition for Lower Semicontinuity.- 2.20 Statement of an Existence Theorem for Lagrange Problems of the Calculus of Variations.- Bibliographical Notes.- 3 Examples and Exercises on Classical Problems.- 3.1 An Introductory Example.- 3.2 Geodesics.- 3.3 Exercises.- 3.4 Fermat's Principle.- 3.5 The Ramsay Model of Economic Growth.- 3.6 Two Isoperimetric Problems.- 3.7 More Examples of Classical Problems.- 3.8 Miscellaneous Exercises.- 3.9 The Integral I = ?(x?2 ? x2)dt.- 3.10 The Integral I = ?xx?2dt.- 3.11 The Integral I = ?x?2(1 + x?)2dt.- 3.12 Brachistochrone, or Path of Quickest Descent.- 3.13 Surface of Revolution of Minimum Area.- 3.14 The Principles of Mechanics.- Bibliographical Notes.- 4 Statement of the Necessary Condition for Mayer Problems of Optimal Control.- 4.1 Some General Assumptions.- 4.2 The Necessary Condition for Mayer Problems of Optimal Control.- 4.3 Statement of an Existence Theorem for Mayer's Problems of Optimal Control.- 4.4 Examples of Transversality Relations for Mayer Problems.- 4.5 The Value Function.- 4.6 Sufficient Conditions.- 4.7 Appendix: Derivation of Some of the Classical Necessary Conditions of Section 2.1 from the Necessary Condition for Mayer Problems of Optimal Control.- 4.8 Appendix: Derivation of the Classical Necessary Condition for Isoperimetric Problems from the Necessary Condition for Mayer Problems of Optimal Control.- 4.9 Appendix: Derivation of the Classical Necessary Condition for Lagrange Problems of the Calculus of Variations with Differential Equations as Constraints.- Bibliographical Notes.- 5 Lagrange and Bolza Problems of Optimal Control and Other Problems.- 5.1 The Necessary Condition for Bolza and Lagrange Problems of Optimal Control.- 5.2 Derivation of Properties (P1?)-(P4?) from (P1)-(P4).- 5.3 Examples of Applications of the Necessary Conditions for Lagrange Problems of Optimal Control.- 5.4 The Value Function.- 5.5 Sufficient Conditions for the Bolza Problem.- Bibliographical Notes.- 6 Examples and Exercises on Optimal Control.- 6.1 Stabilization of a Material Point Moving on a Straight Line under a Limited External Force.- 6.2 Stabilization of a Material Point under an Elastic Force and a Limited External Force.- 6.3 Minimum Time Stabilization of a Reentry Vehicle.- 6.4 Soft Landing on the Moon.- 6.5 Three More Problems on the Stabilization of a Point Moving on a Straight Line.- 6.6 Exercises.- 6.7 Optimal Economic Growth.- 6.8 Two More Classical Problems.- 6.9 The Navigation Problem.- Bibliographical Notes.- 7 Proofs of the Necessary Condition for Control Problems and Related Topics.- 7.1 Description of the Problem of Optimization.- 7.2 Sketch of the Proofs.- 7.3 The First Proof.- 7.4 Second Proof of the Necessary Condition.- 7.5 Proof of Boltyanskii's Statements (4.6.iv-v).- Bibliographical Notes.- 8 The Implicit Function Theorem and the Elementary Closure Theorem.- 8.1 Remarks on Semicontinuous Functionals.- 8.2 The Implicit Function Theorem.- 8.3 Selection Theorems.- 8.4 Convexity, Caratheodory's Theorem, Extreme Points.- 8.5 Upper Semicontinuity Properties of Set Valued Functions.- 8.6 The Elementary Closure Theorem.- 8.7 Some Fatou-Like Lemmas.- 8.8 Lower Closure Theorems with Respect to Uniform Convergence.- Bibliographical Notes.- 9 Existence Theorems: The Bounded, or Elementary, Case.- 9.1 Ascoli's Theorem.- 9.2 Filippov's Existence Theorem for Mayer Problems of Optimal Control.- 9.3 Filippov's Existence Theorem for Lagrange and Bolza Problems of Optimal Control.- 9.4 Elimination of the Hypothesis that A Is Compact in Filippov's Theorem for Mayer Problems.- 9.5 Elimination of the Hypothesis that A Is Compact in Filippov's Theorem for Lagrange and Bolza Problems.- 9.6 Examples.- Bibliographical Notes.- 10 Closure and Lower Closure Theorems under Weak Convergence.- 10.1 The Banach-Saks-Mazur Theorem.- 10.2 Absolute Integrability and Related Concepts.- 10.3 An Equivalence Theorem.- 10.4 A Few Remarks on Growth Conditions.- 10.5 The Growth Property (?) Implies Property (Q).- 10.6 Closure Theorems for Orientor Fields Based on Weak Convergence.- 10.7 Lower Closure Theorems for Orientor Fields Based on Weak Convergence.- 10.8 Lower Semicontinuity in the Topology of Weak Convergence.- 10.9 Necessary and Sufficient Conditions for Lower Closure.- Bibliographical Notes.- 11 Existence Theorems: Weak Convergence and Growth Conditions.- 11.1 Existence Theorems for Orientor Fields and Extended Problems.- 112 Elimination of the Hypothesis that A Is Bounded in Theorems (11.1. i-iv).- 11.3 Examples.- 11.4 Existence Theorems for Problems of Optimal Control with Unbounded Strategies.- 11.5 Elimination of the Hypothesis that A Is Bounded in Theorems (11.4.i-v).- 11.6 Examples.- 11.7 Counterexamples.- Bibliographical Notes.- 12 Existence Theorems: The Case of an Exceptional Set of No Growth.- 12.1 The Case of No Growth at the Points of a Slender Set. Lower Closure Theorems..- 12.2 Existence Theorems for Extended Free Problems with an Exceptional Slender Set.- 12.3 Existence Theorems for Problems of Optimal Control with an Exceptional Slender Set.- 12.4 Examples.- 12.5 Counterexamples.- Bibliographical Notes.- 13 Existence Theorems: The Use of Lipschitz and Tempered Growth Conditions.- 13.1 An Existence Theorem under Condition (D).- 13.2 Conditions of the F, G, and H Types Each Implying Property (D) and Weak Property (Q).- 13.3 Examples.- Bibliographical Notes.- 14 Existence Theorems: Problems of Slow Growth.- 14.1 Parametric Curves and Integrals.- 14.2 Transformation of Nonparametric into Parametric Integrals.- 14.3 Existence Theorems for (Nonparametric) Problems of Slow Growth.- 14.4 Examples.- Bibliographical Notes.- 15 Existence Theorems: The Use of Mere Pointwise Convergence on the Trajectories.- 15.1 The Helly Theorem.- 15.2 Closure Theorems with Components Converging Only Pointwise.- 15.3 Existence Theorems for Extended Problems Based on Pointwise Convergence.- 15.4 Existence Theorems for Problems of Optimal Control Based on Pointwise Convergence.- 15.5 Exercises.- Bibliographical Notes.- 16 Existence Theorems: Problems with No Convexity Assumptions.- 16.1 Lyapunov Type Theorems.- 16.2 The Neustadt Theorem for Mayer Problems with Bounded Controls.- 16.3 The Bang-Bang Theorem.- 16.4 The Neustadt Theorem for Lagrange and Bolza Problems with Bounded Controls.- 16.5 The Case of Unbounded Controls.- 16.6 Examples for the Unbounded Case.- 16.7 Problems of the Calculus of Variations without Convexity Assumptions.- Bibliographical Notes.- 17 Duality and Upper Semicontinuity of Set Valued Functions.- 17.1 Convex Functions on a Set.- 17.2 The Function T(x z).- 17.3 Seminormality.- 17.4 Criteria for Property (Q).- 17.5 A Characterization of Property (Q) for the Sets $$\tilde Q$$(t, x) in Terms of Seminormality.- 17.6 Duality and Another Characterization of Property (Q) in Terms of Duality.- 17.7 Characterization of Optimal Solutions in Terms of Duality.- 17.8 Property (Q) as an Extension of Maximal Monotonicity.- Bibliographical Notes.- 18 Approximation of Usual and of Generalized Solutions.- 18.1 The Gronwall Lemma.- 18.2 Approximation of AC Solutions by Means of C1 Solutions.- 18.3 The Brouwer Fixed Point Theorem.- 18.4 Further Results Concerning the Approximation of AC Trajectories by Means of C1 Trajectories.- 18.5 The Infimum for AC Solutions Can Be Lower than the One for C1 Solutions.- 18.6 Approximation of Generalized Solutions by Means of Usual Solutions.- 18.7 The Infimum for Generalized Solutions Can Be Lower than the One for Usual Solutions.- Bibliographical Notes.- Author Index.
2,371 citations
TL;DR: A technique for empirical optimisation is presented in which a sequence of experimental designs each in the form of a regular or irregular simplex is used, each simplex having all vertices but one in common with the preceding simplex, and being completed by one new point.
Abstract: A technique for empirical optimisation is presented in which a sequence of experimental designs each in the form of a regular or irregular simplex is used, each simplex having all vertices but one in common with the preceding simplex, and being completed by one new point. Reasons for the choice of design are outlined, and a formal procedure given. The performance of the technique in the presence and absence of error is studied and it is shown (a) that in the presence of error the rate of advance is inversely proportional to the error standard deviation, so that replication of observations is not beneficial, and (b) that the “efficiency” of the technique appears to increase in direct proportion to the number of factors investigated. It is also noted that, since the direction of movement from each simplex is dependent solely on the ranking of the observations, the technique may be used even in circumstances when a response cannot be quantitatively assessed. Attention is drawn to the ease with which second-o...
1,303 citations
TL;DR: This book discusses the construction of mental ray, a model for synthetic lighting, and some of the techniques used to design and implement such models.
Abstract: Introduction. Chapter 1: Introduction to mental ray. What Is mental ray? Why Use mental ray? The Structure of mental ray. mental ray Integration. Command-Line Rendering and the Stand-Alone Renderer. mental ray Shaders and Shader Libraries. Indirect Illumination. Chapter 2: Rendering Algorithms. Introduction to Synthetic Lighting. Rendering under the Hood. mental ray Rendering Algorithms. Scanline Rendering in Depth. Raytrace Rendering in Depth. Hardware Rendering. Chapter 3: mental ray Output. mental ray Data Types. The Frame Buffer. Frame Buffer Options. mental ray Cameras. Output Statements. Chapter 4: Camera Fundamentals. Camera Basics and Aspect Ratios. Camera Lenses. Host Application Settings. Chapter 5: Quality Control. Sampling and Filtering in Host Applications. Raytrace Acceleration. Diagnostic and BSP Fine-Tuning. Chapter 6: Lights and Soft Shadows. mental ray Lights. Area Lights. Host Application Settings. Light Profiles. Chapter 7: Shadow Algorithms. Shadow Algorithms. Raytrace Shadows. Depth-Based Shadows. Stand-Alone and Host Settings. Chapter 8: Motion Blur. mental ray Motion Blur. Motion-Blur Options. Motion-Blur Render Algorithms. Host Settings. Chapter 9: The Fundamentals of Light and Shading Models. The Fundamentals of Light. Light Transport and Shading Models. mental ray Shaders. Chapter 10: mental ray Shaders and Shader Trees. Installing Custom Shaders. DGS and Dielectric Shading Models. Glossy Reflection and Refraction Shaders. Brushed Metals with the Glossy and Anisotropic Shaders. The Architectural (mia) Material. Chapter 11: mental ray Textures and Projections. Texture Space and Projections. mental ray Bump Mapping. mental ray Projection and Remapping Shaders. Host Application Settings. Memory Mapping, Pyramid Images, and Image Filtering. Chapter 12: Indirect Illumination. mental ray Indirect Illumination. Photon Shaders and Photon-Casting Lights. Indirect Illumination Options and Fine-Tuning. Participating Media (PM) Effects. Chapter 13: Final Gather and Ambient Occlusion. Final Gather Fundamentals. Final Gather Options and Techniques. Advanced Final Gather Techniques. Ambient Occlusion. Chapter 14: Subsurface Scattering. Advanced Shading Models. Nonphysical Subsurface Scattering. An Advanced Shader Tree. Physical Subsurface Scattering. Appendix: About the Companion CD. Index.
1,022 citations
TL;DR: In this paper, Monte Carlo, simplex and spiral search techniques were found suitable for minimum zone evaluation of spherical, cylindrical and flat surfaces for sphericity, circularity, flatness etc.
216 citations
TL;DR: In this article, the authors discuss different methods that are useful for assessing the errors on the dimensions, form and position of geometric features, and a new approach called the median technique, which gives minimum values of errors is introduced.
Abstract: SUMMARY Measurements are carried out on engineering components and analysed to check the conformity of the components to specification. This paper discusses different methods that are useful for assessing the errors on the dimensions, form and position of geometric features. A new approach called the median technique, which gives minimum values of errors is introduced.
117 citations