Differential Geometry, Kreyszig 10. Tangent and Normal Plane. Let C Be an Arbitrary Curve in R 3 and X(S) Be a Parametric Repres
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
Surface Regularized Geometry Estimation from a Single Image
SURGE: Surface Regularized Geometry Estimation from a Single Image Peng Wang1 Xiaohui Shen2 Bryan Russell2 Scott Cohen2 Brian Price2 Alan Yuille3 1University of California, Los Angeles 2Adobe Research 3Johns Hopkins University Abstract This paper introduces an approach to regularize 2.5D surface normal and depth predictions at each pixel given a single input image. The approach infers and reasons about the underlying 3D planar surfaces depicted in the image to snap predicted normals and depths to inferred planar surfaces, all while maintaining fine detail within objects. Our approach comprises two components: (i) a four- stream convolutional neural network (CNN) where depths, surface normals, and likelihoods of planar region and planar boundary are predicted at each pixel, followed by (ii) a dense conditional random field (DCRF) that integrates the four predictions such that the normals and depths are compatible with each other and regularized by the planar region and planar boundary information. The DCRF is formulated such that gradients can be passed to the surface normal and depth CNNs via backpropagation. In addition, we propose new planar-wise metrics to evaluate geometry consistency within planar surfaces, which are more tightly related to dependent 3D editing applications. We show that our regularization yields a 30% relative improvement in planar consistency on the NYU v2 dataset [24]. 1 Introduction Recent efforts to estimate the 2.5D layout of a depicted scene from a single image, such as per-pixel depths and surface normals, have yielded high-quality outputs respecting both the global scene layout and fine object detail [2, 6, 7, 29]. Upon closer inspection, however, the predicted depths and normals may fail to be consistent with the underlying surface geometry. -
CS 468 (Spring 2013) — Discrete Differential Geometry 1 the Unit Normal Vector of a Surface. 2 Surface Area
CS 468 (Spring 2013) | Discrete Differential Geometry Lecture 7 Student Notes: The Second Fundamental Form Lecturer: Adrian Butscher; Scribe: Soohark Chung 1 The unit normal vector of a surface. Figure 1: Normal vector of level set. • The normal line is a geometric feature. The normal direction is not (think of a non-orientable surfaces such as a mobius strip). If possible, you want to pick normal directions that are consistent globally. For example, for a manifold, you can pick normal directions pointing out. • Locally, you can find a normal direction using tangent vectors (though you can't extend this globally). • Normal vector of a parametrized surface: E1 × E2 If TpS = spanfE1;E2g then N := kE1 × E2k This is just the cross product of two tangent vectors normalized by it's length. Because you take the cross product of two vectors, it is orthogonal to the tangent plane. You can see that your choice of tangent vectors and their order determines the direction of the normal vector. • Normal vector of a level set: > [DFp] N := ? TpS kDFpk This is because the gradient at any point is perpendicular to the level set. This makes sense intuitively if you remember that the gradient is the "direction of the greatest increase" and that the value of the level set function stays constant along the surface. Of course, you also have to normalize it to be unit length. 2 Surface Area. • We want to be able to take the integral of the surface. One approach to the problem may be to inegrate a parametrized surface in the parameter domain. -
Curl, Divergence and Laplacian
Curl, Divergence and Laplacian What to know: 1. The definition of curl and it two properties, that is, theorem 1, and be able to predict qualitatively how the curl of a vector field behaves from a picture. 2. The definition of divergence and it two properties, that is, if div F~ 6= 0 then F~ can't be written as the curl of another field, and be able to tell a vector field of clearly nonzero,positive or negative divergence from the picture. 3. Know the definition of the Laplace operator 4. Know what kind of objects those operator take as input and what they give as output. The curl operator Let's look at two plots of vector fields: Figure 1: The vector field Figure 2: The vector field h−y; x; 0i: h1; 1; 0i We can observe that the second one looks like it is rotating around the z axis. We'd like to be able to predict this kind of behavior without having to look at a picture. We also promised to find a criterion that checks whether a vector field is conservative in R3. Both of those goals are accomplished using a tool called the curl operator, even though neither of those two properties is exactly obvious from the definition we'll give. Definition 1. Let F~ = hP; Q; Ri be a vector field in R3, where P , Q and R are continuously differentiable. We define the curl operator: @R @Q @P @R @Q @P curl F~ = − ~i + − ~j + − ~k: (1) @y @z @z @x @x @y Remarks: 1. -
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions. -
AN INTRODUCTION to the CURVATURE of SURFACES by PHILIP ANTHONY BARILE a Thesis Submitted to the Graduate School-Camden Rutgers
AN INTRODUCTION TO THE CURVATURE OF SURFACES By PHILIP ANTHONY BARILE A thesis submitted to the Graduate School-Camden Rutgers, The State University Of New Jersey in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Mathematics written under the direction of Haydee Herrera and approved by Camden, NJ January 2009 ABSTRACT OF THE THESIS An Introduction to the Curvature of Surfaces by PHILIP ANTHONY BARILE Thesis Director: Haydee Herrera Curvature is fundamental to the study of differential geometry. It describes different geometrical and topological properties of a surface in R3. Two types of curvature are discussed in this paper: intrinsic and extrinsic. Numerous examples are given which motivate definitions, properties and theorems concerning curvature. ii 1 1 Introduction For surfaces in R3, there are several different ways to measure curvature. Some curvature, like normal curvature, has the property such that it depends on how we embed the surface in R3. Normal curvature is extrinsic; that is, it could not be measured by being on the surface. On the other hand, another measurement of curvature, namely Gauss curvature, does not depend on how we embed the surface in R3. Gauss curvature is intrinsic; that is, it can be measured from on the surface. In order to engage in a discussion about curvature of surfaces, we must introduce some important concepts such as regular surfaces, the tangent plane, the first and second fundamental form, and the Gauss Map. Sections 2,3 and 4 introduce these preliminaries, however, their importance should not be understated as they lay the groundwork for more subtle and advanced topics in differential geometry. -
Surface Normals and Tangent Planes
Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, …nding the equation of a tangent plane to a surface at a given point requires the calculation of a surface normal vector. In this section, we explore the concept of a normal vector to a surface and its use in …nding equations of tangent planes. To begin with, a level surface U (x; y; z) = k is said to be smooth if the gradient U = Ux;Uy;Uz is continuous and non-zero at any point on the surface. Equivalently,r h we ofteni write U = Uxex + Uyey + Uzez r where ex = 1; 0; 0 ; ey = 0; 1; 0 ; and ez = 0; 0; 1 : Supposeh now thati r (t) =h x (ti) ; y (t) ; z (t) hlies oni a smooth surface U (x; y; z) = k: Applying the derivative withh respect to t toi both sides of the equation of the level surface yields dU d = k dt dt Since k is a constant, the chain rule implies that U v = 0 r where v = x0 (t) ; y0 (t) ; z0 (t) . However, v is tangent to the surface because it is tangenth to a curve on thei surface, which implies that U is orthogonal to each tangent vector v at a given point on the surface. r That is, U (p; q; r) at a given point (p; q; r) is normal to the tangent plane to r 1 the surface U(x; y; z) = k at the point (p; q; r). -
A Quick and Dirty Introduction to Differential Geometry
CHAPTER 2 A Quick and Dirty Introduction to Differential Geometry 2.1. The Geometry of Surfaces There are many ways to think about the geometry of a surface (using charts, for instance) but here’s a picture that is well-suited to the way we work with surfaces in the discrete setting. Consider a little patch of material floating in space, as depicted below. Its geometry can be described via a map f : M R3 from a region M in the Euclidean plane R2 to a subset f (M) of R3: ! f N df(X) f (M) X M The differential of such a map, denoted by df , tells us how to map a vector X in the plane to the corresponding vector df(X) on the surface. Loosely speaking, imagine that M is a rubber sheet and X is a little black line segment drawn on M. As we stretch and deform M into f (M), the segment X also gets stretched and deformed into a different segment, which we call df(X). Later on we can talk about how to explicitly express df(X) in coordinates and so on, but it’s important to realize that fundamentally there’s nothing deeper to know about the differential than the picture you see here—the differential simply tells you how to stretch out or “push forward” vectors as you go from one space to another. For example, the length of a tangent vector X pushed forward by f can be expressed as df(X) df(X), · q where is the standard inner product (a.k.a. -
Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives
Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives By Andrew Grossfield, Ph. D., P. E. Abstract Many engineering students encounter and algebraically manipulate partial derivatives in their fluids, thermodynamics or electromagnetic wave theory courses. However it is possible that unless these students were properly introduced to these symbols, they may lack the insight that could be obtained from a geometric or visual approach to the equations that contain these symbols. We accept the approach that just as the direction of a curve at a point in two-dimensional space is described by the slope of the straight line tangent to the curve at that point, the orientation of a surface at a point in 3-dimensional space is determined by the orientation of the plane tangent to the surface at that point. A straight line has only one direction described by the same slope everywhere along its length; a tilted plane has the same orientation everywhere but has many slopes at each point. In fact the slope in almost every direction leading away from a point is different. How do we conceive of these differing slopes and how can they be evaluated? We are led to conclude that the derivative of a multivariable function is the gradient vector, and that it is wrong and misleading to define the gradient in terms of some kind of limiting process. This approach circumvents the need for all the unnecessary delta-epsilon arguments about obvious results, while providing visual insight into properties of multi-variable derivatives and differentials. This paper provides the visual connection displaying the remarkably simple and beautiful relationships between the gradient, the directional derivatives and the partial derivatives. -
Against 3 N-Dimensional Space
OUP UNCORRECTED PROOF – FIRST-PROOF, 07/18/12, NEWGEN 7 A g a i n s t 3N - D i m e n s i o n a l S p a c e Bradley Monton 1 Quantum Mechanics Is False Question 1: How many dimensions does space have? I maintain that the answer is “three.” (I recognize the possibility, though, that we live in a three-dimen- sional hypersurface embedded in a higher-dimensional space; I’ll set aside that possibility for the purposes of this chapter.) Why is the answer “three”? I have more to say about this later, but the short version of my argument is that our everyday commonsense constant experience is such that we’re living in three spatial dimensions, and nothing from our experience provides powerful enough reason to give up that prima facie obvious epistemic starting point. (My foils, as you presumably know from reading this volume, are those such as David Albert [1996] who hold that actually space is 3N -dimensional, where N i s t h e n u m - ber of particles [falsely] thought to exist in [nonexistent] three-dimensional space.) Question 2: How many dimensions does space have, according to quantum mechanics? If quantum mechanics were a true theory of the world, then the answer to Question 2 would be the same as the answer to Question 1. But quan- tum mechanics is not true, so the answers need not be the same. Why is quantum mechanics false? Well, our two most fundamental worked-out physical theories, quantum mechanics and general relativity, are incompatible, and the evidence in favor of general relativity suggests that quantum mech- anics is false. -
Normal Curvature of Surfaces in Space Forms
Pacific Journal of Mathematics NORMAL CURVATURE OF SURFACES IN SPACE FORMS IRWEN VALLE GUADALUPE AND LUCIO LADISLAO RODRIGUEZ Vol. 106, No. 1 November 1983 PACIFIC JOURNAL OF MATHEMATICS Vol. 106, No. 1, 1983 NORMAL CURVATURE OF SURFACES IN SPACE FORMS IRWEN VALLE GUADALUPE AND LUCIO RODRIGUEZ Using the notion of the ellipse of curvature we study compact surfaces in high dimensional space forms. We obtain some inequalities relating the area of the surface and the integral of the square of the norm of the mean curvature vector with topological invariants. In certain cases, the ellipse is a circle; when this happens, restrictions on the Gaussian and normal curvatures give us some rigidity results. n 1. Introduction. We consider immersions /: M -> Q c of surfaces into spaces of constant curvature c. We are going to relate properties of the mean curvature vector H and of the normal curvature KN to geometric properties, such as area and rigidity of the immersion. We use the notion of the ellipse of curvature studied by Little [10], Moore and Wilson [11] and Wong [12]. This is the subset of the normal space defined as (α(X, X): X E TpM, \\X\\ — 1}, where a is the second fundamental form of the immersion and 11 11 is the norm of the vectors. Let χ(M) denote the Euler characteristic of the tangent bundle and χ(N) denote the Euler characteristic of the plane bundle when the codimension is 2. We prove the following generalization of a theorem of Wintgen [13]. n THEOREM 1. Let f\M-*Q cbean isometric immersion of a compact oriented surf ace into an orientable n-dimensional manifold of constant curva- ture c. -
Differential Geometry
ALAGAPPA UNIVERSITY [Accredited with ’A+’ Grade by NAAC (CGPA:3.64) in the Third Cycle and Graded as Category–I University by MHRD-UGC] (A State University Established by the Government of Tamilnadu) KARAIKUDI – 630 003 DIRECTORATE OF DISTANCE EDUCATION III - SEMESTER M.Sc.(MATHEMATICS) 311 31 DIFFERENTIAL GEOMETRY Copy Right Reserved For Private use only Author: Dr. M. Mullai, Assistant Professor (DDE), Department of Mathematics, Alagappa University, Karaikudi “The Copyright shall be vested with Alagappa University” All rights reserved. No part of this publication which is material protected by this copyright notice may be reproduced or transmitted or utilized or stored in any form or by any means now known or hereinafter invented, electronic, digital or mechanical, including photocopying, scanning, recording or by any information storage or retrieval system, without prior written permission from the Alagappa University, Karaikudi, Tamil Nadu. SYLLABI-BOOK MAPPING TABLE DIFFERENTIAL GEOMETRY SYLLABI Mapping in Book UNIT -I INTRODUCTORY REMARK ABOUT SPACE CURVES 1-12 13-29 UNIT- II CURVATURE AND TORSION OF A CURVE 30-48 UNIT -III CONTACT BETWEEN CURVES AND SURFACES . 49-53 UNIT -IV INTRINSIC EQUATIONS 54-57 UNIT V BLOCK II: HELICES, HELICOIDS AND FAMILIES OF CURVES UNIT -V HELICES 58-68 UNIT VI CURVES ON SURFACES UNIT -VII HELICOIDS 69-80 SYLLABI Mapping in Book 81-87 UNIT -VIII FAMILIES OF CURVES BLOCK-III: GEODESIC PARALLELS AND GEODESIC 88-108 CURVATURES UNIT -IX GEODESICS 109-111 UNIT- X GEODESIC PARALLELS 112-130 UNIT- XI GEODESIC