Geometry from a Differentiable Viewpoint
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A Law and Order Question
ALawandorder Relation preserving maps. Symmetric relations, asymmetric relations, and antisymmetry. The Cartesian product and the Venn diagram. Euler circles, power sets, partitioning or fibering. A-1 Law and order: Cartesian product, power sets In daily life, we make comparisons of type ... is higher than ... is smarter than ... is faster than .... Indeed, we are dealing with order. Mathematically speaking, order is a binary relation of type −1 xR1y : x<y (smaller than) , xR3y : x>y (larger than, R3 = R ) , 1 (A.1) ≤ ≥ −1 xR2y : x y (smaller-equal) , xR4y : x y (larger-equal, R4 = R2 ) . Example A.1 (Real numbers). x and y, for instance, can be real numbers: x, y ∈ R. End of Example. Question: “What is a relation?” Answer: “Let us explain the term relation in the frame of the following example.” Question. Example A.2 (Cartesian product). Define the left set A = {a1,a2,a3} as the set of balls in a left basket, and {a1,a2,a3} = {red,green,blue}. In contrast, the right set B = {b1,b2} as the set of balls in a right basket, and {b1,b2} = {yellow,pink}. Sequentically, we take a ball from the left basket as well as from the right basket such that we are led to the combinations A × B = (a1,b1) , (a1,b2) , (a2,b1) , (a2,b2) , (a3,b1) , (a3,b2) , (A.2) A × B = (red,yellow) , (red,pink) , (green,yellow) , (green,pink) , (blue,yellow) , (green,pink) . (A.3) End of Example. Definition A.1 (Reflexive partial order). Let M be a non-empty set. The binary relation R2 on M is called reflexive partial order if for all x, y, z ∈ M the following three conditions are fulfilled: (i) x ≤ x (reflexivity) , (ii) if x ≤ y and y ≤ x,thenx = y (antisymmetry) , (A.4) (iii) if x ≤ y and y ≤ z,thenx ≤ z (transitivity) . -
Page 1 of 1 Geometry, Student Text and Homework Helper 11/7/2016
Geometry, Student Text and Homework Helper Page 1 of 1 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-5 Parallel Lines and Triangles 3-5 Parallel Lines and Triangles Teks Focus TEKS (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(G) Vocabulary • Auxiliary line – a line that you add to a diagram to help explain relationships in proofs • Exterior angle of a polygon – an angle formed by a side and an extension of an adjacent side • Remote interior angles – the two nonadjacent interior angles corresponding to each exterior angle of a triangle • Analyze – closely examine objects, ideas, or relationships to learn more about their nature ESSENTIAL UNDERSTANDING The sum of the angle measures of a triangle is always the same. take note Postulate 3-2 Parallel Postulate Through a point not on a line, there is one and only one line parallel to the given line. There is exactly one line through P parallel to ℓ. take note Theorem 3-11 Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180. m∠A + m∠B + m∠C = 180 For a proof of Theorem 3-11, see Problem 1. -
Neutral Geometry
Neutral geometry And then we add Euclid’s parallel postulate Saccheri’s dilemma Options are: Summit angles are right Wants Summit angles are obtuse Was able to rule out, and we’ll see how Summit angles are acute The hypothesis of the acute angle is absolutely false, because it is repugnant to the nature of the straight line! Rule out obtuse angles: If we knew that a quadrilateral can’t have the sum of interior angles bigger than 360, we’d be fine. We’d know that if we knew that a triangle can’t have the sum of interior angles bigger than 180. Hold on! Isn’t the sum of the interior angles of a triangle EXACTLY180? Theorem: Angle sum of any triangle is less than or equal to 180º Suppose there is a triangle with angle sum greater than 180º, say anglfle sum of ABC i s 180º + p, w here p> 0. Goal: Construct a triangle that has the same angle sum, but one of its angles is smaller than p. Why is that enough? We would have that the remaining two angles add up to more than 180º: can that happen? Show that any two angles in a triangle add up to less than 180º. What do we know if we don’t have Para lle l Pos tul at e??? Alternate Interior Angle Theorem: If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are parallel. Converse of AIA Converse of AIA theorem: If two lines are parallel then the aliilblternate interior angles cut by a transversa l are congruent. -
Geometry Ch 5 Exterior Angles & Triangle Inequality December 01, 2014
Geometry Ch 5 Exterior Angles & Triangle Inequality December 01, 2014 The “Three Possibilities” Property: either a>b, a=b, or a<b The Transitive Property: If a>b and b>c, then a>c The Addition Property: If a>b, then a+c>b+c The Subtraction Property: If a>b, then a‐c>b‐c The Multiplication Property: If a>b and c>0, then ac>bc The Division Property: If a>b and c>0, then a/c>b/c The Addition Theorem of Inequality: If a>b and c>d, then a+c>b+d The “Whole Greater than Part” Theorem: If a>0, b>0, and a+b=c, then c>a and c>b Def: An exterior angle of a triangle is an angle that forms a linear pair with an angle of the triangle. A In ∆ABC, exterior ∠2 forms a linear pair with ∠ACB. The other two angles of the triangle, ∠1 (∠B) and ∠A are called remote interior angles with respect to ∠2. 1 2 B C Theorem 12: The Exterior Angle Theorem An Exterior angle of a triangle is greater than either remote interior angle. Find each of the following sums. 3 4 26. ∠1+∠2+∠3+∠4 2 1 6 7 5 8 27. ∠1+∠2+∠3+∠4+∠5+∠6+∠7+ 9 12 ∠8+∠9+∠10+∠11+∠12 11 10 28. ∠1+∠5+∠9 31. What does the result in exercise 30 indicate about the sum of the exterior 29. ∠3+∠7+∠11 angles of a triangle? 30. ∠2+∠4+∠6+∠8+∠10+∠12 1 Geometry Ch 5 Exterior Angles & Triangle Inequality December 01, 2014 After proving the Exterior Angle Theorem, Euclid proved that, in any triangle, the sum of any two angles is less than 180°. -
M2.1 - Transformation Geometry
RHODES UNIVERSITY Grahamstown 6140, South Africa Lecture Notes CCR M2.1 - Transformation Geometry Claudiu C. Remsing DEPT. of MATHEMATICS (Pure and Applied) 2006 ‘Beauty is truth, truth beauty’ – that is all Ye know on earth, and all ye need to know. John Keats Imagination is more important than knowledge. Albert Einstein Do not just pay attention to the words; Instead pay attention to meaning behind the words. But, do not just pay attention to meanings behind the words; Instead pay attention to your deep experience of those meanings. Tenzin Gyatso, The 14th Dalai Lama Where there is matter, there is geometry. Johannes Kepler Geometry is the art of good reasoning from poorly drawn figures. Anonymous What is a geometry ? The question [...] is metamathematical rather than mathematical, and consequently mathematicians of unquestioned competence may (and, indeed, do) differ in the answer they give to it. Even among those mathematicians called geometers there is no generally accepted definition of the term. It has been observed that the abstract, postulational method that has permeated nearly all parts of modern mathematics makes it difficult, if not meaningless, to mark with precision the boundary of that mathematical domain which should be called geometry. To some, geometry is not so much a subject as it is a point of view – a way of loking at a subject – so that geometry is the mathematics that a geometer does ! To others, geometry is a language that provides a very useful and suggestive means of discussing almost every part of mathematics (just as, in former days, French was the language of diplomacy); and there are, doubtless, some mathematicians who find such a query without any real significance and who, consequently, will disdain to vouchsafe any an- swer at all to it. -
Chapter 4 Euclidean Geometry
Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles De¯nition 4.1 Two distinct lines ` and m are said to be parallel ( and we write `km) i® they lie in the same plane and do not meet. Terminologies: 1. Transversal: a line intersecting two other lines. 2. Alternate interior angles 3. Corresponding angles 4. Interior angles on the same side of transversal 56 Yi Wang Chapter 4. Euclidean Geometry 57 Theorem 4.2 (Parallelism in absolute geometry) If two lines in the same plane are cut by a transversal to that a pair of alternate interior angles are congruent, the lines are parallel. Remark: Although this theorem involves parallel lines, it does not use the parallel postulate and is valid in absolute geometry. Proof: Assume to the contrary that the two lines meet, then use Exterior Angle Inequality to draw a contradiction. 2 The converse of above theorem is the Euclidean Parallel Postulate. Euclid's Fifth Postulate of Parallels If two lines in the same plane are cut by a transversal so that the sum of the measures of a pair of interior angles on the same side of the transversal is less than 180, the lines will meet on that side of the transversal. In e®ect, this says If m\1 + m\2 6= 180; then ` is not parallel to m Yi Wang Chapter 4. Euclidean Geometry 58 It's contrapositive is If `km; then m\1 + m\2 = 180( or m\2 = m\3): Three possible notions of parallelism Consider in a single ¯xed plane a line ` and a point P not on it. -
MA3D9: Geometry of Curves and Surfaces
Geometry of Curves and Surfaces Weiyi Zhang Mathematics Institute, University of Warwick November 20, 2020 2 Contents 1 Curves 5 1.1 Course description . .5 1.1.1 A bit preparation: Differentiation . .6 1.2 Methods of describing a curve . .8 1.2.1 Fixed coordinates . .8 1.2.2 Moving frames: parametrized curves . .8 1.2.3 Intrinsic way(coordinate free) . .9 n 1.3 Curves in R : Arclength Parametrization . 11 1.4 Curvature . 13 1.5 Orthonormal frame: Frenet-Serret equations . 16 1.6 Plane curves . 18 1.7 More results for space curves . 20 1.7.1 Taylor expansion of a curve . 21 1.7.2 Fundamental Theorem of the local theory of curves . 21 1.8 Isoperimetric Inequality . 23 1.9 The Four Vertex Theorem . 25 3 2 Surfaces in R 29 2.1 Definitions and Examples . 29 2.1.1 Compact surfaces . 32 2.1.2 Level sets . 33 2.2 The First Fundamental Form . 34 2.3 Length, Angle, Area . 35 2.3.1 Length: Isometry . 36 2.3.2 Angle: conformal . 37 2.3.3 Area: equiareal . 37 2.4 The Second Fundamental Form . 38 2.4.1 Normals and orientability . 39 2.4.2 Gauss map and second fundamental form . 40 2.5 Curvatures . 42 2.5.1 Definitions and first properties . 42 2.5.2 Calculation of Gaussian and mean curvatures . 45 2.5.3 Principal curvatures . 47 3 4 CONTENTS 2.6 Gauss's Theorema Egregium . 49 2.6.1 Compatibility equations . 52 2.6.2 Gaussian curvature for special cases . -
Basic Theorems of Euclidean Geometry
Basic Theorems of Euclidean Geometry We finally adopt the Euclidean Parallel Postulate and explore geometry under this assumption. The result is Euclidean Geometry and its major results are similarity, the Pythagorean Theorem, and trigonometry, although there is of course much more. Axiom (The Euclidean Parallel Postulate): For every line l and for every point P that does not lie on l, there is exactly one line m such that P lies on m and m is parallel to l. We now have as theorems all those statements that we proved equivalent to EPP earlier, including: 1. If two parallel lines are cut by a transversal, alternate interior angles are congruent. 2. (Euclid’s Postulate V) If two lines in the same plane are cut by a transversal so that the sum of the measures of a pair of interior angles on the same side of the transversal is less than 180, then the lines will meet on that side of the transversal. 3. (Proclus’s Axiom) A third line intersecting one of two parallel lines intersects the other. 4. A line perpendicular to one of two parallel lines is perpendicular to the other. 5. The perpendicular bisectors of the sides of a triangle are concurrent. 6. There exists a circle passing through any three noncollinear points. 7. (Angle Sum Postulate) The sum of measures of the angles of a triangle is 180. 8. There exists one triangle such that the sum of measures of the angles is 180. 9. There exists a rectangle. 10. There exist two lines l and m such that l is equidistant from m. -
Neutral Geometry
NEUTRAL GEOMETRY If only it could be proved ... that "there is a Triangle whose angles are together not less than two right angles"! But alas, that is an ignis fatuus that has never yet been caught! c. L. DODGSON (LEWIS CARROLL) GEOMETRY WITHOUT THE PARALLELL AXIOM In the exercises of the previous chapter you gained experience in proving some elementary results from Hilbert's axioms. Many of these results were taken for granted by Euclid. You can see that filling in the gaps and rigorously proving every detail is a long task. In any case, we must show that Euclid's postulates are consequences of Hilbert's. We have seen that Euclid's first postulate is the same as Hilbert's Axiom 1-1. In our new language, Euclid's second postulate says the following: given segments AB and CD, there exists a point E such that A * B * E and CD == BE. This follows immediately from Hilbert's Axiom C-1 applied to the ray emanating from B opposite to -+ . BA (see Figure 4.1). The third postulate of Euclid becomes a definition in Hilbert's system. The circle with center 0 and radius OA is defined as the set of all points P such that OP is congruent to OA. Axiom C-1 then guaran tees that on every ray emanating from 0 there exists such a point P. The fourth postulate of Euclid-all right angles are congruent becomes a theorem in Hilbert's system, as was shown in Proposition 3.23. 116 III Neutral Geometry c o • • • • • FIGURE 4.1 A B E Euclid's parallel postulate is discussed later in this chapter. -
Graph Minor from Wikipedia, the Free Encyclopedia Contents
Graph minor From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Profile ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 Abelian group 5 2.1 Definition ............................................... 5 2.2 Facts ................................................. 5 2.2.1 Notation ........................................... 5 2.2.2 Multiplication table ...................................... 6 2.3 Examples ............................................... 6 2.4 Historical remarks .......................................... 6 2.5 Properties ............................................... 6 2.6 Finite abelian groups ......................................... 7 2.6.1 Classification ......................................... 7 2.6.2 Automorphisms ....................................... 7 2.7 Infinite abelian groups ........................................ 8 2.7.1 Torsion groups ........................................ 9 2.7.2 Torsion-free and mixed groups ................................ 9 2.7.3 Invariants and classification .................................. 9 2.7.4 Additive groups of rings ................................... 9 2.8 Relation to other mathematical topics ................................. 10 2.9 A note on the typography ...................................... -
Equiareal Shape-From-Template
Noname manuscript No. (will be inserted by the editor) Equiareal Shape-from-Template David Casillas-Perez · Daniel Pizarro · David Fuentes-Jimenez · Manuel Mazo · Adrien Bartoli Received: date / Accepted: date Abstract This paper studies the 3D reconstruction of give reconstruction results with both synthetic and real a deformable surface from a single image and a reference examples. surface, known as the template. This problem is known as Shape-from-Template and has been recently shown to be well-posed for isometric deformations, for which 1 Introduction the surface bends without altering geodesics. This pa- per studies the case of equiareal deformations. They The reconstruction of 3D objects from images is an are elastic deformations where the local area is pre- important goal in Computer Vision with numerous served and thus include isometry as a special case. Elas- scientific and engineering applications. Over the last tic deformations have been studied before in Shape- few decades, the reconstruction of rigid objects has from-Template, yet no theoretical results were given on been thoroughly studied and solved by Structure-from- the existence or uniqueness of solutions. The equiareal Motion (SfM) [23], which uses multiple 2D images of model is much more widely applicable than isome- the same scene. Rigidity ensures the problem's well- try. This paper brings Monge's theory, widely used for posedness in the general case, which means that only studying the solutions of non-linear first-order PDEs, one solution is obtained up to scale. to the field of 3D reconstruction. It uses this theory to This paper studies the reconstruction of objects un- establish a theoretical framework for equiareal Shape- dergoing deformations where SfM solutions cannot be from-Template and answers the important question of applied. -
Erik W. Grafarend Friedrich W. Krumm Map Projections Erik W
Erik W. Grafarend Friedrich W. Krumm Map Projections Erik W. Grafarend Friedrich W. Krumm Map Projections Cartographic Information Systems With 230 Figures 123 Professor Dr. Erik W. Grafarend Universität Stuttgart Institute of Geodesy Geschwister-Scholl-Str. 24 D 70174 Stuttgart Germany E-mail: [email protected] Dr. Friedrich W. Krumm Universität Stuttgart Institute of Geodesy Geschwister-Scholl-Str. 24 D 70174 Stuttgart Germany E-mail: [email protected] Library of Congress Control Number: 2006929531 ISBN-10 3-540-36701-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-36701-7 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+BusinessScience+Business Media Springeronline.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: E. Kirchner, Heidelberg Design, layout, and software manuscript by Dr.