Co-Incidence Problems and Methods 1
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Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
Robot Vision: Projective Geometry
Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations -
Definition Concurrent Lines Are Lines That Intersect in a Single Point. 1. Theorem 128: the Perpendicular Bisectors of the Sides
14.3 Notes Thursday, April 23, 2009 12:49 PM Definition 1. Concurrent lines are lines that intersect in a single point. j k m Theorem 128: The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter of the triangle. D E F Theorem 129: The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides of the triangle. This point is called the incenter of the triangle. A B Notes Page 1 C A B C Theorem 130: The lines containing the altitudes of a triangle are concurrent. This point is called the orthocenter of the triangle. A B C Theorem 131: The medians of a triangle are concurrent at a point that is 2/3 of the way from any vertex of the triangle to the midpoint of the opposite side. This point is called the centroid of the of the triangle. Example 1: Construct the incenter of ABC A B C Notes Page 2 14.4 Notes Friday, April 24, 2009 1:10 PM Examples 1-3 on page 670 1. Construct an angle whose measure is equal to 2A - B. A B 2. Construct the tangent to circle P at point A. P A 3. Construct a tangent to circle O from point P. Notes Page 3 3. Construct a tangent to circle O from point P. O P Notes Page 4 14.5 notes Tuesday, April 28, 2009 8:26 AM Constructions 9, 10, 11 Geometric mean Notes Page 5 14.6 Notes Tuesday, April 28, 2009 9:54 AM Construct: ABC, given {a, ha, B} a Ha B A b c B C a Notes Page 6 14.1 Notes Tuesday, April 28, 2009 10:01 AM Definition: A locus is a set consisting of all points, and only the points, that satisfy specific conditions. -
Special Isocubics in the Triangle Plane
Special Isocubics in the Triangle Plane Jean-Pierre Ehrmann and Bernard Gibert June 19, 2015 Special Isocubics in the Triangle Plane This paper is organized into five main parts : a reminder of poles and polars with respect to a cubic. • a study on central, oblique, axial isocubics i.e. invariant under a central, oblique, • axial (orthogonal) symmetry followed by a generalization with harmonic homolo- gies. a study on circular isocubics i.e. cubics passing through the circular points at • infinity. a study on equilateral isocubics i.e. cubics denoted 60 with three real distinct • K asymptotes making 60◦ angles with one another. a study on conico-pivotal isocubics i.e. such that the line through two isoconjugate • points envelopes a conic. A number of practical constructions is provided and many examples of “unusual” cubics appear. Most of these cubics (and many other) can be seen on the web-site : http://bernard.gibert.pagesperso-orange.fr where they are detailed and referenced under a catalogue number of the form Knnn. We sincerely thank Edward Brisse, Fred Lang, Wilson Stothers and Paul Yiu for their friendly support and help. Chapter 1 Preliminaries and definitions 1.1 Notations We will denote by the cubic curve with barycentric equation • K F (x,y,z) = 0 where F is a third degree homogeneous polynomial in x,y,z. Its partial derivatives ∂F ∂2F will be noted F ′ for and F ′′ for when no confusion is possible. x ∂x xy ∂x∂y Any cubic with three real distinct asymptotes making 60◦ angles with one another • will be called an equilateral cubic or a 60. -
Delaunay Triangulations of Points on Circles Arxiv:1803.11436V1 [Cs.CG
Delaunay Triangulations of Points on Circles∗ Vincent Despr´e † Olivier Devillers† Hugo Parlier‡ Jean-Marc Schlenker‡ April 2, 2018 Abstract Delaunay triangulations of a point set in the Euclidean plane are ubiq- uitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alter- native, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm. 1 Introduction Let P be a set of points in the Euclidean plane. If we assume that P is in general position and in particular do not contain 4 concyclic points, then the Delaunay triangulation DT (P ) is the unique triangulation over P such that the (open) circumdisk of each triangle is empty. DT (P ) has a number of interesting properties. The one that we focus on is called the max-min angle property. For a given triangulation τ, let A(τ) be the list of all the angles of τ sorted from smallest to largest. DT (P ) is the triangulation which maximizes A(τ) for the lexicographical order [4,7] . In dimension 2 and for points in general position, this max-min angle arXiv:1803.11436v1 [cs.CG] 30 Mar 2018 property characterizes Delaunay triangulations and highlights one of their most ∗This work was supported by the ANR/FNR project SoS, INTER/ANR/16/11554412/SoS, ANR-17-CE40-0033. -
Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry
Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry 16 Notes ANGLES IN A CIRCLE AND CYCLIC QUADRILATERAL You must have measured the angles between two straight lines. Let us now study the angles made by arcs and chords in a circle and a cyclic quadrilateral. OBJECTIVES After studying this lesson, you will be able to • verify that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle; • prove that angles in the same segment of a circle are equal; • cite examples of concyclic points; • define cyclic quadrilterals; • prove that sum of the opposite angles of a cyclic quadrilateral is 180o; • use properties of cyclic qudrilateral; • solve problems based on Theorems (proved) and solve other numerical problems based on verified properties; • use results of other theorems in solving problems. EXPECTED BACKGROUND KNOWLEDGE • Angles of a triangle • Arc, chord and circumference of a circle • Quadrilateral and its types Mathematics Secondary Course 395 MODULE - 3 Angles in a Circle and Cyclic Quadrilateral Geometry 16.1 ANGLES IN A CIRCLE Central Angle. The angle made at the centre of a circle by the radii at the end points of an arc (or Notes a chord) is called the central angle or angle subtended by an arc (or chord) at the centre. In Fig. 16.1, ∠POQ is the central angle made by arc PRQ. Fig. 16.1 The length of an arc is closely associated with the central angle subtended by the arc. Let us define the “degree measure” of an arc in terms of the central angle. -
Application of Nine Point Circle Theorem
Application of Nine Point Circle Theorem Submitted by S3 PLMGS(S) Students Bianca Diva Katrina Arifin Kezia Aprilia Sanjaya Paya Lebar Methodist Girls’ School (Secondary) A project presented to the Singapore Mathematical Project Festival 2019 Singapore Mathematic Project Festival 2019 Application of the Nine-Point Circle Abstract In mathematics geometry, a nine-point circle is a circle that can be constructed from any given triangle, which passes through nine significant concyclic points defined from the triangle. These nine points come from the midpoint of each side of the triangle, the foot of each altitude, and the midpoint of the line segment from each vertex of the triangle to the orthocentre, the point where the three altitudes intersect. In this project we carried out last year, we tried to construct nine-point circles from triangulated areas of an n-sided polygon (which we call the “Original Polygon) and create another polygon by connecting the centres of the nine-point circle (which we call the “Image Polygon”). From this, we were able to find the area ratio between the areas of the original polygon and the image polygon. Two equations were found after we collected area ratios from various n-sided regular and irregular polygons. Paya Lebar Methodist Girls’ School (Secondary) 1 Singapore Mathematic Project Festival 2019 Application of the Nine-Point Circle Acknowledgement The students involved in this project would like to thank the school for the opportunity to participate in this competition. They would like to express their gratitude to the project supervisor, Ms Kok Lai Fong for her guidance in the course of preparing this project. -
Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used
Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the first 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Definitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent. -
The Dual Theorem Concerning Aubert Line
The Dual Theorem concerning Aubert Line Professor Ion Patrascu, National College "Buzeşti Brothers" Craiova - Romania Professor Florentin Smarandache, University of New Mexico, Gallup, USA In this article we introduce the concept of Bobillier transversal of a triangle with respect to a point in its plan; we prove the Aubert Theorem about the collinearity of the orthocenters in the triangles determined by the sides and the diagonals of a complete quadrilateral, and we obtain the Dual Theorem of this Theorem. Theorem 1 (E. Bobillier) Let 퐴퐵퐶 be a triangle and 푀 a point in the plane of the triangle so that the perpendiculars taken in 푀, and 푀퐴, 푀퐵, 푀퐶 respectively, intersect the sides 퐵퐶, 퐶퐴 and 퐴퐵 at 퐴푚, 퐵푚 and 퐶푚. Then the points 퐴푚, 퐵푚 and 퐶푚 are collinear. 퐴푚퐵 Proof We note that = 퐴푚퐶 aria (퐵푀퐴푚) (see Fig. 1). aria (퐶푀퐴푚) 1 Area (퐵푀퐴푚) = ∙ 퐵푀 ∙ 푀퐴푚 ∙ 2 sin(퐵푀퐴푚̂ ). 1 Area (퐶푀퐴푚) = ∙ 퐶푀 ∙ 푀퐴푚 ∙ 2 sin(퐶푀퐴푚̂ ). Since 1 3휋 푚(퐶푀퐴푚̂ ) = − 푚(퐴푀퐶̂ ), 2 it explains that sin(퐶푀퐴푚̂ ) = − cos(퐴푀퐶̂ ); 휋 sin(퐵푀퐴푚̂ ) = sin (퐴푀퐵̂ − ) = − cos(퐴푀퐵̂ ). 2 Therefore: 퐴푚퐵 푀퐵 ∙ cos(퐴푀퐵̂ ) = (1). 퐴푚퐶 푀퐶 ∙ cos(퐴푀퐶̂ ) In the same way, we find that: 퐵푚퐶 푀퐶 cos(퐵푀퐶̂ ) = ∙ (2); 퐵푚퐴 푀퐴 cos(퐴푀퐵̂ ) 퐶푚퐴 푀퐴 cos(퐴푀퐶̂ ) = ∙ (3). 퐶푚퐵 푀퐵 cos(퐵푀퐶̂ ) The relations (1), (2), (3), and the reciprocal Theorem of Menelaus lead to the collinearity of points 퐴푚, 퐵푚, 퐶푚. Note Bobillier's Theorem can be obtained – by converting the duality with respect to a circle – from the theorem relative to the concurrency of the heights of a triangle. -
The Generalization of Miquels Theorem
THE GENERALIZATION OF MIQUEL’S THEOREM ANDERSON R. VARGAS Abstract. This papper aims to present and demonstrate Clifford’s version for a generalization of Miquel’s theorem with the use of Euclidean geometry arguments only. 1. Introduction At the end of his article, Clifford [1] gives some developments that generalize the three circles version of Miquel’s theorem and he does give a synthetic proof to this generalization using arguments of projective geometry. The series of propositions given by Clifford are in the following theorem: Theorem 1.1. (i) Given three straight lines, a circle may be drawn through their intersections. (ii) Given four straight lines, the four circles so determined meet in a point. (iii) Given five straight lines, the five points so found lie on a circle. (iv) Given six straight lines, the six circles so determined meet in a point. That can keep going on indefinitely, that is, if n ≥ 2, 2n straight lines determine 2n circles all meeting in a point, and for 2n +1 straight lines the 2n +1 points so found lie on the same circle. Remark 1.2. Note that in the set of given straight lines, there is neither a pair of parallel straight lines nor a subset with three straight lines that intersect in one point. That is being considered all along the work, without further ado. arXiv:1812.04175v1 [math.HO] 11 Dec 2018 In order to prove this generalization, we are going to use some theorems proposed by Miquel [3] and some basic lemmas about a bunch of circles and their intersections, and we will follow the idea proposed by Lebesgue[2] in a proof by induction. -
Linear Features in Photogrammetry
Linear Features in Photogrammetry by Ayman Habib Andinet Asmamaw Devin Kelley Manja May Report No. 450 Geodetic Science and Surveying Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University Columbus, Ohio 43210-1275 January 2000 Linear Features in Photogrammetry By: Ayman Habib Andinet Asmamaw Devin Kelley Manja May Report No. 450 Geodetic Science and Surveying Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University Columbus, Ohio 43210-1275 January, 2000 ABSTRACT This research addresses the task of including points as well as linear features in photogrammetric applications. Straight lines in object space can be utilized to perform aerial triangulation. Irregular linear features (natural lines) in object space can be utilized to perform single photo resection and automatic relative orientation. When working with primitives, it is important to develop appropriate representations in image and object space. These representations must accommodate for the perspective projection relating the two spaces. There are various options for representing linear features in the above applications. These options have been explored, and an optimal representation has been chosen. An aerial triangulation technique that utilizes points and straight lines for frame and linear array scanners has been implemented. For this task, the MSAT (Multi Sensor Aerial Triangulation) software, developed at the Ohio State University, has been extended to handle straight lines. The MSAT software accommodates for frame and linear array scanners. In this research, natural lines were utilized to perform single photo resection and automatic relative orientation. In single photo resection, the problem is approached with no knowledge of the correspondence of natural lines between image space and object space. -
CIRCLE and SPHERE GEOMETRY. [July
464 CIRCLE AND SPHERE GEOMETRY. [July, CIRCLE AND SPHERE GEOMETRY. A Treatise on the Circle and the Sphere. By JULIAN LOWELL COOLIDGE. Oxford, the Clarendon Press, 1916. 8vo. 603 pp. AN era of systematizing and indexing produces naturally a desire for brevity. The excess of this desire is a disease, and its remedy must be treatises written, without diffuseness indeed, but with the generous temper which will find room and give proper exposition to true works of art. The circle and the sphere have furnished material for hundreds of students and writers, but their work, even the most valuable, has been hitherto too scattered for appraisement by the average student. Cyclopedias, while indispensable, are tan talizing. Circles and spheres have now been rescued from a state of dispersion by Dr. Coolidge's comprehensive lectures. Not a list of results, but a well digested account of theories and methods, sufficiently condensed by judicious choice of position and sequence, is what he has given us for leisurely study and enjoyment. The reader whom the author has in mind might be, well enough, at the outset a college sophomore, or a teacher who has read somewhat beyond the geometry expected for en trance. Beyond the sixth chapter, however, or roughly the middle of the book, he must be ready for persevering study in specialized fields. The part most read is therefore likely to be the first four chapters, and a short survey of these is what we shall venture here. After three pages of definitions and conventions the author calls "A truce to these preliminaries!" and plunges into inversion.